# Quantum metrology with full and fast quantum control

P. Sekatski  Institut für Theoretische Physik, Universität Innsbruck,
Technikerstr. 21a, A-6020 Innsbruck, Austria  M. Skotiniotis  Institut für
Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020
Innsbruck, Austria  Física Teòrica: Informació i Fenòmens Quàntics,
Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellatera
(Barcelona) Spain  J. Kołodyński  ICFO-Institut de Ciencies Fotoniques, The
Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain  W. Dür  Institut für Theoretische Physik, Universität
Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria

###### Abstract

We establish general limits on how precise a parameter, e.g. frequency or the
strength of a magnetic field, can be estimated with the aid of _full and fast
quantum control_. We consider uncorrelated noisy evolutions of N qubits and
show that fast control allows to fully restore the Heisenberg scaling
($\sim\\!1/N^{2}$) for all rank-one Pauli noise except dephasing. For all
other types of noise the _asymptotic_ quantum enhancement is unavoidably
limited to a constant-factor improvement over the standard quantum limit
($\sim\\!1/N$) even when allowing for the full power of fast control. The
latter holds both in the single-shot and infinitely-many repetitions
scenarios. However, even in this case allowing for fast quantum control helps
to increase the improvement factor. Furthermore, for frequency estimation with
_finite_ resource we show how a parallel scheme utilizing any fixed number of
entangled qubits but no fast quantum control can be outperformed by a simple,
easily implementable, sequential scheme which only requires entanglement
between one sensing and one auxiliary qubit.

###### pacs:

03.67.-a, 03.65.Ud, 03.65.Yz, 03.65.Ta

##  I Introduction

Precision measurements play a fundamental role in physics and beyond as they
constitute the main ingredient for many state-of-the-art applications and
experiments Dowling and Seshadreesan (2015). When investigating the limits of
known or speculative theories experimentally one quickly enters into regimes
where quantities and parameters need to be measured with unprecedented
precision. In this context it is of utmost importance to know the ultimate
limits nature sets on how precise any given quantity can be determined and how
to achieve this.

These questions are at the focus of quantum metrology Giovannetti  _et al._
(2011). Quantum mechanics is a probabilistic theory and the intrinsically
stochastic nature of measurements ultimately limits the achievable precision.
When considering classical probes independently sensing a physical parameter,
such as phase or frequency, the maximum attainable precision (as quantified by
the Mean Squared Error—MSE) follows the _standard scaling_ (SS), $1/N$, where
$N$ is the number of probes Giovannetti  _et al._ (2004). In turn it was shown
that quantum entanglement allows one to achieve the so-called _Heisenberg
scaling_ in precision (HS), $1/N^{2}$, a quadratic improvement as compared to
classical approaches Giovannetti  _et al._ (2006). These precision limits
apply to both single-shot protocols Sanders and Milburn (1995); Berry and
Wiseman (2000); Peres and Scudo (2001); Bagan  _et al._ (2004); Chiribella
_et al._ (2004a); *Chiribella2004b; *Chiribella2005; Higgins  _et al._ (2007)
as well as protocols utilizing many repetitions Yurke (1986); Giovannetti  _et
al._ (2004, 2006). Still, it remains unclear to what extent such an
improvement can be harnessed in practice under non-idealized conditions, i.e.,
when taking unavoidable noise and imperfections into account Huelga  _et al._
(1997); Banaszek  _et al._ (2009); Maccone and Giovannetti (2011).

Due to the difficulty of obtaining exact precision limits in the presence of
noise several asymptotic lower bounds have been established for particular
noise models Fujiwara and Imai (2008); Escher  _et al._ (2011, 2012);
Demkowicz-Dobrzański  _et al._ (2012); Kołodyński and Demkowicz-Dobrzański
(2013); Alipour  _et al._ (2014); Knysh  _et al._ (2011, 2014). These lower
bounds are often not only cumbersome to evaluate but also hard to optimize,
relying on educated guesses, numerical methods employing semi-definite
programming or a combination of the two. Nevertheless, these bounds show that
for typical uncorrelated noise processes the possible gain due to the usage of
quantum resources is limited to a constant-factor improvement over the SS, as
opposed to a different scaling. On the other hand, it was shown that for some
types of noise—namely noise perpendicular to the Hamiltonian that encodes the
parameter of interest—the restriction to SS can be circumvented, and HS can be
fully restored, by allowing for additional resources, such as perfectly
protected auxiliary particles used to perform quantum error correction Dür
_et al._ (2014); Kessler  _et al._ (2014); Arrad  _et al._ (2014); Ozeri
(2013); Lu  _et al._ (2015); Herrera-Martí  _et al._ (2015).

In this work we develop a general framework that provides us with analytic
results for all types of noise processes described by a time-homogeneous
master equation. In order to establish general limits, we account for the
possibility of ancillary resources that do not take part in the sensing
process together with _full and fast quantum control_ (FFQC) of the system and
the ancillae 111We remark that we do not allow for additional control over the
environment as has been recently considered in Gefen  _et al._ (2015); Plenio
and Huelga (2015).. Such FFQC allows one to effectively modify the Lindblad
operator that describes the noise process, a possibility which has hitherto
not been considered 222To the best of our knowledge, quantum control has been
used to modify the overall system-environment dynamics but not the dissipative
evolution itself Arrad  _et al._ (2014); Sekatski  _et al._ (2015).. Our
approach allows us to identify all types of qubit noise that can be fully
corrected, and thus restore HS in precision. In addition, for all other types
of uncorrelated noise we are able to provide fully analytic bounds applicable
to the entire hierarchy of metrology schemes. In contrast to the available
methods Escher  _et al._ (2011); Demkowicz-Dobrzański  _et al._ (2012), our
bounds can be generally determined without need to explicitly solve the system
dynamics.

Our main results can be summarized as follows:

  * (i)

FFQC allows to restore HS by completely eliminating any Pauli rank-one noise
(that is not parallel to the Hamiltonian) at the cost of slowing down the
unitary evolution by a constant factor;

  * (ii)

All other noise processes unavoidably limit the quantum gain to a constant
factor improvement over SS despite full quantum control. We obtain analytic
bounds for the achievable improvement factor.

  * (iii)

For SS-limited noise processes FFQC may yet allow for significant improvement
of precision in case of limited resources. We provide explicit examples
demonstrating the advantage of FFQC-assisted schemes over ones without
intermediate control for both continuous and discrete processes.

We stress that our results do not pertain only to the frequentist approach to
parameter estimation, which assumes an infinite number of repetitions, but
also to the Bayesian approach, where one is restricted to a finite number of
repetitions, or in the extreme case, a single run of the experiment. Moreover,
the most general protocol involving FFQC lends itself naturally to adaptive
strategies, where ones choice of the strategy to adopt at a given time depends
on the record of previous measurement (a single FFQC cirquit can contain
inremediate measurements with feed-forward). We will discuss the implications
of our most general FFQC strategy, both for the frequentist and Bayesian
approaches, in Sec. V.

The paper is organized as follows. In Sec. II we recall some recent results in
quantum metrology, and compare them through a hierarchy of quantum metrology
schemes accounting for various levels of control. We show that the FFQC
scheme, that tops the hierarchy, is the most powerful one allowed by quantum
mechanics. Sec. III introduces the model of noisy quantum processes described
by a time-homogeneous master equation, and reviews the concept of quantum
Fisher information (QFI). In Sec. IV, we review upper bounds on the QFI for
metrology schemes, accounting for intermediate control, and generalize them to
incorporate FFQC (Sec. IV.2.1) demonstrating (i) and (ii). Our extension of
the bounds to the limit of short evolution times is crucial to show (ii), and
is the key to obtain analytic bounds (as it removes the necessity to solve
system dynamics). In Sec. V we discuss the performance of FFQC-assisted
metrology both in single-shot and asymptotically many repetition scenarios.
Sec. VI considers protocols with limited resources, where we show that FFQC-
assisted schemes outperform parallel schemes without intermediate control (see
(iii)). This is demonstrated for noisy frequency estimation in Sec. VI.1, and
for noisy phase estimation in Sec. VI.2. We summarize and conclude our results
in Sec. VII. A reader who is familiar with quantum metrology, or who is
primarily interested in the results, can directly proceed to sections IV.1 and
IV.2.4 where the first two of our main results ((i) and (ii)) are presented.

##  II Quantum metrology protocols

In a standard quantum metrology protocol, a system consisting of $N$ probes
(photons, atoms etc.) is carefully engineered, so that each probe may be used
to independently sense a parameter of interest $\omega$. As depicted in Fig.
1(a), the experimentalist prepares then a suitably entangled state of the $N$
probes, which undergo the sensing process _in parallel_ for a time $t$, during
which the unknown parameter $\omega$ is imprinted on the state of the probes
independently. The final state of the system is measured in order to most
precisely retrieve information regarding $\omega$.

If the sensing process is noiseless, the strategy (a) of Fig. 1 is known to
optimally achieve the HS Giovannetti  _et al._ (2006). However, various
uncorrelated noise types have been shown to constrain the precision to follow
the SS in the asymptotic $N$ limit, even if one optimally prepares the probes
in an entangled but noise robust state Huelga  _et al._ (1997); Fujiwara and
Imai (2008); Escher  _et al._ (2011); Demkowicz-Dobrzański  _et al._ (2012).

A more powerful metrological protocol is depicted in Fig. 1(b). Here, in
addition to the $N$ sensing probes, the experimenter is equipped with
ancillary particles that do not take part in the sensing process, and may be
used to implement single-step _error correction_ (EC) at the final measurement
stage Kołodyński and Demkowicz-Dobrzański (2013). Notice that the protocol of
Fig. 1(b) is more general than that of Fig. 1(a); indeed one recovers the
latter by choosing not to entangle the sensing probes with the ancillae.
Consequently any bound on precision valid for scheme (b) also applies to
scenario (a). Methods have been recently proposed that allow to derive bounds
on the precision for protocols (b) from a Kraus representation of a single
channel ${\mathcal{E}}_{\omega,t}$ Demkowicz-Dobrzański  _et al._ (2012);
Kołodyński and Demkowicz-Dobrzański (2013). Although these methods allowed to
prove the asymptotic SS for various noise types, the derived bounds cannot be
guaranteed to be tight (even in the asymptotic $N$ limit). Nevertheless, in
case of dephasing and particle-loss noise types, the corresponding bounds have
been shown to be asymptotically achievable already within scheme (a) Ulam-
Orgikh and Kitagawa (2001); Knysh  _et al._ (2011, 2014). For amplitude-
damping noise, however, scheme (b) has been shown to give a strictly better
scaling than scheme Fig. 1(a) Demkowicz-Dobrzański and Maccone (2014).

An interesting situation is the case of the X-noise: a noise similar to
dephasing but with the generator perfectly transversal to the Hamiltonian
encoding the parameter. For a fixed sensing time $t$ the bounds Escher  _et
al._ (2011); Demkowicz-Dobrzański  _et al._ (2012) impose asymptotic SS.
However, in frequency estimation one strongly benefits from decreasing the
sensing time while increasing the number of probes $N$. Indeed, by optimizing
the sensing time of the protocols (a) and (b) for each given $N$, it was shown
that one may asymptotically beat the SS and achieve the $1/N^{5/3}$ precision
scaling Chaves  _et al._ (2013).

Moreover, it has been shown that one can even restore the HS of precision for
such strictly transversal noise after allowing for fast possibly multi-step EC
Dür  _et al._ (2014); Kessler  _et al._ (2014); Arrad  _et al._ (2014); Ozeri
(2013). This is a particular case of the metrological scenario depicted in
Fig. 1(c). Here, in addition to employing ancillary particles, the
experimenter is capable of freely interjecting the sensing process with $m$
control pulses that may act on both probes and ancillae representing, e.g. EC
steps Dür  _et al._ (2014); Kessler  _et al._ (2014); Arrad  _et al._ (2014);
Ozeri (2013), dynamical-decoupling pulses Viola and Lloyd (1998); *Viola:99;
*Viola:03; *Viola:09; *Viola:10; *West:10 or any general adaptive feedback
scheme Wiseman and Milburn (2009). Notice that protocol (c) is more general
than Fig. 1(b) and one can obtain the latter from the former simply by
allowing the intermediate operations to be the identity.

Figure 1: Hierarchy of quantum metrology protocols for sensing a frequency-
like parameter $\omega$ that is encoded on each probe during its noisy
evolution ${\mathcal{E}}_{\omega,t}$. Within the standard schemes of type (a),
$N$ probes independently sense the parameter for a time $t$. In (b),
additionally an unlimited number of ancillae is allowed to perform a _single
error-correcting step_ at the end of the protocol. In (c), the sensing process
is interspersed at time intervals, $\delta t$, with quantum control operations
acting on both the sensing probes and ancillae. Finally in (d), a single probe
sequentially senses the parameter for a time $t^{\prime}\\!=\\!Nt$. Fast
quantum operations (FFQC) are applied frequently on the global state of the
probe and ancillae such that at any given time the probe spends an
infinitesimal amount of time $\mathrm{d}t$ sensing the parameter. We note that
the schemes form the following hierarchy:
(a)$\subseteq$(b)$\subseteq$(c)$\subseteq$(d), when ordered in terms of their
ultimate power.

The most general and powerful metrology protocol, and the main focus of the
current work, is the one depicted in Fig. 1(d). Here, the experimenter
prepares a suitably entangled state between a _single_ probe and many
ancillae, and is capable of frequently interjecting the evolution with FFQC—an
arbitrary number, $k$, of most general intermediate control operations acting
on the overall state of probe-plus-ancillae. Moreover, by choosing $k$
sufficiently large one can ensure the sensing time spent by the probe in
between successive FFQC steps to be _infinitesimal_. It is this additional
power of frequently interjecting the sensing process with _fast_ quantum
control that we exploit throughout the remainder of this work to establish the
ultimate bounds on precision. This intermediate control allows one in fact to
modify the noise process, a possibility that has not been considered in
previous approaches.

To see that this is indeed the most general strategy, we note that for any
protocol (c), with $N$ parallel probes and total sensing time $t$, there
exists a protocol of type (d) with total sensing time $t^{\prime}=Nt$ whose
metrological performance is just as good as that of (c) or better. We note in
passing that protocols (c) and (d) are not equivalent; not only are the time
steps in protocol (d) assumed to be infinitesimally small, but protocol (d)
allows us to use the same resources (namely $N$ and $t$) in a ”sequentialized”
fashion.

The above metrological schemes apply equally to the case of Bayesian parameter
estimation, i.e., when considering only a finite number of experimental
repetitions and a prior probability $p(\omega)$ representing our knowledge
about the parameter $\omega$. Examples of such Bayesian scenarios include
reference frame alignment Bagan  _et al._ (2004); Chiribella  _et al._ (2004a,
b, 2005), adaptive Bayesian estimation Berry and Wiseman (2000); Higgins  _et
al._ (2007); Sergeevich  _et al._ (2011), and single-shot estimation Tsang
(2012); Gill and Levit (1995).

####  II.0.1 A remark on error correction

Let us briefly remark that the most general EC step (encoding+syndrome
readout+correction) only requires the dimension of the space describing the
ancillae to be the same as the system one. Indeed, given that the system
containing the sensing probes has an overall dimension $d$ that is smaller
than the dimension of the ancillary space, any bipartite pure state of the two
can be written using the Schmidt decomposition as
$\sum_{i=1}^{d}\\!\lambda_{i}\left|i\right\rangle_{\textrm{S}}\\!\otimes\\!\left|\psi_{i}\right\rangle_{\textrm{A}}$.
Here, $\\{\left|i\right\rangle\\}_{i=1}^{d}$ denotes a given fixed basis for
the system, while $\\{\left|\psi_{i}\right\rangle\\}_{i=1}^{d}$ is a set of
orthonormal states of ancillae that spans a $d$-dimensional subspace. Thus,
since any EC protocol may be implemented by restricting to states of the above
form, it suffices to choose the overall space describing the ancillae to be of
same dimension as the system.

##  III Background

In order to assess the performance of any of the protocols introduced in Sec.
II, we review below the crucial tool of quantum metrology—the QFI. In
addition, in Sec. III.2 we describe the general time-homogeneous qubit noise
processes we consider throughout this work.

However, before doing so, let us first introduce the notation we utilize. In
what follows, we denote the derivative with respect to the estimated parameter
$\omega$ (whenever applied to operators, vectors or matrices; differentiating
adequately the entries) as
$\dot{\bullet}\\!:=\\!\frac{\mathrm{d}\bullet}{\mathrm{d}\omega}$; and shorten
the trigonometric functions to $\mathrm{c}_{\theta}\\!:=\\!\cos(\theta)$ and
$\mathrm{s}_{\theta}\\!:=\\!\sin(\theta)$.
$||A||\\!:=\\!\max_{\psi}|\left\langle\psi\right|A\left|\psi\right\rangle|$
stands for the operator norm, whereas $\mathbbm{1}$ and $\mathcal{I}$ for the
identity operator and identity linear map respectively. We use bold face to
denote vectors, so that ${\bf r}\\!:=\\!(x,y,z)^{T}$ represents a
3-dimensional complex vector, while ${\bf x}$, ${\bf y}$ and ${\bf z}$ the
Cartesian unit vectors. We denote the standard Pauli 4- and 3-vectors as
${\bm{\varsigma}}\\!:=\\!(\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3})^{T}$
and ${\bm{\sigma}}\\!:=\\!(\sigma_{1},\sigma_{2},\sigma_{3})^{T}$ respectively
($\sigma_{0}\\!:=\\!\mathbbm{1}$). The scalar and outer products of vectors
are then defined in the usual way irrespectively of the type of vector
entries, i.e., ${\bf v}^{\dagger}{\bf
v}^{\prime}\\!:=\\!\sum_{i}\\!v_{i}^{\dagger}v^{\prime}_{i}$ and $[{\bf
v}^{\prime}{\bf v}^{\dagger}]_{ij}\\!:=\\!v^{\prime}_{i}v_{j}^{\dagger}$. We
write $\sigma_{{\bf n}}\\!:=\\!{\bf n}^{T}{\bm{\sigma}}$ to denote a Pauli
operator in the spatial direction ${{\bf
n}}=(s_{\theta}c_{\phi},s_{\theta}s_{\phi},c_{\theta})^{T}$. We reserve the
special font, $\mathsf{M}$, for matrices whose entries, $\mathsf{M}_{ij}$, are
(complex) numbers, so that the multiplication of all types of vectors can then
be defined in the standard way: $\mathsf{M}{\bf
v}\\!=\\!(\sum_{j}\mathsf{M}_{1j}v_{j},\sum_{j}\mathsf{M}_{2j}v_{j},\dots)^{T}$.
For example, any qubit operator $O$ can be unambiguously represented by
constructing its corresponding matrix $\mathsf{O}$ (in the Pauli basis) such
that
$O\\!=\\!{\bm{\varsigma}}^{\dagger}\mathsf{O}{\bm{\varsigma}}\\!=\\!\sum_{\mu,\nu=0}^{3}\\!\sigma_{\mu}\mathsf{O}_{\mu\nu}\sigma_{\nu}$.

