# Quantum control of bosonic modes with superconducting circuits

Wen-Long Ma  State Key Laboratory for Superlattices and Microstructures,
Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083,
China  Pritzker School of Molecular Engineering, University of Chicago,
Illinois 60637, USA  Shruti Puri  Department of Applied Physics and Physics,
Yale University, New Haven, Connecticut 06511, USA  Yale Quantum Institute,
Yale University, New Haven, Connecticut 06511, USA  Robert J. Schoelkopf
Department of Applied Physics and Physics, Yale University, New Haven,
Connecticut 06511, USA  Yale Quantum Institute, Yale University, New Haven,
Connecticut 06511, USA  Michel H. Devoret  Department of Applied Physics and
Physics, Yale University, New Haven, Connecticut 06511, USA  Yale Quantum
Institute, Yale University, New Haven, Connecticut 06511, USA  S. M. Girvin
Department of Applied Physics and Physics, Yale University, New Haven,
Connecticut 06511, USA  Yale Quantum Institute, Yale University, New Haven,
Connecticut 06511, USA  Liang Jiang<EMAIL_ADDRESS>Pritzker School
of Molecular Engineering, University of Chicago, Illinois 60637, USA

###### Abstract

Bosonic modes have wide applications in various quantum technologies, such as
optical photons for quantum communication, magnons in spin ensembles for
quantum information storage and mechanical modes for reversible microwave-to-
optical quantum transduction. There is emerging interest in utilizing bosonic
modes for quantum information processing, with circuit quantum electrodynamics
(circuit QED) as one of the leading architectures. Quantum information can be
encoded into subspaces of a bosonic superconducting cavity mode with long
coherence time. However, standard Gaussian operations (e.g., beam splitting
and two-mode squeezing) are insufficient for universal quantum computing. The
major challenge is to introduce additional nonlinear control beyond Gaussian
operations without adding significant bosonic loss or decoherence. Here we
review recent advances in universal control of a single bosonic code with
superconducting circuits, including unitary control, quantum feedback control,
driven-dissipative control and holonomic dissipative control. Various
approaches to entangling different bosonic modes are also discussed.

Key words: bosonic modes, circuit QED, unitary dynamics, quantum feedback
control, driven-dissipative processes, holonomic quantum computation

##  I Introduction

Quantum computation holds the promise of solving some specific problems, such
as factorization of large integers and simulation of quantum many-body
problems Nielsen2000 , much faster than any known classical computer. To build
such a quantum computer, the physical platform should work in the quantum
regime with long coherence time, fast quantum operations and good scalability,
which are daunting obstacles for current technologies. The promising
strategies to overcome such obstacles are quantum error correction (QEC)
Shor1995 ; Knill1997 ; Lidar2013  and fault-tolerant (FT) quantum computation
Preskill1998 , where the coherence time of the quantum memories can be
extended and the quantum operations can tolerate some low-probability errors
(including errors in the QEC circuit) below a certain threshold.

In the prototypical model for quantum computation - the quantum circuit model,
a quantum bit of information (qubit) is encoded into a two-level system,
called a physical qubit, and the usual approach for QEC is to encode a logical
qubit into some subspace of multiple physical qubits, so that different error
processes lead to distinguishable syndromes and can therefore be corrected.
However, the increased number of physical qubits for a logical qubit
introduces more decoherence for the system to correct. Moreover, the logical
gate operations become quite complicated since multiple physical systems need
to be addressed simultaneously. Hence, it is still an outstanding experimental
challenge to build a more robust quantum register using multiple physical
qubits.

An alternative scheme is to encode the quantum information into bosonic modes
such as harmonic oscillators Braunstein2005 ; Weedbrook2012 . A single bosonic
mode already provides an infinitely large Hilbert space, from which we choose
a logical subspace for an error-correcting code Chuang1997 ; Braunstein1998 ;
Gottesman2001 ; Cochrane1999 ; Michael2016 ; Albert2018b . Such bosonic QEC
modes can be hardware-efficient compared to the conventional QEC codes based
on multiple qubits. Moreover, the bosonic modes often have relatively simple
decoherece processes (mainly bosonic excitation loss channel) during which the
bosonic excitations are lost one by one Blais2021 . There have been several
error-correcting encoding schemes in a single bosonic mode proposed to date,
including the Gottesman-Kitaev-Preskill (GKP) codes Gottesman2001 ; Noh2018 ;
Royer2020 , cat codes Cochrane1999 ; Li2017 ; Bergmann2016 , binomial codes
Michael2016 , rotation-symmetric codes Grimsmo2020  and other variations
Albert2018b ; Li2019 . The GKP codes, consisting of superpositions of highly
squeezed states, are not only protected against small shifts in position but
also have been shown to perform well against the more realistic amplitude
damping channel Albert2018b . The cat codes use superpositions of coherent
states evenly distributed around a circle in phase space, which can be
protected against (single or multiple) bosonic excitation loss and dephasing
errors. The binomial codes exploit superpositions of Fock states weighted with
binomial coefficients, which can exactly correct the bosonic excitation loss,
gain and dephasing errors up to a specific degree.