###  III.1 QFI and its role in quantum metrology

In all metrology schemes discussed in Sec. II the information about the
estimated parameter of interest is encoded in the final state of the system
consisting generally of both probes and ancillae. Thus, we denote the final
state of the system in any of the schemes (a,b,c,d) of Fig. 1 as
$\varrho_{\omega}$. It is then $\varrho_{\omega}$ that is measured, in order
to most precisely construct an estimate of the parameter of interest,
$\omega$.

How best to quantify the precision of this estimate depends crucially on how
the information about $\omega$ has been collected. If we are freely allowed to
repeat the protocol sufficiently many times and collect statistics, then the
best precision of estimation, as quantified by the _mean squared error_ (MSE),
is determined by the well-known Cramér-Rao bound (CRB) Helstrom (1976); *H80

$\delta^{2}\omega\geq\frac{1}{\nu\mathcal{F}(\varrho_{\omega})},$ (1)

where $\mathcal{F}(\varrho_{\omega})$ is the QFI of the state and $\nu$ is the
number of repetitions. On the other hand, if the information about $\omega$ is
inferred from the outcome of just a single repetition of the protocol, then
the single-shot MSE has to be averaged over the prior distribution
representing the knowledge we possess about $\omega$ prior to its estimation.
Then, the _average mean squared error_ (AvMSE ) can be lower-bounded by either
the Bayesian Cramér-Rao bound (BCRB) Gill and Levit (1995) (for well behaved
prior distributions) or the Ziv-Zakai bound (ZZB) Tsang (2012).

Whether considering the CRB, BCRB, or ZZB, the crucial quantity of interest is
the QFI: $\mathcal{F}(\varrho_{\omega})$. Given the spectral decomposition
$\varrho_{\omega}\\!=\\!\sum_{i}p_{i}(\omega)\left|\psi_{i}(\omega)\right\rangle\\!\left\langle\psi_{i}(\omega)\right|$,
the QFI of the state $\varrho_{\omega}$ evaluated with respect to the
estimated parameter $\omega$ generally reads

${{\mathcal{F}}}(\varrho_{\omega}):=2\\!\\!\sum_{\underset{p_{j}+p_{j}\neq
0}{i,j}}\frac{1}{p_{i}(\omega)+p_{j}(\omega)}|\left\langle\psi_{i}(\omega)\right|\dot{\varrho}_{\omega}\left|\psi_{j}(\omega)\right\rangle|^{2}.$
(2)

On the other hand, the QFI may be equivalently defined by using its relation
with the fidelity between quantum states Braunstein and Caves (1994),
$F(\rho,\sigma)\\!:=\\!\mathrm{tr}\sqrt{\sqrt{\sigma}\rho\sqrt{\sigma}}$:

${\mathcal{F}}(\varrho_{\omega}):=8\lim_{\mathrm{d}\omega\to
0}\frac{1-F(\varrho_{\omega},\varrho_{\omega+\mathrm{d}\omega})}{\mathrm{d}\omega^{2}}.$
(3)

The QFI satisfies the following important properties. It is _additive_ , so
that if the final state of the system is a _product_ state, i.e.,
$\varrho_{\omega}\\!=\\!\rho^{\otimes N}_{\omega}$, then its QFI satisfies
$\mathcal{F}(\rho^{\otimes N}_{\omega})\\!=\\!N\mathcal{F}(\rho_{\omega})$ and
thus gives rise to SS Giovannetti  _et al._ (2004). If the final state of the
probes is _pure_ ,
$\varrho_{\omega}\\!=\\!\left|\psi_{\omega}\right\rangle\\!\left\langle\psi_{\omega}\right|$,
then

$\mathcal{F}(\varrho_{\omega})=4\left(\left\langle\dot{\psi}_{\omega}|\dot{\psi}_{\omega}\right\rangle-\left|\left\langle\dot{\psi}_{\omega}|\psi_{\omega}\right\rangle\right|^{2}\right).$
(4)

Moreover, if the parameter is also encoded via a unitary, so that
$\left|\psi_{\omega}\right\rangle\\!=\\!\mathrm{e}^{-\mathrm{i}\omega
H}\left|\psi\right\rangle$ where $H$ is some Hamiltonian, then
$\mathcal{F}(\varrho_{\omega})\\!=\\!4(\left\langle\psi\right|H^{2}\left|\psi\right\rangle-\left\langle\psi\right|H\left|\psi\right\rangle^{2})=:4\,\mathrm{Var}_{\left|\psi\right\rangle}(H)$.
In this case the QFI is maximized by preparing the probes in the state
$\left|\psi\right\rangle\\!=\\!\frac{1}{\sqrt{2}}\left(\left|\lambda_{\textrm{max}}\right\rangle\\!+\\!\left|\lambda_{\textrm{min}}\right\rangle\right)$,
where $\left|\lambda_{\textrm{max/min}}\right\rangle$ are the eigenstates
corresponding to the maximum/minimum eigenvalues of $H$. Furthermore, if the
$N$ probes are subjected to a unitary evolution generated by a _local_
Hamiltonian, e.g., $H=\frac{1}{2}\sum_{n=0}^{N}\sigma_{3}^{(n)}$, then such a
state corresponds to the GHZ state and attains the HS by virtue of the CRB (1)
Giovannetti  _et al._ (2004). However, note that the QFI is important not only
for metrology but also for entanglement detection Pezzé and Smerzi (2009);
Lücke  _et al._ (2014); Strobel  _et al._ (2014); defines a natural geometric
distance between quantum states Pires  _et al._ (2015), thus serving as a tool
to derive speed limits on quantum evolution Pires  _et al._ (2015); Taddei
_et al._ (2013), as well as constitutes a measure of macroscopicity Fröwis and
Dür (2012). The QFI is the primary focus of this work and we further discuss
its role in quantifying the precision of estimation in metrology protocols in
more detail in Sec. V.

###  III.2 Time-homogeneous qubit evolution

The interaction dynamics via which a probe senses the parameter is generally
described by a quantum channel Nielsen and Chuang (2010) (see Fig. 1),
${\mathcal{E}}_{\omega,t}$, that encodes the parameter $\omega$ onto the state
of the probe over the interrogation time $t$:
$\rho_{\omega}(t)\\!=\\!{\mathcal{E}}_{\omega,t}(\rho)$. The channel consists
of a unitary evolution part, encoding the parameter of interest, and a non-
unitary (noise) part describing additional system-environment interactions. In
this work we shall consider that the non-unitary part arises due to
uncorrelated noise processes described by a _time-homogeneous master equation
of Lindblad form_ , which provides an appropriate description for most
physically relevant noise processes. In particular, in such a typical setting
the environment is assumed to be sufficiently large such that it disturbs the
system in the same fashion independently of the system state and the time
instance Breuer and Petruccione (2002). Thus, the time-homogeneity of
evolution allows us to unambiguously apply the FFQC techniques as any
operation applied on the system does not affect the environment. Moreover, one
may then freely swap the sensing particles at any time with the ancillary
ones, which experience afterwards the same fixed noisy dynamics. In contrast,
noise processes described by time-inhomogeneous master equations have been
recently considered within the context of quantum metrology in Matsuzaki  _et
al._ (2011); Chin  _et al._ (2012); Macieszczak (2015); Smirne  _et al._
(2016), where time-inhomogeneity has been shown to be beneficial at short
time-scales— the so-called Zeno regime—where the bath can no longer be assumed
to be uncorrelated from the system. Let us stress that such a regime, however,
does not allow for the general control operations considered here to be
unambiguously applied without explicitly modelling the environment Addis  _et
al._ (2016); Sekatski  _et al._ (2015).

We describe the evolution of a single qubit probe by following time-
homogeneous master equation

$\frac{\mathrm{d}\rho_{\omega}(t)}{\mathrm{d}t}=-\mathrm{i}\frac{\omega}{2}[\sigma_{3},\rho_{\omega}(t)]+{\mathcal{L}}(\rho_{\omega}(t)),$
(5)

where $\omega$ is the parameter to be estimated and the noise is given by the
Lindblad super-operator, $\mathcal{L}$. The Lindbladian is then generally
defined as

${\mathcal{L}}(\rho):=\frac{1}{2}\sum_{\mu,\nu=0}^{3}\mathsf{L}_{\mu\nu}\left(\left[\sigma_{\mu}\rho,\sigma_{\nu}\right]+\left[\sigma_{\mu},\rho\sigma_{\nu}\right]\right),$
(6)

where $\mathsf{L}$ is a Hermitian, positive semi-definite matrix whose entries
are independent of $t$ and $\omega$. We note that all the noise terms
$\mathsf{L}_{0i}$ and $\mathsf{L}_{i0}$ can be straightforwardly corrected
within FFQC as they lead to an additional Hamiltonian evolution term in the
Eq. (5), sometimes referred to as Lamb-shift. This $\omega$-independent
Hamiltoanian term can be canceled by continuously applying the inverse unitary
rotation. In what follows we assume that the control operations already
incorporate such unitary and will, thus, be concerned with the restriction of
$\mathsf{L}$ to the subspace spanned by
$\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}$, which we denote as
$\bar{\mathsf{L}}$.

We can group all the relevant noise processes into three important families:

  * •

_Rank-one Pauli_ noise of strength $\gamma$

$\mathcal{L}^{\textrm{1P}}_{\bf n}(\rho):=\frac{\gamma}{2}(\sigma_{\bf
n}\rho\sigma_{\bf n}-\rho),$ (7)

with the particular case of _dephasing_ noise $\mathcal{L}^{\textrm{1P}}_{\bf
z}$ for which $\sigma_{\bf z}=\sigma_{3}$.

  * •

_General rank-one_ noise $\mathcal{L}^{\textrm{1G}}_{\bf r}$ defined by the
matrix

$\bar{\mathsf{L}}^{\textrm{1G}}_{\bf r}:={\bf r}\,{\bf r}^{\dagger},$ (8)

where ${\bf r}=(x,y,z)^{T}$ and the strength of the noise is denoted
$\gamma/2\\!=\\!|{\bf r}|^{2}$. The special case $\mathrm{Re}({\bf
r})\\!\times\\!\mathrm{Im}({\bf r})\\!=\\!0$ corresponds to the rank-one Pauli
noise of Eq. (7), whereas whenever $|\mathrm{Re}({\bf
r})|\\!=\\!|\mathrm{Im}({\bf r})|$ and $\mathrm{Re}({\bf
r})^{T}\mathrm{Im}({\bf r})\\!=\\!0$ an _amplitude damping_ channel is
recovered, which represents spontaneous emission along some particular
direction.

  * •

_Rank-two Pauli_ noise $\mathcal{L}^{\textrm{2P}}_{\bf\Omega}$ defined by the
matrix

$\displaystyle\bar{\mathsf{L}}^{\textrm{2P}}_{\bf\Omega}:=\frac{1}{2}\,R_{\bf\Omega}^{T}\small\left(\begin{array}[]{ccc}\gamma_{1}&&\\\
&\gamma_{2}&\\\ &&0\end{array}\right)R_{\bf\Omega},$ (12)

where ${\bf\Omega}\\!\in\\!\mathrm{SO}(3)$ and
$R_{\bf\Omega}\\!=\\!R_{z}(\varphi)R_{y}(\theta)R_{z}(\xi)$ is its matrix
representation written in the Euler form with angles $(\varphi,\theta,\xi)$.
The special case of $R_{\bf\Omega}\\!=\\!\mathbbm{1}$ corresponds to
_asymmetric X-Y noise_ , which we conveniently parametrize with
$\gamma_{1}\\!=\\!\gamma\,p$ and $\gamma_{2}\\!=\\!\gamma(1-p)$ for
$0\\!\leq\\!p\\!\leq\\!1$ to define

$\displaystyle\mathcal{L}_{\textrm{X-Y}}(\rho):=\frac{\gamma}{2}(p\,\sigma_{1}\rho\,\sigma_{1}+(1-p)\sigma_{2}\rho\,\sigma_{2}-\rho).$
(13)

In order to explicitly determine the form of the quantum channel describing
the qubit evolution, one has to integrate Eq. (5) Andersson  _et al._ (2007).
The resulting dynamics can then be expressed, e.g, with help of the
corresponding dynamical matrix, $\mathsf{S}$ (specified in the Pauli operator
basis), or via a Kraus representation as follows:

$\displaystyle{\mathcal{E}}_{\omega,t}(\rho)$ $\displaystyle=$
$\displaystyle\sum_{\mu,\nu=0}^{3}\mathsf{S}_{\mu\nu}(\omega,t)\,\sigma_{\mu}\,\rho\,\sigma_{\nu}$
(14) $\displaystyle=$
$\displaystyle\;\sum_{i}K_{i}(\omega,t)\,\rho\,K_{i}(\omega,t)^{\dagger}.$
(15)

Note that the Kraus representation of Eq. (15) is not unique as starting from
a set of $r$ linearly independent Kraus operators, ${\bf
K}\\!=\\!(K_{1},\dots,K_{r})^{T}$, we may simply construct another valid set
${\bf K}^{\prime}\\!=\\!\mathsf{u}{\bf K}$ choosing $\mathsf{u}$ to be any
(potentially $\omega$-,$t$\- dependent) $r\\!\times\\!r$ unitary matrix.

##  IV Noisy metrology with full and fast control

In this section we establish analytic upper bounds on the QFI for uncorrelated
qubit noise using the most general metrological protocol employing FFQC (Fig.
1(d)). We first outline the optimal FFQC protocol suitable for correcting
rank-one Pauli noise that is not parallel to the Hamiltonian (Sec. IV.1). In
Sec. IV.2.1 we describe the channel extension (CE) method used to upper bound
the QFI Fujiwara and Imai (2008); Demkowicz-Dobrzański  _et al._ (2012);
Kołodyński and Demkowicz-Dobrzański (2013); Demkowicz-Dobrzański and Maccone
(2014) and then review the bounds it gives for strategies (a), (b) and (c) of
Fig. 1. However, the CE method fails to yield a meaningful bound when
considering the limit in which the sensing process lasts an infinitesimal
time-step. In Sec. IV.2.3 we resolve this issue by ameliorating the CE method
in order to obtain an upper bound on the QFI for strategies that employ FFQC.
Using the latter, we show that all rank-one non-Pauli noise, rank-two, and
rank-three noise processes result in a QFI that scales linearly with the total
number of resources even with FFQC (Sec. IV.2.4). In Sec. IV.2.5 we explicitly
derive the optimal bounds on the QFI using FFQC for the noise types introduced
in Sec. III.2. Finally, in Sec. IV.3 we discuss the relation between the
bounds obtained when allowing for FFQC to the ones applicable for scenarios
without fast control. We stress that we are able to derive analytic bounds by
extending the CE method to the limit $\mathrm{d}t\to 0$, in which the problem
is actually simplified: one does not have to solve the master equation and
determine the full dynamics! This is even more remarkable as these bounds
apply to any metrological scheme, see the hierarchy in Fig. 1.

###  IV.1 Removing rank-one Pauli noise

Here we construct a general FFQC strategy that allows one to correct for _any_
rank-one Pauli noise $\mathcal{L}^{\textrm{1P}}_{{\bf n}}$ introduced in Eq.
(7) with ${\bf n}\\!\neq\\!{\bf z}$. For convenience, we perform a change of
basis so that the noise is generated by $\sigma_{1}$ (${\bf
n}\\!\\!=\\!\\!{\bf x}$), whereas $\omega$ is encoded via a rotated
Hamiltonian
$\sigma_{\theta}\\!:=\\!\mathrm{s}_{\theta}\sigma_{3}+\mathrm{c}_{\theta}\sigma_{1}$.
We consider the qubit probe to be aided by an ancillary qubit and define the
two-qubit _code space_ as a subset of
${\mathcal{H}}_{\textrm{S}}\otimes{\mathcal{H}}_{\textrm{A}}\supset{\mathcal{H}}_{\textrm{C}}\\!:=\\!\mathrm{span}\\{\left|00\right\rangle,\left|11\right\rangle\\}$,
with corresponding projector
$\Pi_{\textrm{C}}\\!:=\\!\left|00\right\rangle\\!\left\langle
00\right|+\left|11\right\rangle\\!\left\langle 11\right|$, and the _error
space_ ,
${\mathcal{H}}_{\textrm{E}}\\!:=\\!\mathrm{span}\\{\left|01\right\rangle,\left|10\right\rangle\\}$,
with $\Pi_{\textrm{E}}\\!:=\\!\mathbbm{1}-\Pi_{\textrm{C}}$.

Let the probe-plus-ancilla be prepared in a pure state
$\varrho\\!=\\!\left|\psi\right\rangle\\!\left\langle\psi\right|$ with
$\left|\psi\right\rangle\\!=\\!\alpha\left|00\right\rangle+\beta\left|11\right\rangle\in{\mathcal{H}}_{\textrm{C}}$.
The dynamics on the probe-plus-ancilla is then described by Eq. (7) with only
the probe system evolving

$\frac{\mathrm{d}\varrho}{\mathrm{d}t}=-\mathrm{i}\frac{\omega}{2}[\sigma_{\theta}\otimes\mathbbm{1},\varrho]+\mathcal{L}^{\textrm{1P}}_{\bf
x}\\!\otimes\\!\mathcal{I}\,(\varrho).$ (16)

Integrating Eq. (16) over an elementary time step $\mathrm{d}t$, we may write
the probe-plus-ancilla state, up to first order in $\mathrm{d}t$, as

$\displaystyle\varrho(\mathrm{d}t)$ $\displaystyle=$
$\displaystyle\varrho(1-\frac{\gamma}{2}\mathrm{d}t)-\mathrm{i}\frac{\omega}{2}[\sigma_{\theta}\otimes\mathbbm{1},\varrho]\mathrm{d}t$
$\displaystyle+\,\frac{\gamma}{2}(\sigma_{1}\otimes\mathbbm{1})\varrho(\sigma_{1}\otimes\mathbbm{1})\mathrm{d}t+O(\mathrm{d}t^{2}).$

Projecting $\varrho(\mathrm{d}t)$ onto the code and error subspaces yields the
unnormalized states
$\varrho_{\textrm{C}}(\mathrm{d}t)\\!=\\!\Pi_{\textrm{C}}\varrho(\mathrm{d}t)\Pi_{\textrm{C}}$
and
$\varrho_{\textrm{E}}(\mathrm{d}t)\\!=\\!\Pi_{\textrm{E}}\varrho(\mathrm{d}t)\Pi_{\textrm{E}}$
given by

$\displaystyle\varrho_{\textrm{C}}(\mathrm{d}t)$
$\displaystyle=\varrho(1-\frac{\gamma}{2}\mathrm{d}t)-\mathrm{i}\,\mathrm{s}_{\theta}\frac{\omega}{2}[\sigma_{3}\otimes\mathbbm{1},\varrho]\mathrm{d}t+O(\mathrm{d}t^{2}),$
$\displaystyle\varrho_{\textrm{E}}(\mathrm{d}t)$
$\displaystyle=\frac{\gamma}{2}(\sigma_{1}\otimes\mathbbm{1})\varrho(\sigma_{1}\otimes\mathbbm{1})\,\mathrm{d}t+O(\mathrm{d}t^{2})$
(18)

respectively.

If an error is detected, we simply apply $\sigma_{1}$ on the sensing probe (so
after correction
$\bar{\varrho}_{\textrm{E}}(\mathrm{d}t)\\!:=\\!(\sigma_{1}\otimes\mathbbm{1})\varrho_{\textrm{E}}(\mathrm{d}t)(\sigma_{1}\otimes\mathbbm{1})$),
and otherwise do nothing. Hence, the state of probe-plus-ancilla after an
infinitesimal time-step followed by fast EC is the mixture

$\displaystyle\bar{\varrho}(\mathrm{d}t)$ $\displaystyle:=$
$\displaystyle\varrho_{\textrm{C}}(\mathrm{d}t)+\bar{\varrho}_{\textrm{E}}(\mathrm{d}t)$
(19) $\displaystyle=$
$\displaystyle\varrho-\mathrm{i}\,\mathrm{s}_{\theta}\frac{\omega}{2}[\sigma_{3}\otimes\mathbbm{1},\varrho]\,\mathrm{d}t+O(\mathrm{d}t^{2}),$

which, to first order in $\mathrm{d}t$, is equivalent to a unitary evolution
under the projected Hamiltonian $\frac{\omega}{2}s_{\theta}\sigma_{3}$. As the
measurement may always be adjusted to compensate for a known rotation of the
Hamiltonian encoding the parameter, the above strategy perfectly corrects
rank-one Pauli noise (7) at the price of slowing the evolution down by a
factor $s_{\theta}=\sqrt{1-({{\bf n}}^{T}{\bf z})^{2}}$.

We summarize the above result in the following observation:

######  Observation 1.

A general FFQC strategy allows one to completely eliminate any rank-one Pauli
noise $\mathcal{L}^{\mathrm{1P}}_{{\bf n}}$ of Eq. (7) that is not exactly
parallel to the parameter-encoding Hamiltonian. The resulting dynamics of the
system are then described by a noiseless evolution with a rotated Hamiltonian
and the estimated parameter being rescaled to $\sqrt{1-({\bf n}^{T}{\bf
z})^{2}}\,\omega$, which still yields HS in precision.

Note that that in the derivation above nothing forbids to replace a scalar
parameter $\omega$ by any operator $\hat{\omega}$ acting on an additional
system described by the Hilbert space ${\mathcal{H}}_{\textrm{B}}$.
Consequently, with the help of one ancillary qubit the FFQC strategy described
above allows to effectively modify the noisy evolution

$\frac{\mathrm{d}\varrho_{\textrm{BS}}}{\mathrm{d}t}=-\mathrm{i}\frac{1}{2}[\hat{\omega}\otimes\sigma_{\theta},\varrho_{\textrm{BS}}]+\mathcal{I}_{\textrm{B}}\\!\otimes\\!\mathcal{L}^{\textrm{1P}}_{\bf
x}\,(\varrho_{\textrm{BS}}),$ (20)

with
$\varrho_{\textrm{BS}}\in\mathcal{B}({\mathcal{H}}_{\textrm{B}}\otimes{\mathcal{H}}_{\textrm{S}})$,
to

$\frac{\mathrm{d}\varrho_{\textrm{BS}}}{\mathrm{d}t}=-\mathrm{i}\frac{\sqrt{1-({{\bf
n}}^{T}{\bf
z})^{2}}}{2}[\hat{\omega}\otimes\sigma_{3},\varrho_{\textrm{BS}}],$ (21)

with a modified interaction Hamiltonian but without the local rank-one Pauli
noise acting on the system qubit. In the particular case where the system B is
also a qubit affected by some rank-one Pauli noise and
$\hat{\omega}=\sigma_{\varphi}$ (again we chose a basis in which the noise is
in the ${\bf x}$ direction) one may use the same FFQC on both qubits and get
rid of both local noise terms. This can be used for gate implementation, where
the slow-down factor can be compensated by prolonging the interaction time and
the original coupling $\sigma_{\varphi}\otimes\sigma_{\theta}$ can be
recovered from the effective one $\sigma_{3}\otimes\sigma_{3}$ by means of
initial and final local unitaries333Remark that the constuction of our FFQC
strategy assumes that one can perform noiseless parity measurement on the
probe-plus-ancillary system, which is a two-qubit operation. However in
practice the ancilla may correspond to an internal degree of freedom, while
the gate is constructed on two different systems.. In general, one does not
need to know $\hat{\omega}$ to arrive to the effective evolution Eq. (21).

###  IV.2 Other noise-types: unavoidable linear scaling

####  IV.2.1 Upper bounds on the QFI

We would like to compare the maximum achievable QFI in the various metrology
schemes introduced in Sec. II and, in particular, quantify the improvement (if
any) between protocols (b), (c) and (d). To this end we use variations of the
CE method Fujiwara and Imai (2008); Demkowicz-Dobrzański  _et al._ (2012);
Kołodyński and Demkowicz-Dobrzański (2013); Demkowicz-Dobrzański and Maccone
(2014) that allow one to upper-bound the QFI depending on the scenario
considered. Crucially, they constrain the QFI using solely the properties of
the quantum channel responsible for encoding the parameter.

##### CE bound.

The QFI of a state representing the output of a quantum channel encoding the
parameter may always be upper-bounded purely by the channel properties after
performing a channel extension (CE). Given a probe in a state $\rho$ and a
quantum channel ${\mathcal{E}}_{\omega,t}$ one may always construct the _CE
bound_ on the QFI of the output state ${\mathcal{E}}_{\omega,t}(\rho)$
Fujiwara and Imai (2008):

$\mathcal{F}({\mathcal{E}}_{\omega,t}(\rho))\;\leq\;\max_{\varrho}\mathcal{F}({\mathcal{E}}_{\omega,t}\otimes\mathcal{I}_{\textrm{A}}(\varrho))\;\leq\;4\,||\alpha(t)||,$
(22)

where
$\varrho\\!\in\\!{\mathcal{B}}({\mathcal{H}}_{\textrm{S}}\otimes{\mathcal{H}}_{\textrm{A}})$
represents a joined (_extended_) state of the probe and an ancilla, and the
operator

$\displaystyle\alpha(t)$ $\displaystyle:=$
$\displaystyle\sum_{i}\\!\dot{K}_{i}^{\dagger}\dot{K}_{i}=\dot{{\bf
K}}^{\dagger}\,\dot{{\bf K}}$ (23)

for any of the Kraus representation Eq. (15) of the probe channel. And
$\dot{K}_{i}$ are the derivatives of the Kraus operators with respect to
$\omega$ (in what follows, we write explicit dependencies of the Kraus
operators on $t$ and $\omega$ only if necessary) 444Note that alternatively to
the derivation given in Fujiwara and Imai (2008), the second inequality in Eq.
(22) can be understood as a consequence of the strong concavity of the
fidelity in Eq. (3)..

Although, the first inequality in Eq. (22) may not be tight—entangling the
probe and ancilla may increase the achievable QFI—the second is. A valid Kraus
representation of ${\mathcal{E}}_{\omega,t}$ for which the second inequality
is saturated is guaranteed to exist as long as
$\dim({\mathcal{H}}_{\textrm{A}})\\!\geq\\!\dim({\mathcal{H}}_{\textrm{S}})$
Fujiwara and Imai (2008). In consequence, the CE bound (22) directly
determines the maximal attainable QFI for the parallel scheme with EC depicted
in Fig. 1(b) with $N\\!=\\!1$ Kołodyński and Demkowicz-Dobrzański (2013).

##### Parallel CE bound.

The CE bound (22) may be directly applied to upper-bound the ultimate
attainable QFI of any parallel scheme of Fig. 1(b) that employs $N$ probes and
ancillae. In scenario (b), the final state generally reads
$\varrho^{\textrm{(}b)}_{N}(t)\\!=\\!{\mathcal{E}}^{\otimes
N}_{\omega,t}\otimes\mathcal{I}_{\textrm{A}}(\varrho_{N}^{\textrm{(b)}})$.
Hence, applying the CE bound (22) to channel ${\mathcal{E}}^{\otimes
N}_{\omega,t}\otimes\mathcal{I}_{\textrm{A}}$ that describes the overall
evolution and fixing a tensor product Kraus representation such that

${\mathcal{E}}^{\otimes
N}_{\omega,t}(\rho^{N})=\\!\\!\\!\\!\sum_{i_{1},\dots,i_{N}}\\!\\!\\!\\!K_{i_{1}}\\!\otimes...\otimes
K_{i_{N}}\;\rho^{N}\,K_{i_{1}}^{\dagger}\\!\otimes...\otimes
K_{i_{N}}^{\dagger},$ (24)

one obtains the _parallel CE bound_ Fujiwara and Imai (2008); Demkowicz-
Dobrzański  _et al._ (2012); Kołodyński and Demkowicz-Dobrzański (2013):

$\displaystyle\mathcal{F}\Big{(}\varrho^{\textrm{(b)}}_{N}(t)\Big{)}$
$\displaystyle\leq$
$\displaystyle\max_{\varrho_{N}^{\textrm{(b)}}}\,\mathcal{F}\Big{(}{\mathcal{E}}^{\otimes
N}_{\omega,t}\otimes\mathcal{I}_{\textrm{A}}(\varrho^{(\textrm{b})}_{N})\Big{)}$
(25) $\displaystyle\leq$ $\displaystyle
4N||\alpha(t)||+4N(N-1)||\beta(t)||^{2},$

where
$\varrho_{N}^{\textrm{(b)}}\\!\in\\!{\mathcal{B}}({\mathcal{H}}_{\textrm{S}}^{\otimes
N}\otimes{\mathcal{H}}_{\textrm{A}})$ describes the initial probes-plus-
ancillae state and

$\beta(t):=\mathrm{i}\sum_{i}\\!\dot{K}_{i}^{\dagger}K_{i}=\mathrm{i}\,\dot{{\bf
K}}^{\dagger}\,{{\bf K}}$ (26)

which depends solely on Kraus operators and their derivatives. In order to
obtain Eq. (25) from Eq. (22) note that due to the tensor product structure
the term $\dot{K}_{{\bf i}}^{\dagger}\dot{K}_{{\bf
i}}=\frac{\mathrm{d}}{\mathrm{d}\omega}(K_{i_{1}}^{\dagger}\otimes...\otimes
K_{i_{N}}^{\dagger})\,\frac{\mathrm{d}}{\mathrm{d}\omega}(K_{i_{1}}\otimes...\otimes
K_{i_{N}})$ in Eq. (22) contains $N$ terms for which in both parenthesis the
derivative applies on the Kraus operator acting on the same probe and $N(N-1)$
terms where it applies on the Kraus operators acting on two different probes
Fujiwara and Imai (2008). Yet, in contrast to Eq. (22), the parallel CE bound
(25) is then not generally attainable, even after optimizing Eq. (25) over all
Kraus representations of the single-probe channel. This is because the Kraus
decomposition for the map ${\mathcal{E}}^{\otimes
N}_{\omega,t}\otimes\mathcal{I}_{\textrm{A}}$ that makes Eq. (22) saturable
might not be of a tensor-product form, as assumed in Eq. (24).

##### Sequential CE bound.

Now, we would like to apply the CE methods to the two strategies involving
control, i.e., (c) and (d) of Fig. 1. However, as argued in Sec. II, any
scheme of type (c) employing $N$ probes with $m$ control steps (each lasting
$\delta t$) can always be mimicked by a scheme Fig. 1(d) employing a single
probe but lasting $N$-times longer, i.e., with $t^{\prime}\\!=\\!Nt$ and
involving more $k\\!=\\!Nm$ steps each still of duration $\delta t$. Hence, we
may always upper-bound the QFI of _any_ scheme depicted in Fig. 1(c) as

$\mathcal{F}\Big{(}\varrho^{\textrm{(c)}}_{N,m}(t)\Big{)}\leq\max_{\underset{k=Nm,\,t^{\prime}=Nt}{\textrm{schemes\,(d)}}}\mathcal{F}\Big{(}\varrho^{\textrm{(d)}}_{k}(t^{\prime})\Big{)}$
(27)

where the maximization is over all sequential schemes of type (d) with step
duration $\delta t\\!=\\!N/m$. Thus, in what follows, we focus on applying the
CE methods to the scenario of Fig. 1(d). Note that by decreasing $\delta t$
further, or equivalently by raising the number of control steps $k$, we may
only increase the right hand side of Eq. (27). This manifests the power of
FFQC and the optimality of the $\delta t\\!\to\\!0$ limit.

For sequential strategies employing intermediate control a CE-based upper
bound on the corresponding QFI has recently been derived in Ref. Demkowicz-
Dobrzański and Maccone (2014). Let the output state for the sequential
protocol of Fig. 1(d) be given as

$\varrho^{\textrm{(d)}}_{k}(t^{\prime})\\!=\\!\sum_{{\bm{i}}^{(k)}}K_{{\bm{i}}^{(k)}}\\!(t^{\prime})\;\,\varrho^{\textrm{(d)}}\,\;K^{\dagger}_{{\bm{i}}^{(k)}}\\!(t^{\prime})$
(28)

with
$\varrho^{\textrm{(d)}}\\!\in\\!{\mathcal{B}}({\mathcal{H}}_{\textrm{S}}\otimes{\mathcal{H}}_{\textrm{A}})$
being the initial state. However, in contrast to Eq. (24), the overall Kraus
operators

$K_{{\bm{i}}^{(k)}}\\!(t)\\!:=\\!(K_{i_{k}}\\!(\delta
t)\otimes\mathbbm{1}_{\textrm{A}})U_{k}\dots(K_{i_{1}}\\!(\delta
t)\otimes\mathbbm{1}_{\textrm{A}})U_{1}$ (29)

have a composition rather than tensor-product structure, where $U_{\ell}$
stands for the a control operation acting on both probe and ancillae at the
$\ell$th step , while $K_{i_{\ell}}$ are the Kraus operators of the channel
${\mathcal{E}}_{\omega,\delta t}$ acting solely on the probe at the $\ell$th
step. An application of the general CE bound (22) to this case was derived in
Ref. Demkowicz-Dobrzański and Maccone (2014), leading to the _sequential CE
bound_ on the QFI attainable using _any_ sequential scheme with FFQC (with
time step $\delta t$):

$\displaystyle\mathcal{F}\Big{(}\varrho^{\textrm{(d)}}_{k}(t^{\prime})\Big{)}$
$\displaystyle\leq$ $\displaystyle 4k||\alpha(\delta t)||+$ $\displaystyle+$
$\displaystyle 4k(k-1)||\beta(\delta t)||(||\alpha(\delta t)||+||\beta(\delta
t)||+1),$

which applies to _any_ protocol (d) of Fig. 1 with time-steps of duration
$\delta t$, and hence also to _all_ schemes (c) of Fig. 1 upon substituting
$k\\!=\\!Nm$.

Notice that, just as in the case of the parallel CE bound (25), there may not
exist a Kraus representation of the single-probe channel such that the
sequential CE bound of Eq. (IV.2.1) is guaranteed to be tight. This holds also
for the optimal sequential strategy (d) that maximizes the corresponding QFI
(i.e., the one maximizing the right hand side of Eq. (27)).

####  IV.2.2 Optimization of the CE bounds

All the CE bounds presented this far:  _standard_ (Eq. (22)), _parallel_ (Eq.
(25)) and _sequential_ (Eq. (IV.2.1)); rely solely on the structure of the
single-probe channel ${\mathcal{E}}_{\omega,t}$ encoding the parameter and its
particular Kraus representation (15). Thus, in order to obtain the tightest
versions of these bounds one should minimize them over all equivalent Kraus
representations, ${\bf K^{\prime}}\\!=\\!\mathsf{u}{\bf K}$ (with
$\mathsf{u}^{\dagger}\mathsf{u}\\!=\\!\mathbbm{1}$), describing the single-
probe dynamics. Although such a gauge freedom is generally parameter
dependent, i.e., $\mathsf{u}\\!=\\!\mathsf{u}(\omega)$, the operators $\alpha$
(23) and $\beta$ (26) depend solely on
$\mathsf{h}\\!:=\\!\mathrm{i}\,\mathsf{u}^{\dagger}\dot{\mathsf{u}}$. Hence,
it always suffices to search only through Kraus representations satisfying
${\bf K}^{\prime}\\!=\\!{\bf K}$ and $\dot{{\bf K}}^{\prime}\\!=\\!\dot{{\bf
K}}\\!-\\!\mathrm{i}\mathsf{h}{\bf K}$ or, in other words, any CE-based bound
considered may always be optimized by simply performing its minimization over
all Hermitian matrices $\mathsf{h}$ Fujiwara and Imai (2008).

In contrast to the sequential CE bound (IV.2.1), the standard (22) and
parallel (25) CE bounds are quadratic (convex) in $\mathsf{h}$, and thus may
be minimized numerically in a systematic manner by means of semi-definite
programming (SDP) given some ${\bf K}$ Demkowicz-Dobrzański  _et al._ (2012);
Kołodyński and Demkowicz-Dobrzański (2013). On the other hand, when
considering t (or $\delta t$ in case of Eq. (IV.2.1)) to be fixed and the
asymptotic limit of $N$ (25) ($Nm$ (27), or $k$ (IV.2.1)), it is always
optimal to set $\beta(t)\\!=\\!0$ in the CE bounds if possible 555Note that
for any $A=A^{\dagger}$, $||A||=0\Leftrightarrow A=0$, so that A is the null
operator., so that they _asymptotically_ become linear in $N$ ($Nm$, or $k$
respectively). In such a regime, as the constraint $\beta(t)\\!=\\!0$ is
linear in $\mathsf{h}$, one may always apply the SDP methods to minimize the
remaining coefficient $4||\alpha(t)||$, which is then emergent in all the
bounds Demkowicz-Dobrzański  _et al._ (2012); Kołodyński and Demkowicz-
Dobrzański (2013).

Equivalently, the same argumentation can be made starting from the dynamical
matrix $\mathsf{S}$ of the channel ${\mathcal{E}}_{\omega,t}$, defined in Eq.
(14), instead of its Kraus representation. The two are directly related (see
also App. A), as any valid vector of Kraus operators ${\bf K}$ can always be
expressed in a complete operator basis, e.g. the Pauli basis in case of a
single-qubit maps, via a matrix $\mathsf{M}$ satisfying

$\displaystyle{\bf
K}=\mathsf{M}{\bm{\varsigma}}\quad\text{and}\quad\mathsf{M}^{\dagger}\mathsf{M}=\mathsf{S}^{T}.$
(31)

As a result, one may rewrite the operators $\alpha$ (23) and $\beta$ (26) with
help of the matrix $\mathsf{M}$ and a general Hermitian matrix $\mathsf{h}$ to
be optimized as

$\displaystyle\alpha(t)$ $\displaystyle=$
$\displaystyle{\bm{\varsigma}}^{\dagger}\,(\dot{\mathsf{M}}^{\dagger}+\mathrm{i}{\mathsf{M}}^{\dagger}{\mathsf{h}})\,(\dot{\mathsf{M}}-\mathrm{i}{\mathsf{h}}{\mathsf{M}})\,{\bm{\varsigma}},$
(32) $\displaystyle\beta(t)$ $\displaystyle=$
$\displaystyle{\bm{\varsigma}}^{\dagger}\,(\dot{\mathsf{M}}^{\dagger}+\mathrm{i}{\mathsf{M}}^{\dagger}{\mathsf{h}})\,{\mathsf{M}}\,{\bm{\varsigma}}.$
(33)

In what follows, we always choose
$\mathsf{M}\\!=\\!\mathsf{M}^{\dagger}\\!=\\!\sqrt{\mathsf{S}}^{T}$ and thus
unambiguously fix the starting Kraus representation ${\bf K}$ in Eq. (31). We
then make use of Eqs. (32) and (33) in order to _analytically_ optimize over
$\mathsf{h}$ the infinitesimal time-step version of the sequential CE bound
(IV.2.1) as we now explain.

####  IV.2.3 The infinitesimal time-step CE bound

In the most powerful FFQC setting of Fig. 1(d), one allows for the control
operations to be _arbitrarily fast_ and thus assumes the steps of the protocol
to last an _infinitesimal_ duration $\mathrm{d}t$. Hence, one deals then with
a sequential strategy of infinitesimal step duration, for which the sequential
bound of Eq. (IV.2.1) can be directly applied after setting
$k\\!\to\\!\frac{t^{\prime}}{\mathrm{d}t}$ and taking the limit $\delta
t\\!\equiv\\!\mathrm{d}t\\!\to\\!0$ (see App. B). In order for the bound to
remain meaningful, we adjust the derivation of Ref. Demkowicz-Dobrzański and
Maccone (2014) in App. B showing that the bound still holds for any
$x\\!>\\!0$:

$\displaystyle\;\;\mathcal{F}\Big{(}\varrho^{\textrm{(d)}}_{t^{\prime}/\mathrm{d}t}(t^{\prime})\Big{)}\leq$
(34) $\displaystyle\qquad\leq
4\frac{t^{\prime}}{\mathrm{d}t}||\alpha(\mathrm{d}t)||+$
$\displaystyle\qquad\;\;+4\Big{(}\frac{t^{\prime}}{\mathrm{d}t}\Big{)}^{2}||\beta(\mathrm{d}t)||\\!\left(x||\alpha(\mathrm{d}t)||\\!+\\!||\beta(\mathrm{d}t)||\\!+\\!\frac{1}{x}\right).$

In particular, we term Eq. (34) the _infinitesimal time-step CE bound_ after
setting $x\\!=\\!1/\mathrm{d}t$.

In order to derive the ultimate precision bounds valid in the presence of
noise and FFQC, we compute the infinitesimal time-step CE bound (34) after
substituting the expressions for $\alpha$ and $\beta$ given by Eqs. (32) and
(33). We explicitly minimize Eq. (34) over Kraus representations as described
in Sec. IV.2.2, by expanding the Kraus operators, ${\bf K}(\mathrm{d}t)$, and
the Hermitian matrix, $\mathsf{h}(\mathrm{d}t)$, up to small orders of
$\sqrt{\mathrm{d}t}$. As we are interested in the regime of _asymptotic
resources_ , i.e., the limit $t^{\prime}\\!\gg\\!\mathrm{d}t$ (or equivalently
$Nt\\!\gg\\!\delta t$ in case of the scheme (c)), we may assume without loss
of generality that it is always optimal to make the second term in Eq. (34)
vanish by setting $\beta(\mathrm{d}t)\\!=\\!0$. We outline the methods we use
below, with detailed calculations deferred to App. C.

Consider the dynamical matrix $\mathsf{S}$, defined in Eq. (14), of the
infinitesimal time channel ${\mathcal{E}}_{\omega,\mathrm{d}t}$. Expanding it
in $\mathrm{d}t$ around $t\\!=\\!0$ gives
$\mathsf{S}(\mathrm{d}t)\\!:=\\!\mathsf{S}^{(0)}\\!+\\!\mathsf{S}^{(1)}\mathrm{d}t\\!+\\!O(\mathrm{d}t^{2})$.
The evolution of the probe after an infinitesimally short time is then given
by

$\displaystyle{\mathcal{E}}_{\omega,\mathrm{d}t}(\rho)$ $\displaystyle=$
$\displaystyle\sum_{\mu,\nu=0}^{3}\mathsf{S}_{\mu\nu}(\mathrm{d}t)\;\sigma_{\mu}\,\rho\,\sigma_{\nu}.$
$\displaystyle=$
$\displaystyle\sum_{\mu,\nu=0}^{3}(\mathsf{S}^{(0)}_{\mu\nu}+\mathsf{S}^{(1)}_{\mu\nu}\mathrm{d}t)\,\sigma_{\mu}\,\rho\,\sigma_{\nu}+O(\mathrm{d}t^{2}).$

The above expansion is unambiguously specified up to $O(\mathrm{d}t^{2})$ by
the master equation in Eq. (5)

${\mathcal{E}}_{\omega,\mathrm{d}t}(\rho)=\rho+\left(-\mathrm{i}\frac{\omega}{2}[\sigma_{3},\rho]+{\mathcal{L}}(\rho)\right)\mathrm{d}t+O(\mathrm{d}t^{2}),$
(36)

so we may directly relate the short-time expansion of $\mathsf{S}$ to the
Lindbladian of Eq. (6) without explicitly integrating Eq. (5):

$\displaystyle\mathsf{S}^{(0)}$ $\displaystyle=$
$\displaystyle\mathrm{diag}(1,0,0,0),$ (37) $\displaystyle\mathsf{S}^{(1)}$
$\displaystyle=$
$\displaystyle{\small\left(\begin{array}[]{c|ccc}-\sum_{i=1}^{3}\bar{\mathsf{L}}_{ii}&\mathrm{Im}\bar{\mathsf{L}}_{23}&\mathrm{Im}\bar{\mathsf{L}}_{31}&\mathrm{Im}\bar{\mathsf{L}}_{12}+\mathrm{i}\frac{\omega}{2}\\\
\hline\cr\mathrm{Im}\bar{\mathsf{L}}_{23}&&&\\\
\mathrm{Im}\bar{\mathsf{L}}_{31}&&\bar{\mathsf{L}}&\\\
\mathrm{Im}\bar{\mathsf{L}}_{12}-\mathrm{i}\frac{\omega}{2}\end{array}\right)}\\!.$
(42)

We remark that in the limit $\mathrm{d}t\to 0$ these are the only meaningful
terms, as higher orders in the expansion do not affect the final state (which
is precisely the reason why the master equation formalism is valid).

Now, when minimizing the bound of Eq. (34), we may utilize the conveniently
chosen relation
$\mathsf{M}\\!=\\!\mathsf{M}^{\dagger}\\!=\\!\sqrt{\mathsf{S}}^{T}$, so that
the expansion of $\mathsf{M}$ at small times is then also directly specified
by Eqs. (37) and (42). In particular, we expand the infinitesimal time-step CE
bound (34) in orders of $\sqrt{\mathrm{d}t}$ after adequately defining
expansions for operators $\alpha(\mathrm{d}t)\\!:=\\!\sum_{\ell\geq
0}\alpha^{(\ell)}(\mathrm{d}t)^{\ell/2}$ and similarly for
$\beta(\mathrm{d}t)$. Then, stemming from the expressions given by Eqs. (32)
and (33), we may relate such expansions “order by order” to the ones of the
matrix $\mathsf{M}$ (fixed by Eqs. (37) and (42)) and the Hermitian matrix
$\mathsf{h}$, which we still can optimize in order to obtain the tightest
infinitesimal time-step CE bound (34) in the limit $\mathrm{d}t\\!\to\\!0$.

We include the full details in App. B, yet let us summarize here that a non-
trivial bound in Eq. (34) can only be obtained if the expansion terms satisfy
$\alpha^{(0)}\\!=\\!\alpha^{(1)}\\!=\\!\beta^{(0)}\\!=\\!\beta^{(1)}\\!=\\!0$,
which constrains the low-order terms in the expansion of the matrix
$\mathsf{h}$. In addition, if these terms can be chosen such that also
$\beta^{(2)}=0$, then by taking the limit $\mathrm{d}t\\!\to\\!0$ with
$k=t^{\prime}/\mathrm{d}t$ (and $x=1/\mathrm{d}t$) the infinitesimal time-step
CE bound of Eq. (34) yields a linear scaling in $t^{\prime}$:

$\mathcal{F}\Big{(}\varrho^{\textrm{(d)}}_{t^{\prime}/\mathrm{d}t}(t^{\prime})\Big{)}\leq
4||\alpha^{(2)}||\,t^{\prime}+O(\sqrt{dt}).$ (43)

Note that as the sequential scheme with FFQC is the most powerful one (see
Sec. II), Eq. (43) holds for all protocols depicted in Fig. 1 after
accordingly setting $t^{\prime}\\!=\\!Nt$. In particular, Eq. (43) also
applies in case of control with finite time-step $\delta t$ (scheme (c)) and
in the absence of quantum control when $\delta t\\!=\\!t$ (schemes (a) and
(b)). Finally, let us remark that although in the case of parallel strategies
with ancillae of Fig. 1(b) it is the parallel CE bound (25) that must yield
tighter limits on precision for finite $t$, one may show (following exact
argumentation as in the previous paragraphs) that the linearly scaling
parallel CE bound must always converge to its infinitesimal time-step
equivalent of Eq. (43) when fixing $t^{\prime}\\!=\\!Nt$ and considering the
$t\\!\to\\!0$ limit, in which the parallel protocol becomes infinitesimally
short but involves an infinite number of probes.

####  IV.2.4 Universal asymptotic linear bound

Thanks to the infinitesimal time-step CE bound (34), we may formulate a
general observation about the scaling of the QFI applicable to any of the
schemes depicted in Fig. 1:

######  Observation 2.

For all noise processes described by Eq. (6) except rank-one Pauli noise-types
that are not parallel to the parameter-encoding Hamiltonian, the QFI in any
scheme of Fig. 1 is upper-bounded by

$\mathcal{F}_{\mathcal{L}}\leq 4\,\alpha_{\mathcal{L}}\,t^{\prime},$ (44)

where $\alpha_{\mathcal{L}}$ is a constant that solely depends on the
particular form of the Lindbladian $\mathcal{L}$ defined in Eq. (6), while
$t^{\prime}$ is the effective protocol time as defined via the most powerful
scheme (d) with FFQC in Fig. 1.

We explicitly prove Obs. 2 in App. C but outline the derivation here. In App.
C, we first show that dephasing noise, as well as any rank-one noise which is
not of Pauli type, imposes a linear scaling of the QFI due to Eq. (43). Next,
we consider noise processes described by Lindbladians whose rank is strictly
greater than one. Any such noise process corresponds to a Lindblad matrix (6)
that may be written as a direct sum of Lindblad matrices of lower rank. We
show that the QFI of noise processes of rank greater than one is always
smaller than the QFI of any of its orthogonal components. Thus, the only rank-
two noise processes that, in principle, could still allow for quadratic
scaling of the QFI are the ones corresponding to a direct sum of two
orthogonal rank-one Pauli noises, i.e., the rank-two Pauli noise of Eq. (45c).
For this case, however, we explicitly show that the QFI scales linearly with
$t^{\prime}$. Moreover, this also settles the case of the rank-three noise
types as any such noise can be expressed as a direct sum that contains either
a rank-one non-Pauli or a rank-two Pauli noise, or both.

####  IV.2.5 Exemplary noise-types

Although we have shown the validity of Eq. (44) for most types of noise
processes, it still remains to compute $\alpha_{{\mathcal{L}}}$ for a general
Lindbladian (see Eq. (6)) by adequately minimizing Eq. (43). In App. E we
perform such minimization in a completely analytic manner. In particular, we
minimize $||\alpha^{(2)}||$ in Eq. (43) over Hermitian matrices $\mathsf{h}$
(under the constraint of $\beta^{(2)}\\!=\\!0$ and
$\alpha^{(\ell)}\\!=\\!\beta^{(\ell)}\\!=\\!0$ for $\ell\\!\leq\\!1$) for all
the exemplary Lindbladians stated in Sec. III.2. The corresponding CE bounds
for dephasing, general rank-one, and rank-two Pauli noise types respectively
read

$\displaystyle\mathcal{F}^{\textrm{1P}}_{\bf z}$
$\displaystyle\leq\frac{t^{\prime}}{2\gamma}$ (45a)
$\displaystyle\mathcal{F}^{\textrm{1G}}_{{\bf r}}$
$\displaystyle\leq\frac{t^{\prime}}{4}\,\frac{\max\\{|x+\mathrm{i}y|,|x-\mathrm{i}y|\\}^{2}}{(|\mathrm{Re}({\bf
r})|^{2}|\mathrm{Im}({\bf r})|^{2}-(\mathrm{Re}({\bf r})^{T}\mathrm{Im}({\bf
r}))^{2})}$ (45b) $\displaystyle\mathcal{F}^{\textrm{2P}}_{\bf\Omega}$
$\displaystyle\leq\frac{t^{\prime}}{2\gamma_{1}\gamma_{2}}\Big{(}c_{\theta}^{2}(\gamma_{1}+\gamma_{2})+s_{\theta}^{2}(\gamma_{1}s_{\varphi}^{2}+\gamma_{2}c_{\varphi}^{2})\Big{)}$
(45c)
$\displaystyle\implies\;\mathcal{F}_{\textrm{X-Y}}\leq\frac{t^{\prime}}{2\gamma\,p(1-p)}$
(45d)

We note that in the case of general rank-one noise, yielding
$\mathcal{F}^{\textrm{1G}}_{{\bf r}}$ in Eq. (45b), we analytically find the
optimal $\mathsf{h}$ for the cases where one of the components of ${\bf r}$
($x$, $y$, or $z$) vanishes (see App. E). This allows us to use the structure
of the optimal $\mathsf{h}$ as an ansatz for the case of general rank-one
noise ${\mathcal{L}}^{\textrm{1G}}$. However, although the minimization of
$||\alpha^{(2)}||$ subject to the constraint $\beta^{(2)}=0$ can be done for
any fixed Lindbladian, it generally contains a large number of parameters to
be optimized. Hence, we derive a valid, but not provably tightest, analytical
bound for the case of general noise.

###  IV.3 The gain allowed by fast control

We now discuss the impact of fast control, in particular the gap in the
achievable QFI between the scheme (b) and the schemes with FFQC, i.e., (c) and
(d) (with $\delta t\to 0$ throughout this section) in Fig. 1. To do the
comparison we focus on the example of the X-Y noise
${\mathcal{L}}_{\textrm{X-Y}}$ in Eq. (13), and the rank-one transversal Pauli
noise ${\mathcal{L}}_{\textrm{{\bf x}}}^{\textrm{1P}}$ in Eq. (7) with
$\sigma_{{\bf n}}=\sigma_{1}$, which is also the limiting case of X-Y noise
for $p=1$. For X-Y noise the master equation (5) can be analytically solved,
hence the parallel CE bound Eq. (25) allows one to upper-bound the QFI of the
parallel scheme $\mathcal{F}^{\textrm{(b)}}(t)$ (see App. D).

We begin with the rank-one transversal Pauli noise
${\mathcal{L}}_{\textrm{{\bf
x}}}^{\textrm{1P}}\\!=\\!{\mathcal{L}}_{\textrm{X-Y}}|_{p=1}$. From Obs. 1 and
from previous results Dür  _et al._ (2014); Kessler  _et al._ (2014); Arrad
_et al._ (2014); Ozeri (2013) we know that this noise can be completely
removed by FFQC, without harming the evolution. Consequently, the strategies
with FFQC attain the QFI given by
$\mathcal{F}^{\textrm{(d)}}(t^{\prime})\\!=\\!(t^{\prime})^{2}$ and
$\mathcal{F}^{\textrm{(c)}}(t)\\!=\\!(Nt)^{2}$ respectively. Remarkably, in
order to attain such QFI within (d) one only requires one qubit and one
ancilla in parallel: this is enough to implement the FFQC strategy of Sec.
IV.1 and get rid of the noise, and it is known that in the noiseless case a
sequential single qubit strategy is optimal Giovannetti  _et al._ (2006). For
the parallel strategy (b) the CE bound reads
$\mathcal{F}^{\textrm{(b)}}\\!\leq\\!4N\alpha_{\omega,\gamma,p=1}^{\textrm{(b)}}(t)$
(see App. D). We plot $4\,\alpha_{\omega,\gamma,p}^{\textrm{(b)}}$ for
$p\\!=\\!\omega\\!=\\!\gamma\\!=\\!1$ in Fig. 2 (thin solid line). This
clearly shows the advantage offered by FFQC: for any fixed $t$ in strategy (b)
the attainable $\mathcal{F}^{\textrm{(b)}}$ is bound to a linear scaling in
$N$. Moreover, for any $\gamma$ and $\omega$ as $t$ increases the bound
eventually starts to decrease (actually it has been shown that
$\mathcal{F}^{\textrm{(b)}}(t)/{t}$ is maximized for $t\propto
1/N^{\frac{1}{3}}$ Chaves  _et al._ (2013)). Interestingly, the bound
$\alpha_{\omega,\gamma,0}$ diverges when $t$ approaches zero. This is to be
expected, as in this limit a super-linear scaling in $N$ is known to be
possible already within scenario (a) Chaves  _et al._ (2013); Brask  _et al._
(2015).

Now let us turn to X-Y noise ${\mathcal{L}}_{\textrm{X-Y}}$ (with
$0\\!<\\!p\\!<\\!1$). For the parallel strategy (b), the same bound
$\mathcal{F}^{\textrm{(b)}}\leq 4N\alpha_{\omega,\gamma,p}^{\textrm{(b)}}(t)$
applies and the function $\alpha_{\omega,\gamma,p}^{\textrm{(b)}}(t)$ is now
well-behaved (it starts at zero for $t=0$). We plot
$\alpha_{\omega,\gamma,p}^{\textrm{(b)}}(t)$ for $\omega\\!=\\!\gamma\\!=\\!1$
and $p=0.1$ in Fig. 2 (thick solid line). The situation for FFQC schemes is a
bit more subtle, and allows to nicely illustrate different aspects in which
fast control is helpful. First there is the ultimate bound of Eq. (45c) which
holds for any FFQC strategy and reads
$\mathcal{F}^{\textrm{(d)}}_{\textrm{X-Y}}\leq\frac{t^{\prime}}{2\gamma
p(1-p)}$ ($\mathcal{F}^{\textrm{(c)}}_{\textrm{X-Y}}\leq\frac{Nt}{2\gamma
p(1-p)}$ for scheme (c)) but might not be attainable (dashed line in Fig. 2).

Now let us consider a particular FFQC strategy, which we shall consider in
more details in Sec. VI.1.2. Using the EC code of Sec. IV.1 one can detect if
an error happens at a given time step. However, as there is no way to tell if
the error corresponds to a $\sigma_{1}$ or $\sigma_{2}$ the noise cannot be
fully corrected. Still, one can chose to correct the most probable error term,
say $p\leq 1/2$ and it is $\sigma_{2}$. Hence, if the guess was right the
error is corrected $\sigma_{2}\sigma_{2}=\mathbbm{1}$, while the other case
leads to a z-error $\sigma_{2}\sigma_{1}=-\mathrm{i}\sigma_{3}$. This strategy
modifies the X-Y noise process to an announced dephasing noise (one knows how
many correction steps were performed), for which the same bound Eq. (45d)
holds as all our control operations are unitaries.

Next, consider a suboptimal variant of the above scheme (we will show that
this is strictly suboptimal in Sec. VI) where we forget the error register,
i.e., the knowledge about the number of correction steps. In this case the
resulting effective noise process is the usual dephasing (${\mathcal{L}}_{{\bf
z}}^{\textrm{1P}}$ of Eq. (7)) with strength $\gamma_{{\bf z}}=p\,\gamma$, and
the bounds $\mathcal{F}^{\textrm{(d)}}\leq\frac{t^{\prime}}{2\gamma p}$ and
$\mathcal{F}^{\textrm{(c)}}\leq\frac{tN}{2\gamma p}$ of Eq. (45a) apply
(dotted line in Fig. 2). Notice that these bounds are attainable; it is known
that the CE bound for dephasing may be asymptotically achieved already within
strategy (a) when considering the limit of short dynamics $t\to 0$ and large
number of probes $N\to\infty$ Ulam-Orgikh and Kitagawa (2001); Knysh  _et al._
(2011, 2014). Crucially, when considering the more powerful strategies (c) and
(d), such a regime can always be mimicked by employing an unbounded number of
ancillary qubits. These are then continuously swapped with the sensing probes
and measured so that the required protocol of type (a) is indeed recovered.

A natural question to ask then is what happens if we keep the number of
ancillary qubits constant, and equal to $N$ in (c) or (d). In this case, the
FFQC strategy described above allows one to mimic the scheme (a) with
dephasing noise ${\mathcal{L}}_{{\bf z}}^{\textrm{1P}}$ of strength
$\gamma_{{\bf z}}=p\,\gamma$ and a fixed number of probes $N$. For such an
effective noise-type, the parallel CE bound is known to yield
$\mathcal{F}^{\textrm{(a)}}(t)\leq N\frac{t^{2}\eta^{2}}{(1-\eta^{2})}$ with
$\eta=\mathrm{e}^{-p\gamma t}$ Kołodyński and Demkowicz-Dobrzański (2013),
which is attainable when $N$ is large (dot-dashed line in Fig. 2). Hence, it
is also attainable by strategy (c) with the number of ancillary qubits equal
the number of probes $N$, or by strategy (d) with $N+1$ qubits in total (only
one qubit is needed as an ancilla to run the EC protocol, while the others can
be used for consecutive preparation of the required initial entangled states).

In summary, we see that FFCQ offers several advantages. It allows to modify
the noise process (dot-dashed line in Fig. 2), it offers the possibility to
prolong the short-time dynamics by “parallelizing the evolution” (dotted line
in Fig. 2), and finally it also gives the possibility to keep the error
register (dashed line in Fig. 2). We explore the latter in more depth in Sec.
VI). In any case, it undoubtedly outperforms the parallel scheme (b) (solid
line in Fig. 2) by large amount.

We showed how the introduction of FFQC leads to an improved scaling of the
QFI. Yet an interesting question is if there exists a gap between the two
strategies (c) and (d). If the frequency with which one applies the controls
in (c) limited, i.e. the duration of each step $\delta t$ is fixed, it is easy
to see that there exists a gap. The bound derived in the App. D also applies
to this case and yields $\mathcal{F}^{\textrm{(c)}}\leq
4Nm\,\alpha^{\textrm{(b)}}_{\omega,\gamma,p}(\delta t)$, and we just
demonstrated that it can be outperformed by FFQC. However, it is not clear if
there is a gap between (c) and (d) when $\delta t\to 0$ and $t=t^{\prime}/N$.
Furthermore, it is not clear these strategies outperform scheme (b) for $t\to
0$ with $N=\frac{t^{\prime}}{t}$, as all the CE bounds coincide in this limit.
On the other hand, the fact that the CE bounds coincide does not disprove the
existence of a gap, as none of the bounds is necessarily tight. Finally, we
note that protocol (d) has following important advantage: one can read the
error syndrome at any time-step before deciding on the optimal strategy for
the future, whereas for protocols (b) and (c) some decisions have to be made
beforehand. We will come back to this particular point, and show for a simple
phase estimation example how this difference allows to boost the attainable
QFI in Sec. VI.2.

Figure 2: X-Y noise: Scaling of the QFI as function of $t$ for various
protocols and $\omega=1$, $\gamma=0.1$ and $p=0.1$. The solid line corresponds
to the CE bound on $\mathcal{F}^{\textrm{(b)}}(t)/N$ for strategy (b). The
dashed straight is the ultimate upper-bound $\frac{t}{2\gamma p(1-p)}$ Eq.
(45c). The dotted straight line corresponds to the attainable bound for the
particular FFQC strategy where one corrects the dominant noise but discards
the error record, it corresponds to $\frac{t}{2\gamma p}$ from Eq. (45a). The
dot-dashed line is the attainable bound for the same strategy but with fixed
number of ancillae equal to $N$ and no intermediate measurements. X-noise:
Finally, the thin line is the CE bound on $\mathcal{F}^{\textrm{(b)}}(t)/N$
for the X-noise with $p\\!=\\!\gamma\\!=\\!\omega\\!=\\!1$, note that it
diverges as $t\to 0$ since strategy (b) allows to super-classical scaling in
$N$ for vanishing $t$. The inset is a zoom on the region close to the origin.

##  V Implications on attainable precision in metrology schemes

Hitherto our analysis focused on the scaling of the QFI evaluated on the final
system state of the metrology protocols depicted in Fig. 1 in presence of
general time-homogeneous Lindbladian noise. We now discuss the implications of
FFQC on the precision with which one can estimate the parameter $\omega$. As
already mentioned in Sec. III.1, how to best quantify the estimation precision
depends crucially on how information about $\omega$ is obtained. In Sec. V.1
we consider the implications of FFQC on precision for the case of
asymptotically many repetitions, whereas Sec. V.2 deals with single-shot
estimation. Regardless of how one chooses to quantify the precision, whenever
a noiseless estimation scenario exhibits a quantum improvement in precision
scaling then this improvement may always be maintained in the presence of
rank-one Pauli noise by employing the FFQC protocol described in Sec. IV.1. We
note again that the ability of our scheme to ameliorate for any rank-one Pauli
noise finds applications beyond metrology, i.e., in the design of high
fidelity gates, or indeed any other protocol where such noise terms may
appear.

The achievable precision in estimating $\omega$ in any metrological protocol
is dictated by the resources at hand. In what follows we shall consider two
different ways of quantifying the resources of any metrological protocol:
time-particles and number of probes. In the _time-particles_ approach the
total resource _per experimental run_ is defined as the product between the
number of probes $N$ and the protocol duration $t$. Thus, for all the types of
protocols depicted Fig. 1 the time-particle resource is equal to
$t^{\prime}=Nt$. In the _number of probes_ approach one is primarily
interested in how estimation precision scales with $N$ only. In particular,
this is how resources are defined in _frequency estimation_ Huelga  _et al._
(1997); Escher  _et al._ (2011); Kołodyński and Demkowicz-Dobrzański (2013);
Chaves  _et al._ (2013), where the time for each experimental run, $t$, is
bounded from above so that one can consistently consider the limit of many
experimental repetitions by letting the overall experiment last sufficiently
long. In all cases we show how our results yield noise dependent upper bounds
on the attainable precision and restrict it to a classical scaling in terms of
the resource.

###  V.1 Precision in the presence of free repetitions

We now consider precision bounds for estimating parameter $\omega$ using FFQC
for the case where the protocol is repeated asymptotically many times so that
CRB is applicable. We first consider the time-particles picture before moving
onto the scenario in which the number of probes $N$ is the resource.

####  V.1.1 Time-particles.

Given $\nu$ repetitions of the protocol and using the CRB, we bound the
ultimate MSE attainable by the most general FFQC strategy in the presence of
rank-one Pauli noise as

$\displaystyle\delta\omega^{2}\,\nu$ $\displaystyle\geq$
$\displaystyle\frac{1}{(1-({\bf n}^{T}{\bf
z})^{2})}\,\frac{1}{{t^{\prime}}^{2}}.$ (46)

On the other hand, stemming from Eq. (44) the MSE for all other noise
processes may be lower-bounded as follows

$\displaystyle\delta\omega^{2}\,\nu$ $\displaystyle\geq$
$\displaystyle\frac{1}{4\alpha_{\mathcal{L}}}\,\frac{1}{t^{\prime}}.$ (47)

Yet, in contrast to Eq. (46), Eq. (47) is not guaranteed to be achievable in
the limit $\nu\to\infty$.

####  V.1.2 Frequency estimation.

We now turn to the setting of frequency estimation in which the resource of
interest is the number of probes $N$. Notice that the FFQC protocol in this
case is the one depicted in Fig. 1(c). Yet, its QFI can always be upper-
bounded by that of scheme (d) after setting $t^{\prime}\\!=\\!Nt$. In order to
ensure a large number of repetitions we fix the time of a single run $t$ and
demand that the total time $T\\!:=\\!\nu t\gg t$. Hence, we can again use the
CRBs of Eqs. (46) and (47) and obtain the precision bounds:

$\displaystyle\delta\omega^{2}\,T$ $\displaystyle\geq$
$\displaystyle\frac{1}{(1-({\bf n}^{T}{\bf z})^{2})}\frac{1}{N^{2}t},$ (48)
$\displaystyle\delta\omega^{2}\,T$ $\displaystyle\geq$
$\displaystyle\frac{1}{4\alpha_{\mathcal{L}}}\frac{1}{N}.$ (49)

Indeed, Eq. (48) proves that HS is achievable for rank-one Pauli noises. On
the other hand, for all other noise processes, thanks to Eq. (44) being linear
in $t^{\prime}$, the bound of Eq. (49) is independent of the single-run
duration $t$. This suggests that FFQC allows us to indefinitely maintain the
ultimate precision normally exhibited only at very short time scales (see Sec.
IV.3).

###  V.2 Single-shot precision bounds

We now consider precision bounds applicable in the single-shot scenario, i.e.,
when $\nu\\!=\\!1$. As a result we are forced to consider the AvMSE as the
figure of merit, where the average is with respect to the probability
distribution $p_{0}(\omega)$ describing our prior knowledge about the
parameter. The AvMSE can then be lower-bounded with the help of BCRB and ZZB
(see Sec. III.1). However, note that the FFQC-assisted sequential strategies
of Fig. 1(d) with sufficiently large $t^{\prime}$ also incorporate all many
repetition protocols. This is because one can always set the intermediate
control gates to implement measurements and state re-preparations. Hence, we
expect all the single-shot bounds to reproduce the ones derived in the regime
of many repetitions when considering $t^{\prime}\to\infty$.

####  V.2.1 Pauli rank-one noise.

As by repeating we may only improve the precision, whenever the precision is
forced to follow the SS in the many-repetitions regime, it must be SS-bounded
in case of the single-shot estimation. Thus, it is most important to ask
whether the HS can still be attained in presence of rank-one Pauli noise
within strategies (c) and (d) with only a single run. In particular any
protocol that achieves the maximum attainable precision in the absence of
noise can maintain this precision in the presence of rank-one Pauli noise by
utilizing FFQC. For example, by incorporating FFQC in the protocol in
Sergeevich  _et al._ (2011) one can maintain its optimal performance for any
rank-one Pauli noise process; the number of measurements $N$ in Sergeevich
_et al._ (2011) correspond to the control-steps $k$ in our protocol (d). In
App. G, we provide an explicit strategy which for a constant prior constant
over a finite interval, and zero elsewhere, leads to the AvMSE satisfying

$\displaystyle\langle\delta\omega^{2}\rangle_{\textrm{FFQC}}\leq\frac{165}{1-({\bf
n}^{T}{\bf z})^{2}}\,\frac{1}{t^{\prime 2}}$ (50)

for (d) and (c) (after substituting $t^{\prime}=Nt$). This shows that, at
least for this prior, quadratic scaling of precision in the _total resource_
${t^{\prime}}^{2}$ (or $(Nt)^{2}$) is attainable by a single shot strategy. In
the context of frequency estimation of Sec. V.1, this demonstrates that if one
can decide whether to divide the total time $T$ in many runs of a fixed
duration $t$ or perform a single shot strategy with FFQC, one should choose
the latter as it achieves precision that scales quadratically
$\langle\delta\omega^{2}\rangle\sim 1/(NT)^{2}$ in $N$ and $T$, and not only
in $N$, as in Eq. (48).

####  V.2.2 Bayesian Cramér-Rao bound.

The BCRB is defined as Gill and Levit (1995)

$\langle\delta\omega^{2}\rangle\geq\frac{1}{\langle\mathcal{F}_{\mathcal{L}}\rangle+F\big{(}p_{0}(\omega)\big{)}},$
(51)

where $F\big{(}p_{0}(\omega)\big{)}\\!=\\!\langle(\partial_{\omega}\log
p_{0}(\omega))^{2}\rangle$ is the classical Fisher information of the prior
distribution and $\langle\dots\rangle$ denotes averaging with respect to the
prior distribution $p_{0}(\omega)$. Hence, using Eq. (44), we obtain the
following lower bound on the AvMSE :

$\langle\delta\omega^{2}\rangle\geq\frac{1}{4\alpha_{\mathcal{L}}t^{\prime}+F\big{(}p_{0}(\omega)\big{)}}.$
(52)

Notice that if $F\big{(}p_{0}(\omega)\big{)}$ is finite, by substituting
$t^{\prime}\\!\to\\!\nu t^{\prime}$ above we adequately recover the many-
repetition bound of Eq. (47).

####  V.2.3 Ziv-Zakai bound,

A more general bound that also allows for irregular priors is the ZZB
introduced in Tsang (2012). In App. D we generalize the bound of Tsang (2012)
so that it can be applied also to mixed states. As a result we can lower bound
the AvMSE for large enough $t^{\prime}$ by

$\displaystyle\langle\delta\omega^{2}\rangle\geq\frac{1}{12\alpha_{\mathcal{L}}t^{\prime}}$
(53)

for any prior that fulfills some very mild regularity conditions (see App. D).
Again, by substituting $t^{\prime}\to\nu t^{\prime}$ we adequately recover the
many-repetition bound of Eq. (47).

##  VI Importance of quantum control for metrology with limited resources

###  VI.1 Frequency estimation with X-Y noise

We study the setting of _frequency estimation_ in which the probes sensing the
parameter are also affected by the _X-Y noise_ specified in Eq. (13). In
accordance to Eq. (45d, 49) the attainable precision for this noise is
asymptotically constrained to a constant improvement over SS, even when
considering the most powerful FFQC scheme of Fig. 1(d) (with the exception of
the limiting cases $p\\!=\\!0$ or $p\\!=\\!1$, where one of the components
vanishes and one recovers the perpendicular dephasing Chaves  _et al._ (2013);
Dür  _et al._ (2014); Kessler  _et al._ (2014); Arrad  _et al._ (2014); Ozeri
(2013) which is an instance of correctable rank-one Pauli noise described by
Eq. (48)). Nevertheless, we showed in Sec. IV.3, that also in this case
strategies with FFQC are better then strategies without, as they give a bigger
improvement factor. In this section we strengthen this point by demonstrating
how the most general parallel scheme of Fig. 1(b) with $N$ qubits and $N$
ancillae is outperformed by a simple and experimentally tangible strategy,
which only requires entanglement between one sensing and one ancillary qubit
but employs fast control.

Figure 3: (b): The most general parallel protocol of Fig. 1(b) that requires
to manipulate entangled states of $N$ probe and $N$ ancillary qubits. (s1): A
control-assisted sequential protocol that only involves one probe and one
ancillary qubit. (sN): A control-assisted sequential protocol that employs $N$
independent probe-plus-ancilla entangled pairs running in parallel.

More precisely, we compare the most general parallel strategy Fig. 3(b) with
simple sequential strategies Fig. 3(s1) and Fig. 3(sN), where one is only
allowed to entangle probes and ancillae two by two rather then manipulate a
global entangled state as required by (b). The two strategies (s1) and (sN)
are actually the same, but correspond to two different ways to treat the total
resources. In (s1) the experimentalist only manipulates one qubit and one
ancilla and does the experiment for a total duration $T$. This protocol has
the same duration than (b) but uses $N$ times less “calls of the master
equation”, i.e one effectively applies the single-qubit evolution for a time
that is $N$ times shorter as compared to (b). The scheme (sN) is equivalent to
(s1) upon replacing $T$ with $NT$, such that it uses the same amount of time-
particles as (b). Equivalently (sN) corresponds to running $N$ protocols (s1)
simultaneously. In this respect it has the same number of qubits and the same
time duration as (b). However, in contrast to (b), the scheme (sN) is granted
with fast control on the one hand, but only requires two-qubit entanglement on
the other. We demonstrate that, even with such restrictions, for any fixed $N$
both strategies (sN) and (s1) outperform any parallel schemes (b) given that
the asymmetry in the X-Y noise is high enough.

Our results show that the use of control and error correction techniques allow
one to attain resolutions with a two-qubit probe-plus-ancillae setup that are
better than ones reached when considering systems containing large-scale
entanglement. We believe that our results may support current state-of-art
quantum metrology experiments with NV-centres Unden  _et al._ (2016). In
particular, as the dominant noise is such systems has been argued to be nearly
transversal Kessler  _et al._ (2014); Arrad  _et al._ (2014) and, hence,
highly asymmetric Brask  _et al._ (2015), our work proves that such systems
are indeed capable of attaining resolutions unreachable by the entanglement-
based schemes employing comparable resources.

####  VI.1.1 Parallel strategy with N qubits and N ancillae

A general parallel frequency estimation scenario is depicted in Fig. 1(b) and
Fig. 3(b). For a regular prior knowledge and sufficiently large $T$ (see Sec.
V.2.2), or with the explicit assumption of sufficiently many repetitions
($T/t\\!\gg\\!1$), the ultimate attainable precision is determined by the CRB:

$T\delta^{2}\omega\;\geq\;\min_{t}\frac{t}{\mathcal{F}\Big{(}\varrho_{N}^{\textrm{(b)}}(t)\Big{)}}.$
(54)

The right hand side in Eq. (54) is the inverse of the _maximal QFI rate_ for a
given scheme of type (b) employing $N$ sensing qubits. It is the optimized for
the single-run duration that we dub $t_{\textrm{opt}}^{\textrm{(b)}}$. The
optimal time in general depends on the system size (the probe number $N$) and
the form of the noise (the Lindbladian in Eq. (6)) 666Note that in the absence
of noise it is optimal to set $t\\!=\\!T$ as large as possible, which makes
the CRB (54) not applicable.. Following the methods described in Sec. IV.2.1
(see also App. D), we can upper-bound the maximal QFI rate by employing the
parallel CE bound (25):

$\displaystyle\frac{\mathcal{F}\Big{(}\varrho_{N}^{\textrm{(b)}}(t_{\textrm{opt}}^{\textrm{(b)}})\Big{)}}{t_{\textrm{opt}}^{\textrm{(b)}}}\;\leq\;N\,\mathfrak{f}^{\textrm{(b)}\uparrow}_{N},\qquad\textrm{where}\qquad$
$\displaystyle\mathfrak{f}^{\textrm{(b)}\uparrow}_{N}:=4\max_{t}\,\min_{\mathsf{h}(t)}\frac{||\alpha(t)||+(N-1)||\beta(t)||^{2}}{t}$
(55)

is the so-obtained upper bound on the _maximal QFI rate per probe_. We
adequately label then as $t_{\textrm{opt}}^{\textrm{(b)}\uparrow}$ the optimal
$t$ maximising Eq. (55). As a result, we may generally lower-bound the
precision dictated by the CRB (54) as follows:

$\text{(b):}\qquad
T\delta^{2}\omega\geq\frac{1}{N\mathfrak{f}^{\textrm{(b)}\uparrow}_{N}},$ (56)

at the price of the saturability that cannot be guaranteed any more, even in
the $T/t_{\textrm{opt}}^{\textrm{(b)}\uparrow}\\!\gg\\!1$ limit.

In order to perform the minimization in (55), we resort to the SDP-based
methods of Ref. Kołodyński and Demkowicz-Dobrzański (2013) reviewed in Sec.
IV.2.2. To this end we solve the master Eq. (13) for the X-Y noise, and
compute the Kraus representation of the resulting channel,
${\mathcal{E}}_{\omega,t}^{\textrm{X-Y}}$, given in App. D. For any fixed $N$
and $t$ this allows us to perform the minimization over the Hamiltonian
$\mathsf{h}(t)$ numerically, while for $N\\!=\\!1$ 777For which the CE bound
(55) is guaranteed to be tight, see Eq. (22). and $N\\!\to\\!\infty$ we obtain
the corresponding expressions analytically for all times. Finally, to obtain
$\mathfrak{f}^{\textrm{(b)}\uparrow}_{N}$ we numerically search for the
optimal $t_{\textrm{opt}}^{\textrm{(b)}\uparrow}$ in Eq. (55). The detailed
analysis on the solution of the master equation and derivation of the bounds
may be found in App. D. Note that the balanced X-Y noise ($p\\!=\\!1/2$) is
phase-covariant, i.e., it commutes with the Hamiltonian, so that for this
particular case the bound (55) can be obtained analytically Smirne  _et al._
(2016).

####  VI.1.2 Simple sequential strategy

For the sequential strategy, all the probe plus ancilla pairs behave
independently, and since the QFI is additive in this case in order to
establish the global performance it is sufficient to consider a single pair up
to the first measurement in Fig. 3(s). We denote the state of a probe-plus-
ancilla at any time by
$\varrho(t)\\!\in\\!{\mathcal{B}}(\mathds{C}^{2}\\!\otimes\\!\mathds{C}^{2})$.

The maximal QFI rate per probe then reads

$\displaystyle\mathfrak{f}^{\textrm{(s)}}\\!:=\\!\max_{t}\frac{\mathcal{F}(\varrho(t))}{t}=\frac{\mathcal{F}(\varrho(t_{\textrm{opt}}^{\textrm{(s)}}))}{t_{\textrm{opt}}^{\textrm{(s)}}}$
(57)

with $t_{\textrm{opt}}^{\textrm{(s)}}$ being now determined solely by the
noise.

Let us carefully explain the sequential FFQC that we already sketched in Sec.
IV.3, and which is inspired by the strategy of Sec. IV.1 for perpendicular
dephasing Kessler  _et al._ (2014). Recall that neglecting the higher orders
in $\mathrm{d}t$ the probe-plus-ancilla state after an evolution of
infinitesimal duration can follow two possible branches, see Eq. (IV.1):

$\displaystyle\varrho(t+\mathrm{d}t)=\varrho_{\textrm{C}}(t+\mathrm{d}t)+\varrho_{\textrm{E}}(t+\mathrm{d}t).$
(58)

Either no error happened
$\varrho_{\textrm{C}}(t+\mathrm{d}t)=(1-\frac{\gamma}{2}\mathrm{d}t)\varrho(t)-\mathrm{i}\,\frac{\omega}{2}\mathrm{d}t[\sigma_{3}\otimes\mathbbm{1},\varrho(t)]$
and the parameter was imprinted on the state; or with probability
$\tfrac{\gamma}{2}\mathrm{d}t$ an error occurred
$\varrho_{\textrm{E}}(t+\mathrm{d}t)=\frac{\gamma}{2}\mathrm{d}t\big{(}p\,(\sigma_{1}\otimes\mathbbm{1})\varrho(t)(\sigma_{1}\otimes\mathbbm{1})+(1-p)(\sigma_{2}\otimes\mathbbm{1})\varrho(t)(\sigma_{2}\otimes\mathbbm{1})\big{)}$.
If in addition the state at time $t$ belongs to the _code space_
$\varrho\in{{\mathcal{B}}(\mathcal{H}_{\textrm{C}})}$ with
$\mathcal{H}_{\textrm{C}}=\textrm{span}(\left|00\right\rangle,\left|11\right\rangle)$
and
$\mathcal{H}_{\textrm{E}}=\textrm{span}(\left|01\right\rangle,\left|10\right\rangle)$,
the two branches of the infinitesimal evolution can be distinguished in a non-
demolition manner by projecting onto the code the error spaces via a parity
measurement (since
$\varrho_{\textrm{C}}(t+\mathrm{d}t)\in{\mathcal{B}}(\mathcal{H}_{\textrm{C}})$
and
$\varrho_{\textrm{E}}(t+\mathrm{d}t)\in{\mathcal{B}}(\mathcal{H}_{\textrm{E}})$).
If the state happens to be in the correct branch we simply leave it as it is.
While if an error is detected our strategy consists of mapping the error
branch back into the code space by correcting for the most probable error term
that we assume to be $\sigma_{2}\otimes\mathbbm{1}$ ($0\\!<\\!p\\!\leq\\!1/2$)
without loss of generality. Thus, after the correction step the error branch
reads
$\bar{\varrho}_{E}(t+\mathrm{d}t)=(\sigma_{2}\otimes\mathbbm{1})\varrho_{\textrm{E}}(t+\mathrm{d}t)(\sigma_{2}\otimes\mathbbm{1})$
with

$\displaystyle\bar{\varrho}_{\textrm{E}}(t+\mathrm{d}t)$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2}\mathrm{d}t\big{(}(1-p)\varrho(t)+p(\sigma_{3}\otimes\mathbbm{1})\varrho(t)(\sigma_{3}\otimes\mathbbm{1})\big{)}+$
(59) $\displaystyle+\;O(\mathrm{d}t^{2}).$

Note that here we assume that the control steps are performed fast enough, so
that Eq. (59) holds up to first order in $\mathrm{d}t$ 888Note that the
control steps need not be ultra fast, just sufficiently fast so that the first
order approximation holds. the duration of these control-pulses depends on the
strength of the decoherence process.. Importantly, after the correction the
final state belongs to the code space again
$\bar{\varrho}(t+\mathrm{d}t)\in{\mathcal{B}}(\mathcal{H}_{\textrm{C}})$, and
we can continue running the strategy. Another important point is that, in
contrast to the strategy in Sec. IV.3, we keep track of the number of errors
that have been detected before a time $t$ in a classical error register.

The total number of errors $m$ detected during a time $t$ follows the
Poissonian distribution $p(m;t)\\!=\\!\mathrm{e}^{-\gamma t/2}(\gamma
t/2)^{m}/m!$. And in the basis
$\\{\left|00\right\rangle,\left|11\right\rangle\\}$ the state at time $t$
conditional to the detection of $m$ errors reads

$\bar{\varrho}_{m}(t)\;:=\;\frac{1}{2}\left(\begin{array}[]{cc}1&(1-2p)^{m}\mathrm{e}^{\mathrm{i}\omega
t}\\\ (1-2p)^{m}\mathrm{e}^{-\mathrm{i}\omega t}&1\end{array}\right).$ (60)

Since one keeps track of the number of errors the overall state at time $t$
possesses a direct-sum structure
$\bar{\varrho}(t)\\!=\\!\bigoplus_{m=0}^{\infty}p(m;t)\bar{\varrho}_{m}(t)$.
Hence, the corresponding QFI may be straightforwardly evaluated:

$\mathcal{F}(\bar{\varrho}(t))=\sum_{m}p(m;t)\mathcal{F}\big{(}\bar{\varrho}_{m}(t)\big{)}=t^{2}\,\mathrm{e}^{-\gamma
t\,2p(1-p)},$ (61)

and yields the following maximal QFI rate per probe in Eq. (57)

$\mathfrak{f}^{\textrm{(s)}}=\frac{1}{2\gamma\mathrm{e}}\frac{1}{p(1-p)},$
(62)

which is attained after optimally setting
${t_{\textrm{opt}}^{\textrm{(s)}}}\\!=\\!1/(2\gamma p(1-p))$ 999A careful
reader notices that Eq. (61) is symmetric in the exchange of $p$ with $1-p$.
Indeed, it turns out that if the error register is kept it does not matter for
which error one corrects in the error branch, as the finals states
$\bar{\varrho}_{m}(t)$ for the two cases are related by a unitary
transformation..

In the regime of long experimental duration
$T/t_{\textrm{opt}}^{\textrm{(s)}}\\!\gg\\!1$, which we are interested in, the
CRB (54) for the two strategies Fig. 4(s1) and Fig. 4(sN) read:

(s1): $\displaystyle\qquad
T\delta^{2}\omega\geq\frac{1}{\mathfrak{f}^{\textrm{(s)}}}$ (63) (sN):
$\displaystyle\qquad
T\delta^{2}\omega\geq\frac{1}{N\mathfrak{f}^{\textrm{(s)}}},$ (64)

and are guaranteed to be attainable.

Before we proceed to the comparison of strategies (s) and (b), let us briefly
comment on the role of the error register, and how it allows to boost the
attainable performance. As we argued in Sec. IV.3 discarding the error
register is equivalent to effectively modifying the X-Y noise to a dephasing
noise of strength $p\,\gamma$. Concretely, if the error register is discarded,
the state at time $t$ is a mere mixture
$\bar{\varrho}(t)\\!=\\!\sum_{m=0}^{\infty}p(m;t)\bar{\varrho}_{m}(t)$ rather
than a direct sum as stated below the Eq. (60). The corresponding maximal QFI
rate per probe is then given by
$\mathfrak{f}^{\textrm{(s)}}\\!=\\!1/(2\gamma\mathrm{e}p)$, which is reduced
by a factor $1/(1-p)$ as compared to the strategy including the error register
and described by Eq. (62). The intuition behind this gap is rather simple.
When inferring the value of $\omega$ after a certain number of runs it is
helpful to know how noisy each run was. This information, contained then by
the error register, allows to properly ponder the data obtained in each run in
order to optimally construct the global estimator.

####  VI.1.3 Comparison of the strategies

In Fig. 4 we explicitly present the maximal QFI rate per probe attained by the
sequential strategy $\mathfrak{f}^{\textrm{(s)}}$ of Eq. (62), and compare it
to the bound limiting the performance of any parallel scheme
$\mathfrak{f}^{\textrm{(b)}\uparrow}_{N}$ of Eq. (55), which we plot as a
function of the number of probes $N$. This corresponds to a comparison of
strategies (sN) and (b) described in Fig. 3. In particular, after fixing
$\omega\\!=\\!1$, $\gamma\\!=\\!0.05$, we show that when the noise generated
by $\sigma_{2}$ is three times more dominant than the one generated by
$\sigma_{1}$, i.e., $p\\!=\\!1/4$, the sequential strategy is guaranteed to be
superior for $N$ less than $5$. However, with the increase of noise asymmetry
the superiority of the sequential strategy drastically improves, so that for
$p\\!=\\!0.05$ only if the probe number $N$ is greater than $316$ the parallel
schemes (b) can potentially beat the sequential one (sN). In this case, in
order to also compare (b) to the strategy (s1), which employs only a _single_
(rather than $N$) probe-plus-ancilla qubit pair for the same duration $T$, we
also plot $\mathfrak{f}^{\textrm{(s)}}/N$. Strikingly, the parallel strategies
(b) are still outperformed by (s1) as long as $N\\!<\\!4$.

Figure 4: Maximal QFI rate per probe $\mathfrak{f}^{\textrm{(s)}}$ attainable
by the sequential strategies (_dashed lines_) of Fig. 3(s1) and (sN), as
compared to the upper bound, $\mathfrak{f}^{\textrm{(b)}\uparrow}_{N}$ (_solid
curves_), valid for all parallel protocols of Fig. 3(b) in presence of X-Y
noise ($\omega\\!=\\!1$, $\gamma\\!=\\!0.05$). For $p\\!=\\!0.25$ (_black_)
the curves indicate the sequential strategy (sN) to be superior for $N\\!<5$,
whereas for $p\\!=\\!0.05$ (_red_) up to $N\\!<\\!316$ (while (s1) is superior
up to $N\\!<\\!4$— _dotted red curve_). The inset depicts dependence on the
probe number of $t_{\textrm{opt}}^{\textrm{(b)}\uparrow}$ that maximizes Eq.
(55). Despite not being smooth at finite $N$,
$t_{\textrm{opt}}^{\textrm{(b)}\uparrow}$ asymptotically follows $1/N^{1/3}$
scaling (_dashed line_). Figure 5: Threshold numbers of probes,
$N_{\textrm{th}}^{\textrm{(s1)}}$ and $N_{\textrm{th}}^{\textrm{(sN)}}$, below
which the two sequential strategies (s1) and (sN) are superior to all parallel
protocols (b) in Fig. 3. as a function of the X-Y noise asymmetry $p$ for
$\omega\\!=\\!1$ and $\gamma\\!=\\!\\{0.1,0.2,0.3\\}$ (_top to bottom_). _Red
lines_ are plotted assuming sequential protocol performance to be evaluated
per probe (sN), while _blue lines_ for the sequential protocol employing only
a _single_ probe-plus-ancilla pair (s1). In the former case,
$N_{\textrm{th}}^{\textrm{(sN)}}\\!\geq\\!2$ with equality always occurring
strictly at $p\\!=\\!1/2$, while in the single-pair scenario there exists a
range of $0.13\lessapprox\\!p\\!\leq 0.5$ for which trivially
$N_{\textrm{th}}^{\textrm{(s1)}}\\!=\\!1$. In both cases, $N_{\textrm{th}}$
diverges with noise asymmetry, as $p\\!\to\\!0$, and eventually follows the
$p^{-3/2}$, $p^{-3/5}$ scalings respectively (_dashed lines_) .

To compare the strategies more directly we explicitly present in Fig. 5 the
threshold probe numbers below which the sequential strategies are guaranteed
to be superior over all parallel schemes. Concretely, to compare (sN) with (b)
we plot $N_{\textrm{th}}^{\textrm{(sN)}}$ such that
$\mathfrak{f}^{\textrm{(s)}}\\!>\\!\mathfrak{f}^{\textrm{(b)}\uparrow}_{N}$
for all $N\\!<\\!N_{\textrm{th}}^{\textrm{(sN)}}$ (the top curves), and to
compare (s1) with (b) we plot $N_{\textrm{th}}^{\textrm{(s1)}}$ such that
$\mathfrak{f}^{\textrm{(s)}}/N\\!>\\!\mathfrak{f}^{\textrm{(b)}\uparrow}_{N}$
for all $N\\!<\\!N_{\textrm{th}}^{\textrm{(s1)}}$ (the bottom curves). Note
that $N_{\textrm{th}}^{\textrm{(sN)}}\\!\geq\\!2$ for any value of $p$, as in
the worst case of balanced noise ($p\\!=\\!1/2$) one may explicitly show that
$\mathfrak{f}^{\textrm{(b)}\uparrow}_{N=2}\\!=\\!\mathfrak{f}^{\textrm{(s)}}$,
see App. D. Regarding the other threshold, although
$N_{\textrm{th}}^{\textrm{(s1)}}\\!=\\!1$ in the range
$0.13\lessapprox\\!p\\!\leq 0.5$, meaning that the sequential strategy (s1)
only outperforms parallel strategies with a single probe-plus-ancilla pair;
$N_{\textrm{th}}^{\textrm{(s1)}}$ still diverges to infinity when
$p\\!\to\\!0$ and the noise approaches the perpendicular dephasing. We observe
that the thresholds scale as $N_{\textrm{th}}^{\textrm{(sN)}}\propto p^{-3/2}$
and $N_{\textrm{th}}^{\textrm{(s1)}}\propto p^{-3/5}$. This proves that both
sequential strategies (s1) and (sN) can outperform the most general parallel
one of type (b) for any $N$, given sufficient noise asymmetry.

###  VI.2 Phase estimation with bit and bit-phase flip noise

We now return to the canonical phase estimation scenario Giovannetti  _et al._
(2006), which may yet be interpreted as the frequency estimation considered
before but with the evolution time $t$ being fixed, such that the parameter of
interest is $\theta\\!=\\!\omega t$. In this case the sensing interaction of a
single probe of duration $t$ constitutes the elementary block, drawn as a
square box in Fig. 6. Each box corresponds to a CPTP map
${\mathcal{E}}_{\theta}$ incorporating the noise Demkowicz-Dobrzański and
Maccone (2014). In this context one is free to use ancillary particles to
benefit from EC techniques, but also to combine the boxes in different ways.
The two schemes depicted in Fig. 6(i) and Fig. 6(ii) are the elementary phase
estimation equivalents of the schemes depicted in Fig. 1(b) and Fig. 1(d)
respectively.

We provide a simple example demonstrating that a sequential strategy
outperforms the parallel one. Specifically, consider the situation where each
line in Fig. 6 corresponds to a qubit and the map ${\mathcal{E}}_{\theta}$ is
a combination of the unitary evolution given by
$U_{\theta}=e^{-i\frac{\theta}{2}\sigma_{3}}$ and of the noise which consists
of a random application of either a bit flip ($\sigma_{1}$) or bit-phase flip
($\sigma_{2}$) with equal probability $\frac{1-p}{2}$. Accordingly,
${\mathcal{E}}_{\theta}$ has a Kraus representation of the form
$\\{K_{0}=\sqrt{p}\,U_{\theta},\,K_{1}=\sqrt{1-p}\,|1\rangle\langle
0|,\,K_{2}=\sqrt{1-p}\,|0\rangle\langle 1|\\}$. Note that this noise channel
corresponds to a Pauli $X-Y$ noise channel which, however, differs from the
solution of the master equation for the balanced $X-Y$ noise described in Sec.
VI.1.

Figure 6: Two different strategies for phase estimation involving two
applications of the noisy channel ${\mathcal{E}}_{\theta}$ presented in Sec.
VI.2. In strategy (i) the two probe and ancillary qubits are prepared in a
suitably entangled state and the channels are applied to the sensing systems
in parallel before they are finally measured. This strategy intrinsically
includes error-correction but only at the final step (see strategy (iii) in
Demkowicz-Dobrzański and Maccone (2014)). (ii) The sequential strategy from
Sec. VI.2, after the application of the first channel one reads out the error
syndrome and decides on the optimal strategy for the next step (see strategy
(iv) in Demkowicz-Dobrzański and Maccone (2014)).

In the parallel strategy of Fig. 6(i) one starts with a suitably entangled
state of two sensing qubits and two ancillae. The channel is applied on both
qubits in parallel, and the final state is measured (this includes error-
correction at the final stage see strategy (iii) in Demkowicz-Dobrzański and
Maccone (2014)). Without loss of generality the input two-qubit plus two-
ancilla state is given by
$\left|\psi\right\rangle=\sum_{i,j=0}^{1}a_{ij}\left|i,j\right\rangle_{Q}\left|i,j\right\rangle_{A}$
with $a_{ij}\in\mathbb{R}$. This encoding is optimal as it allows us to
distinguish between all the different branches of the noisy evolution.
Moreover, within every branch the state remains pure, as the channel
corresponds to either leaving the sensing qubit untouched—with probability
$p$—or projecting it onto the
$\\{\left|0\right\rangle,\,\left|1\right\rangle\\}$ basis (followed by an
irrelevant flip of the qubit) which can be detected by our error correction
code. As all the branches can be detected, and within every branch the state
remains pure, the QFI of the final state is given by the mean variance of
$H=\frac{1}{2}(\sigma_{3}\otimes\mathbbm{1}+\mathbbm{1}\otimes\sigma_{3})$
over all the branches. The mean variance can be straightforwardly maximized by
a brute force numerical optimization over the three dimensional manifold
spanned by the coefficients $a_{ij}$. It turns out that the optimal states are
symmetric under permutations of the two qubits ($a_{01}=a_{10}$) and,
moreover, satisfy $a_{00}=a_{11}$. Using these conditions at the outset one
can analytically determine the optimal state to be

$\displaystyle\left|\psi\right\rangle_{\mathrm{opt}}$
$\displaystyle=\sqrt{\frac{x_{\textrm{opt}}}{2}}(\left|00\right\rangle_{Q}\left|00\right\rangle_{S}+\left|11\right\rangle_{Q}\left|11\right\rangle_{S})$
(65)
$\displaystyle+\sqrt{\frac{1-x_{\textrm{opt}}}{2}}(\left|01\right\rangle_{Q}\left|01\right\rangle_{S}+\left|10\right\rangle_{Q}\left|10\right\rangle_{S}),$

with $x_{\textrm{opt}}=\min(1,\frac{2-p}{4(1-p)})$. The corresponding maximal
QFI is

$\displaystyle\mathcal{F}^{\textrm{(i)}}=\left\\{\begin{array}[]{cc}-\frac{(p-2)^{2}p}{2(p-1)}&p\leq\frac{2}{3}\\\
4p^{2}&\text{otherwise.}\end{array}\right.$ (68)

Now consider the following sequential strategy with intermediate EC involving
two sensing qubits and a single ancilla, see Figure 6(ii). The first sensing
qubit and the ancilla are prepared in the state
$\frac{1}{\sqrt{2}}\left(\left|00\right\rangle_{Q_{1}A}+\left|11\right\rangle_{Q_{1}A}\right)$.
After the application of the first box on the first sensing qubit but before
the second one we perform our error detection scheme. If an error is detected
the state contains no information about the parameter, hence one discard
$Q_{1}$ and prepare the second qubit and the ancilla in the same state
$\frac{1}{\sqrt{2}}\left(\left|00\right\rangle_{Q_{2}A}+\left|11\right\rangle_{Q_{2}A}\right)$.
If no error is detected after the first application of the channel the second
sensing qubit is introduced and entangled with the first qubit such that the
overall state is given by

$\displaystyle\sqrt{\frac{1}{2}}\left(e^{-i\frac{\theta}{2}}\left|0\right\rangle_{Q1}(\sqrt{y}\left|00\right\rangle_{Q2,A}+\sqrt{1-y}\left|11\right\rangle_{Q2,A})\right.$
$\displaystyle\left.+e^{i\frac{\theta}{2}}\left|1\right\rangle_{Q1}(\sqrt{y}\left|11\right\rangle_{Q2,A}+\sqrt{1-y}\left|00\right\rangle_{Q2,A})\right).$
(69)

One easily checks that the state that optimizes the final QFI (after the
application of the second box) is the one with
$y=y_{opt}=\min(1,\frac{1}{2(1-p)})$. Finally the overall QFI of this scheme
is given by

$\mathcal{F}^{\textrm{(ii)}}=\left\\{\begin{array}[]{cc}p(3p+1)&p\geq\frac{1}{2}\\\
\frac{p\left(p^{2}-2p+2\right)}{1-p}&\text{otherwise.}\end{array}\right.$ (70)

In Figure 7, we plot the difference between QFI’s of the strategies (ii) in
Eq. (70) and (i) in Eq. (68). One clearly sees that the strategy with
intermediate control Fig. 6(ii) performs better than the parallel strategy
Fig. 6(i). Moreover there is a gap for all values of $p$ except the trivial
cases: the noiseless case $p=1$ and the case where no information about
$\theta$ is contained in the channel $p=0$.

Figure 7: Comparison of the QFI’s for strategies (i) and (ii). The curve
corresponds to the difference of QFI between the strategies (ii) in Eq. (70)
and (i) Eq. (68).

This simple example shows that intermediate control (FFQC) allows for an
improvement in the achievable accuracy. Notice that such an advantage can be
maintained for any finite number of blocks as one can choose to introduce new
sensing qubits, appropriately entangled with the already existing ones, only
after one has obtained information about the errors from the intermediate
error correcting steps. In the parallel strategy however, all the qubits have
to be initialized in an entangled state, which is highly fragile to subsequent
errors. However the question whether this improvement vanishes in the
asymptotic case—as conjectured in Demkowicz-Dobrzański and Maccone (2014)—or
persists, as one might suspect from the finite $N$ results shown here, remains
open.

##  VII Summary and Outlook

We have considered the general limits of quantum metrology where one is, in
principle, equipped with a full-scale quantum computer to assist in the
sensing process. We have shown that one can use techniques from quantum error
correction to detect or correct for certain kinds of errors while maintaining
the sensing capabilities of the system. In particular, we discover that the
use of FFQC can alleviate all rank-one Pauli noise process, except the rank-
one Pauli noise that is identical to the Hamiltonian, at the cost of a slowing
down the evolution by a constant factor. This allows one to maintain HS. For
all other noise processes, we have shown that the QFI is limited to a linear
scaling with the number of resources. As we demonstrated, this result forbids
the Heisenberg scaling of the estimation precision and provides noise
dependent upper bounds for all metrological scenarios, local or Bayesian.
Remarkably, considering the ultimate metrological scheme (FFQC) actually lead
to a simplification of the problem, as in the limit of short evolution steps
$\mathrm{d}t\to 0$ the CE method does not require to solve the master
equation, this allowed us to obtain fully analytic bounds on the achievable
precision for relevant noise processes.

However, even in the cases where FFQC does not completely alleviate the noise,
one can achieve a significant improvement over parallel strategies that
operate on limited resources and do not utilize FFQC. We have demonstrated
this for the example of asymmetric X-Y noise where the improvement over
parallel schemes employing the same number of resources but do not use FFQC
can be significant. Moreover, we have put forward simple sequential protocols
that make use of only a single sensing and auxiliary system, but that
nevertheless outperform parallel entanglement-based strategies for limited
number of sensing systems. From a practical perspective, the existence of
simple practical schemes that operate with only a single sensing and auxiliary
system, are of high relevance, and may open the way for practical noisy
metrology with a significant quantum advantage. This demonstrates that fast
quantum control and error correction are powerful tools in quantum metrology
and may enable practical sensing with a significant quantum enhancement even
in the presence of noise.

An important question one may ask is how our results generalize to higher
dimensional systems. Note that in higher dimensions the parameter encoding
Hamiltonian may have support only on a subspace of the Hilbert space of the
probe system, which may allow for more involved error correcting codes.

## Acknowledgements

This work has been supported by Austrian Science Fund (FWF: P24273-N16,
P28000-N27), Swiss National Science Foundation Grant (P2GEP2_151964), Spanish
Ministry National plans FOQUS and MINECO (Severo Ochoa Grant No.
SEV-2015-0522), Spanish MINECO FIS2013-40627-P, Generalitat de Catalunya CIRIT
2014 SGR 966 Fundació Privada Cellex, Generalitat de Catalunya Grant (No.
SGR875), as well as received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie Grant:
Q-METAPP (No. 655161).

## References

  * Dowling and Seshadreesan (2015) J. P. Dowling and K. P. Seshadreesan, J. Lightwave Technol. 33, 2359 (2015).
  * Giovannetti  _et al._ (2011) V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photon. 5, 222 (2011).
  * Giovannetti  _et al._ (2004) V. Giovannetti, S. Lloyd, and L. Maccone, Science 306, 1330 (2004).
  * Giovannetti  _et al._ (2006) V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006).
  * Sanders and Milburn (1995) B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944 (1995).
  * Berry and Wiseman (2000) D. W. Berry and H. M. Wiseman, Phys. Rev. Lett. 85, 5098 (2000).
  * Peres and Scudo (2001) A. Peres and P. F. Scudo, Phys. Rev. Lett. 86, 4160 (2001).
  * Bagan  _et al._ (2004) E. Bagan, M. Baig, and R. Muñoz Tapia, Phys. Rev. A 70, 030301 (2004).
  * Chiribella  _et al._ (2004a) G. Chiribella, G. M. D’Ariano, P. Perinotti, and M. F. Sacchi, Phys. Rev. Lett. 93, 180503 (2004a).
  * Chiribella  _et al._ (2004b) G. Chiribella, G. M. D’Ariano, P. Perinotti, and M. F. Sacchi, Phys. Rev. A 70, 062105 (2004b).
  * Chiribella  _et al._ (2005) G. Chiribella, G. M. D’Ariano, and M. F. Sacchi, Phys. Rev. A 72, 042338 (2005).
  * Higgins  _et al._ (2007) B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, Nature 450, 393 (2007).
  * Yurke (1986) B. Yurke, Phys. Rev. Lett. 56, 1515 (1986).
  * Huelga  _et al._ (1997) S. F. Huelga, C. Macchiavello, T. Pellizzari, A. K. Ekert, M. B. Plenio, and J. I. Cirac, Phys. Rev. Lett. 79, 3865 (1997).
  * Banaszek  _et al._ (2009) K. Banaszek, R. Demkowicz-Dobrzański, and I. A. Walmsley, Nature Photon. 3, 673 (2009).
  * Maccone and Giovannetti (2011) L. Maccone and V. Giovannetti, Nature Phys. 7, 376 (2011).
  * Fujiwara and Imai (2008) A. Fujiwara and H. Imai, Journal of Physics A: Mathematical and Theoretical 41, 255304 (2008).
  * Escher  _et al._ (2011) B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406 (2011).
  * Escher  _et al._ (2012) B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, Phys. Rev. Lett. 109, 190404 (2012).
  * Demkowicz-Dobrzański  _et al._ (2012) R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, Nat. Commun. 3, 1063 (2012).
  * Kołodyński and Demkowicz-Dobrzański (2013) J. Kołodyński and R. Demkowicz-Dobrzański, New J. Phys. 15, 073043 (2013).
  * Alipour  _et al._ (2014) S. Alipour, M. Mehboudi, and A. T. Rezakhani, Phys. Rev. Lett. 112, 120405 (2014).
  * Knysh  _et al._ (2011) S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, Phys. Rev. A 83, 021804 (2011).
  * Knysh  _et al._ (2014) S. I. Knysh, E. H. Chen, and G. A. Durkin, arXiv preprint  (2014), arXiv:1402.0495 [quant-ph] .
  * Dür  _et al._ (2014) W. Dür, M. Skotiniotis, F. Fröwis, and B. Kraus, Phys. Rev. Lett. 112, 080801 (2014).
  * Kessler  _et al._ (2014) E. M. Kessler, I. Lovchinsky, A. O. Sushkov, and M. D. Lukin, Phys. Rev. Lett. 112, 150802 (2014).
  * Arrad  _et al._ (2014) G. Arrad, Y. Vinkler, D. Aharonov, and A. Retzker, Phys. Rev. Lett. 112, 150801 (2014).
  * Ozeri (2013) R. Ozeri, arXiv preprint  (2013), arXiv:1310.3432 [quant-ph] .
  * Lu  _et al._ (2015) X.-M. Lu, S. Yu, and C. Oh, Nat. Commun. 6, 7282 (2015).
  * Herrera-Martí  _et al._ (2015) D. A. Herrera-Martí, T. Gefen, D. Aharonov, N. Katz, and A. Retzker, Phys. Rev. Lett. 115, 200501 (2015).
  * Note (1) We remark that we do not allow for additional control over the environment as has been recently considered in Gefen  _et al._ (2015); Plenio and Huelga (2015).
  * Note (2) To the best of our knowledge, quantum control has been used to modify the overall system-environment dynamics but not the dissipative evolution itself Arrad  _et al._ (2014); Sekatski  _et al._ (2015).
  * Ulam-Orgikh and Kitagawa (2001) D. Ulam-Orgikh and M. Kitagawa, Physical Review A 64, 052106 (2001).
  * Demkowicz-Dobrzański and Maccone (2014) R. Demkowicz-Dobrzański and L. Maccone, Phys. Rev. Lett. 113, 250801 (2014).
  * Chaves  _et al._ (2013) R. Chaves, J. B. Brask, M. Markiewicz, J. Kołodyński, and A. Acín, Phys. Rev. Lett. 111, 120401 (2013).
  * Viola and Lloyd (1998) L. Viola and S. Lloyd, Phys. Rev. A 58, 2733 (1998).
  * Viola  _et al._ (1999) L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999).
  * Viola and Knill (2003) L. Viola and E. Knill, Phys. Rev. Lett. 90, 037901 (2003).
  * Khodjasteh and Viola (2009) K. Khodjasteh and L. Viola, Phys. Rev. Lett. 102, 080501 (2009).
  * Khodjasteh  _et al._ (2010) K. Khodjasteh, D. A. Lidar, and L. Viola, Phys. Rev. Lett. 104, 090501 (2010).
  * West  _et al._ (2010) J. R. West, D. A. Lidar, B. H. Fong, and M. F. Gyure, Phys. Rev. Lett. 105, 230503 (2010).
  * Wiseman and Milburn (2009) H. M. Wiseman and G. J. Milburn,  _Quantum measurement and control_ (Cambridge University Press, 2009).
  * Sergeevich  _et al._ (2011) A. Sergeevich, A. Chandran, J. Combes, S. D. Bartlett, and H. M. Wiseman, Phys. Rev. A 84, 052315 (2011).
  * Tsang (2012) M. Tsang, Phys. Rev. Lett. 108, 230401 (2012).
  * Gill and Levit (1995) R. D. Gill and B. Y. Levit, Bernoulli 1(1/2), 59 (1995).
  * Helstrom (1976) C. W. Helstrom,  _Quantum Detection and Estimation Theory_ (Academic Press, 1976).
  * Holevo (1980) A. S. Holevo,  _Probabilistic and Statistical Aspects of Quantum Theory_ (North-Holland Series in Statistics and Probability, 1980).
  * Braunstein and Caves (1994) S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
  * Pezzé and Smerzi (2009) L. Pezzé and A. Smerzi, Phys. Rev. Lett. 102, 100401 (2009).
  * Lücke  _et al._ (2014) B. Lücke, J. Peise, G. Vitagliano, J. Arlt, L. Santos, G. Tóth, and C. Klempt, Phys. Rev. Lett. 112, 155304 (2014).
  * Strobel  _et al._ (2014) H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezzè, A. Smerzi, and M. K. Oberthaler, Science 345, 424 (2014).
  * Pires  _et al._ (2015) D. P. Pires, M. Cianciaruso, L. C. Céleri, G. Adesso, and D. O. Soares-Pinto, (2015), arXiv:1507.05848 [quant-ph] .
  * Taddei  _et al._ (2013) M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Phys. Rev. Lett. 110, 050402 (2013).
  * Fröwis and Dür (2012) F. Fröwis and W. Dür, New J. Phys. 14, 093039 (2012).
  * Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang,  _Quantum computation and quantum information_ (Cambridge university press, 2010).
  * Breuer and Petruccione (2002) H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, 2002).
  * Matsuzaki  _et al._ (2011) Y. Matsuzaki, S. C. Benjamin, and J. Fitzsimons, Phys. Rev. A 84, 012103 (2011).
  * Chin  _et al._ (2012) A. W. Chin, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 109, 233601 (2012).
  * Macieszczak (2015) K. Macieszczak, Phys. Rev. A 92, 010102 (2015).
  * Smirne  _et al._ (2016) A. Smirne, J. Kołodyński, S. F. Huelga, and R. Demkowicz-Dobrzański, Phys. Rev. Lett. 116, 120801 (2016).
  * Addis  _et al._ (2016) C. Addis, E.-M. Laine, C. Gneiting, and S. Maniscalco, (2016), arXiv:1604.07998 [quant-ph] .
  * Sekatski  _et al._ (2015) P. Sekatski, M. Skotiniotis, and W. Dür, arXiv preprint  (2015), arXiv:1512.07476 [quant-ph] .
  * Andersson  _et al._ (2007) E. Andersson, J. D. Cresser, and M. J. W. Hall, J. Mod. Opt. 54, 1695 (2007).
  * Note (3) Remark that the constuction of our FFQC strategy assumes that one can perform noiseless parity measurement on the probe-plus-ancillary system, which is a two-qubit operation. However in practice the ancilla may correspond to an internal degree of freedom, while the gate is constructed on two different systems.
  * Note (4) Note that alternatively to the derivation given in Fujiwara and Imai (2008), the second inequality in Eq. (22) can be understood as a consequence of the strong concavity of the fidelity in Eq. (3).
  * Note (5) Note that for any $A=A^{\dagger}$, $||A||=0\Leftrightarrow A=0$, so that A is the null operator.
  * Brask  _et al._ (2015) J. B. Brask, R. Chaves, and J. Kołodyński, Phys. Rev. X 5, 031010 (2015).
  * Unden  _et al._ (2016) T. Unden, P. Balasubramanian, D. Louzon, Y. Vinkler, M. B. Plenio, M. Markham, D. Twitchen, I. Lovchinsky, A. O. Sushkov, M. D. Lukin, _et al._ , (2016), arXiv:1602.07144 [quant-ph] .
  * Note (6) Note that in the absence of noise it is optimal to set $t\tmspace-{.1667em}=\tmspace-{.1667em}T$ as large as possible, which makes the CRB (54) not applicable.
  * Note (7) For which the CE bound (55) is guaranteed to be tight, see Eq. (22).
  * Note (8) Note that the control steps need not be ultra fast, just sufficiently fast so that the first order approximation holds. the duration of these control-pulses depends on the strength of the decoherence process.
  * Note (9) A careful reader notices that Eq. (61) is symmetric in the exchange of $p$ with $1-p$. Indeed, it turns out that if the error register is kept it does not matter for which error one corrects in the error branch, as the finals states $\mathaccentV{bar}016{\varrho}_{m}(t)$ for the two cases are related by a unitary transformation.
  * Gefen  _et al._ (2015) T. Gefen, D. A. Herrera-Martí, and A. Retzker, arXiv preprint  (2015), arXiv:1510.00049 [quant-ph] .
  * Plenio and Huelga (2015) M. B. Plenio and S. F. Huelga, arXiv preprint  (2015), arXiv:1510.02737 [quant-ph] .
  * Bengtsson and Życzkowski (2006) I. Bengtsson and K. Życzkowski,  _Geometry of Quantum States: An Introduction to Quantum Entanglement_ (Cambridge University Press, 2006).

##  Appendix A Relation between the Kraus representations and the dynamical
matrix

Any Kraus operator, $K_{i}\\!=\\!\sum_{k}\mathsf{M}_{ik}\sigma_{k}$, can be
expressed in the operator basis given by the four Pauli operators in the qubit
case, which we write for the vector of Kraus operators as

${\bf K}=\mathsf{M}{\bm{\varsigma}}.$ (71)

The action of the channel on the density matrix is expressed in the Kraus
representation as

$\displaystyle{\mathcal{E}}(\rho)=\sum_{i}K_{i}\rho
K_{i}^{\dagger}=\sum_{i}(\sum_{k}\mathsf{M}_{ik}\sigma_{k})\rho(\sum_{\ell}\mathsf{M}_{i\ell}^{*}\sigma_{\ell})$
(72) $\displaystyle=\sum_{k\ell}(\sum_{i}\mathsf{M}_{\ell
i}^{\dagger}\mathsf{M}_{ik})\sigma_{k}\rho\sigma_{\ell}=\sum_{k\ell}(\mathsf{M}^{\dagger}\mathsf{M})_{\ell
k}\,\sigma_{k}\rho\sigma_{\ell}.$

Imposing the equality with the dynamical matrix representation (14) one
arrives at

$\mathsf{M}^{\dagger}\mathsf{M}=\mathsf{S}^{T}.$ (73)

From this expression it follows that all Kraus representation satisfying
$\mathsf{M}^{\prime}=\mathsf{u}\mathsf{M}$ (with a unitary $\mathsf{u}$) lead
to the same dynamical matrix and therefore correspond to the same channel.
Remark that in general $\mathsf{M}$ may not be a square but a rectangular
$n\times 4$ matrix, where $n$ is the number of Kraus operators in a given
representation. However, any $\mathsf{M}$ admits a singular value
decomposition of the form

$\mathsf{M}=\mathsf{u}\,\mathsf{D}_{n\times 4}\,\mathsf{v}^{\dagger},$ (74)

with $\mathsf{u}$– a unitary $n\times n$ matrix , $\mathsf{v}$– a unitary
$4\times 4$ matrix and $\mathsf{D}_{n\times 4}$ – a rectangular $n\times 4$
diagonal matrix. Now $\mathsf{M}^{\prime}=\mathsf{D}_{n\times
4}\,\mathsf{v}^{\dagger}$ is also a valid Kraus representation, and so is the
$4\times 4$ matrix $\mathsf{M}^{\prime\prime}=\mathsf{D}_{4\times
4}\,\mathsf{v}^{\dagger}$ (where to get to $\mathsf{D}_{4\times 4}$ from
$\mathsf{D}_{n\times 4}$ we either remove all the $n-4$ zero lines in the case
$n>4$, or add $4-n$ zero lines in the case $n<4$). Finally, we chose the
Hermitian $4\times 4$ matrix

$\mathsf{M}^{\prime\prime\prime}=\mathsf{v}\,\mathsf{D}_{4\times
4}\,\mathsf{v}^{\dagger}$ (75)

to be our canonical Kraus representation for the channel.

##  Appendix B Infinitesimal time-step CE bound

Here we prove the infinitesimal time-step CE bound of Eq. (34) that may be
applied also to schemes that incorporate FFQC, i.e, the ones of type (d) in
Fig. 1. We essentially follow the derivation contained in the Supplementary
Material of Ref. Demkowicz-Dobrzański and Maccone (2014) except the last step,
at which we slightly generalize their bound.

In particular, one should follow the reasoning up to Eq. (S14) of Ref.
Demkowicz-Dobrzański and Maccone (2014), at which a more general upper bound
(for any real $\sqrt{x}$) may be derived via

$\displaystyle||\sum_{k}\dot{K}_{k}^{\dagger}iAK_{k}+h.c.||=||i\sum_{k}(\sqrt{x}\,\dot{K}_{k}^{\dagger})A(\frac{1}{\sqrt{x}}K_{k})$
$\displaystyle-(\frac{1}{\sqrt{x}}K_{k}^{\dagger})A(\sqrt{x}\,\dot{K}_{k})||$
$\displaystyle=||i\sum_{k}(\sqrt{x}\,\dot{K}_{k}+i\frac{1}{\sqrt{x}}K_{k}\big{)}^{\dagger}A(\sqrt{x}\,\dot{K}_{k}+i\frac{1}{\sqrt{x}}K_{k})$
$\displaystyle-\lambda\dot{K}_{k}^{\dagger}A\dot{K}_{k}-\frac{1}{x}K_{k}^{\dagger}AK_{k}||$
(76)

Using the triangle inequality together with the identity stated in Eq. (S12)
of Ref. Demkowicz-Dobrzański and Maccone (2014), we arrive at a generalized
version of Eq. (S15) therein:

$\displaystyle\leq
2||A||\Big{(}x||\sum_{k}\dot{K}_{k}^{\dagger}\dot{K}_{k}||+||\sum_{k}\dot{K}_{k}^{\dagger}K_{k}||+\frac{1}{x}\Big{)},$
(77)

which in turn results in the generalization of Eq. (S18):

$\displaystyle\mathcal{F}\leq
k||\alpha||+k(k-1)||\beta||(x||\alpha||+||\beta||+\frac{1}{x}).$ (78)

By taking the limit of infinitesimal time steps
$k\\!=\\!t^{\prime}/\mathrm{d}t$ we get

$\displaystyle\mathcal{F}\leq\frac{t^{\prime}}{\mathrm{d}t}||\alpha||+\frac{(t^{\prime})^{2}}{\mathrm{d}t^{2}}||\beta||(x||\alpha||+||\beta||+\frac{1}{x}).$
(79)

As we seek a non-trivial bound we are only interested in Kraus representations
such that $\alpha=\alpha^{(2)}\mathrm{d}t+\mathcal{O}(\mathrm{d}t^{3/2})$ and
$\beta=\beta^{(2)}\mathrm{d}t+\mathcal{O}(\mathrm{d}t^{3/2})$. Furthermore, we
are willing to use the bound to show a linear scaling of the QFI in
$t^{\prime}$. In order to do so, we set $x=\mathrm{d}t$ and look for a Kraus
representation such that $\beta^{(2)}=0$. As a result, the upper bound does
not diverge in the $\mathrm{d}t\\!\to\\!0$ limit and yields

$\displaystyle\mathcal{F}\leq
t^{\prime}||\alpha^{(2)}||\quad\text{with}\quad\beta=\mathcal{O}(\mathrm{d}t^{2}),$
(80)

which indeed imposes the SQL-like scaling. Notice that to get this expression
one also formally needs to show that the next order
$\beta^{(3)}\mathrm{d}t^{3/2}$ can be set to zero. However, this can always be
done without affecting the lower orders, as we show in the next section.

##  Appendix C Asymptotic FFQC-valid CE bounds

In this appendix we perform analytical minimization of the infinitesimal time-
step CE bound introduced in Sec. IV.2.3 that lead to the Eq. (43). For ease of
notation we denote the matrix $\mathsf{S}^{(1)}$ in Eq. (42) by

$\mathsf{S}^{(1)}={\small\left(\begin{array}[]{c|c}s^{(1)}&{\bf
s}^{(1){\dagger}}\\\ \hline\cr{\bf
s}^{(1)}&\bar{\mathsf{S}}^{(1)}\end{array}\right)}.$ (81)

Recall that our goal is to minimize over all Kraus decompositions ${\bf
K}=\mathsf{M}{\bm{\varsigma}}$ of the channel
$\mathcal{E}_{\omega,\mathrm{d}t}$ up to first order in $\mathrm{d}t$.
Expanding the matrix $\mathsf{M}$ up to first order in $\mathrm{d}t$,
$\mathsf{M}=\mathsf{M}^{(0)}+\sqrt{\mathrm{d}t}\,\mathsf{M}^{(1)}+\mathrm{d}t\,\mathsf{M}^{(2)}$,
and using Eq. (31) the matrices $\mathsf{M}^{(\ell)}$ satisfy

$\displaystyle\mathsf{M}^{(0)}\mathsf{M}^{(0)}$
$\displaystyle=\mathsf{S}^{(0)}$
$\displaystyle\mathsf{M}^{(0)}\mathsf{M}^{(1)}+\mathsf{M}^{(1)}\mathsf{M}^{(0)}$
$\displaystyle=0$
$\displaystyle\mathsf{M}^{(1)}\mathsf{M}^{(1)}+\mathsf{M}^{(0)}\mathsf{M}^{(2)}+\mathsf{M}^{(2)}\mathsf{M}^{(0)}$
$\displaystyle=\mathsf{S}^{(1)}.$ (82)

These constraints enforce the following structure on the matrices
$\mathsf{M}^{(\ell)}$

$\displaystyle\mathsf{M}^{(0)}$ $\displaystyle=\mathrm{diag}(1,0,0,0),$
$\displaystyle\mathsf{M}^{(1)}$
$\displaystyle={\small\left(\begin{array}[]{c|c}0&{\bf 0}^{T}\\\ \hline\cr{\bf
0}&\bar{\mathsf{M}}^{(1)}=\sqrt{\bar{\mathsf{L}}}\\\
\end{array}\right)},\quad\mathsf{M}^{(2)}={\small\left(\begin{array}[]{c|c}\frac{1}{2}s^{(1)}&{\bf
s}^{(1){\dagger}}\\\ \hline\cr{\bf
s}^{(1)}&\bar{\mathsf{M}}^{(2)}\end{array}\right)},$ (87)

where $\bar{\mathsf{M}}^{(2)}$ does not contribute to the first order
expansion of the channel and can thus be chosen at will.

By performing a power series expansion of the matrices $\alpha$ and $\beta$ in
terms of $\mathrm{d}t$

$\displaystyle\alpha$
$\displaystyle=\alpha^{(0)}+\sqrt{\mathrm{d}t}\,\alpha^{(1)}+\mathrm{d}t\,\alpha^{(2)}+O(\mathrm{d}t^{3/2})$
$\displaystyle\beta$
$\displaystyle=\beta^{(0)}+\sqrt{\mathrm{d}t}\,\beta^{(1)}+\mathrm{d}t\beta^{(2)}+\mathrm{d}t^{3/2}\beta^{(3)}+O(dt^{2}),$
(88)

and using Eq. (32) and Eq. (33) with
$\mathsf{h}=\mathsf{h}^{(0)}+\mathsf{h}^{(1)}\sqrt{\mathrm{d}t}+\mathsf{h}^{(2)}\mathrm{d}t$
the minimization over all equivalent Kraus operators is equivalent to
searching over Hermitian matrices $\mathsf{h}^{(\ell)}$ that minimize the
bound of Eq. (43).

To yield a non-trivial bound a Kraus decomposition has to satisfy
$\alpha^{(0)}=\alpha^{(1)}=\beta^{(0)}=\beta^{(1)}=0$. Expanding Eq. (32) and
Eq. (33) in powers of $\mathrm{d}t$ and minimizing order by order imposes the
following structure for $\mathsf{h}^{(0)},\,\mathsf{h}^{(1)}$

$\displaystyle\mathsf{h}^{(0)}={\small\left(\begin{array}[]{c|c}0&0\\\
\hline\cr
0&{\mathsf{H}}^{(0)}\end{array}\right)},\quad\quad\mathsf{h}^{(1)}={\small\left(\begin{array}[]{c|c}0&{\bf
h}^{(1){\dagger}}\\\ \hline\cr{\bf
h}^{(1)}&{\mathsf{H}}^{(1)}\end{array}\right)},$ (93)

and the coefficients $\alpha^{(2)}$ and $\beta^{(2)}$ are given by

$\displaystyle\alpha^{(2)}={\bm{\varsigma}}^{\dagger}{\small\left(\begin{array}[]{c|c}{\bf
h}^{(1)^{\dagger}}{\bf h}^{(1)}&{\bf
h}^{(1){\dagger}}\,{\mathsf{H}}^{(0)}\bar{\mathsf{M}}^{(1)}\\\
\hline\cr\bar{\mathsf{M}}^{(1)}{\mathsf{H}}^{(0)}{\bf
h}^{(1)}&\bar{\mathsf{M}}^{(1)}{\mathsf{H}}^{(0)}{\mathsf{H}}^{(0)}\bar{\mathsf{M}}^{(1)}\end{array}\right)}{\bm{\varsigma}}$
(96)
$\displaystyle\beta^{(2)}=\mathrm{i}\,\left(\frac{-1}{2}\sigma_{3}+{\bm{\varsigma}}^{\dagger}{\small\left(\begin{array}[]{c|c}h^{(2)}_{00}&{\bf