For bosonic modes, the standard operations (e.g., displacement operation,
phase rotation, one-mode squeezing, beam splitting, and two-mode squeezing)
are all Gaussian operations, which can only transform Gaussian states into
Gaussian states Braunstein2005 ; Lloyd1999 . However, universal control of a
single bosonic mode can be achieved by adding a single nonlinear operation
Lloyd1999 . When such a direct nonlinear operation is difficult to realize
directly, it is still possible to implement an indirect nonlinear interaction
by coupling the bosonic mode to a finite-level ancilla. Moreover, quantum non-
demolition (QND) measurement of the ancilla enables measurement-based feedback
control and therefore arbitrary operation on the bosonic mode. Here, we will
review recent advances in the approaches for universal control and arbitrary
operation of bosonic modes (Table 1), including unitary control, quantum
feedback control, driven-dissipative control and holonomic control (Fig. 1a).
In the first two approaches, an ancilla qubit is coupled to a single bosonic
mode to introduce nonlinear interaction and feedback control, while in the
remaining two approaches, a special coupling between the bosonic mode and
reservoir or a special Hamiltonian of the bosonic mode is engineered to
support some stabilized manifold, consisting of all coherent superpositions of
multiple steady states that are free of any nonunitary effect caused by the
reservoir.

Table 1: Recent theoretical and experimental advances in quantum control of
bosonic modes in circuit QED.

Unitary control |  Unitary & Feedback |  Unitary & Dissipation
---|---|---
Ancilla-induced nonlinearity |  Quantum adaptive control |  Quantum Zeno dynamics
Theory: \- SNAP gate Krastanov2015  \- Optimal control Khaneja2005 ; Heeres2017  \- E-SWAP gate Lau2016  \- CPHASE gate Wang2020  |  Theory: \- CPTP maps Shen2017  \- Teleported gate Gottesman1999  \- ET gate Vy2013 ; Kapit2018  \- PI gate Ma2019  |  Theory: \- Dissipative cat Mirrahimi2014  \- Kerr cat Puri2017  \- FT syndrome detection Puri2019  \- Bias-preserving Kerr cat Puri2019b  \- Bias-preserving dissipative cat Guillaud2019  \- Holonomic gatea Albert2016a
Experiments: \- SNAP gate Heeres2015  \- Optimal control Heeres2017  \- CNOT gate Rosenblum2018a  \- CZ gate Xu2020  \- E-SWAP gate Gao2019  |  Experiments: \- QEC Ofek2016 ; Hu2019 ; Campagne-Ibarcq2019  \- CPTP simulation Hu2018b ; Cai2020  \- Teleported CNOT gate Chou2018  \- FT parity measurement Rosenblum2018b  \- PI SNAP gate Reinhold2019  \- ET phase gate Ma2019b  |  Experiments: \- Dissipative cat Leghtas2015 ; Touzard2018 ; Lescanne2020  \- Kerr catGrimm2020

a This scheme uses only dissipation.

Figure 1: (Color online) (a) Schematic of various approaches for controlling a
quantum system: (i) unitary control on the system alone or both the system and
an ancilla; (ii) quantum feedback control based on measurement of the ancilla;
(iii) driven-dissipative control with either engineered dissipation or
Hamiltonian engineering; (iv) holonomic quantum control based on only
engineered dissipation. (b), (c) Schematic and device photograph of a circuit
QED system modeled as a coupled qubit-oscillator system. The storage cavity
with long coherence time is used to encode quantum information, the transmon
qubit acts as an ancilla for universal control of the storage cavity, and the
readout cavity with short coherence time is used for qubit readout. Both the
storage cavity and transmon qubit can be addressed by microwave control
fields. Reprinted with permission from Refs. Shen2017 ; Kirchmair2013 .

The physical platform we consider is circuit quantum electrodynamics (circuit
QED) Blais2004 ; Wallraff2004 ; Schoelkopf2008 ; Blais2020 ; Cai2021 ;
Joshi2020 ; Blais2021 , which is an analog of cavity QED Haroche2013  using
superconducting circuits Devoret2013 ; Haroche2020 . Cavity QED engineers the
environment of the atoms by placing them in a cavity that supports only
discrete bosonic modes of the electromagnetic field. Examples of cavity QED
systems include alkali atoms in optical cavities Mabuchi2002  and Rydberg
atoms in microwave cavities Nogues1999 . Circuit QED uses superconducting
qubits as artificial atoms coupled to microwave resonators. A key advantage of
circuit QED is the extremely strong coupling between the superconducting
qubits and the cavity. Corresponding to the two encoding schemes based on
qubits or bosonic modes, there are two main routes in superconducting quantum
computing with circuit QED.

Qubits can be encoded into the first two levels of superconducting artificial
atoms, such as the most widely used transmons Koch2007 ; Paik2011 ;
Rigetti2012 ), while the cavity resonators are used for qubit readout.
Arbitrary single qubit rotations can be realized with resonant voltage drives
at the qubit frequencies Blais2004 ; Blais2007 , and gate errors can be
reduced below $10^{-3}$ by pulse shaping techniques Barends2014 ; Chen2016 .
Two-qubit gates can be realized by either capacitive coupling Barends2014  or
using the resonator as a mediator Blais2004 ; Blais2007 ; Majer2007 , with
current error rates being less than one percent Sung2020 ; Samach2021 . Recent
experimental developments include implementations of quantum search algorithms
DiCarlo2009 , quantum teleportation Steffen2013 , simulations of topological
transitions Roushan2014 , digitized adiabatic quantum computing Barends2016 ,
variational quantum algorithms Kandala2017 , supervised learning with quantum-
enhanced feature spaces Havlicek2019 , quantum reservoir engineering MaR2019 ,
quantum walks Gong2021 , advances towards quantum error correction Kelly2015 ;
Takita2017 ; Andersen2020 ; McEwen2021 , building cloud-based devices and
demonstrating quantum supremacy with several tens of qubits Arute2019 .

Alternatively, a storage cavity resonator as a bosonic mode with long
coherence time can encode the quantum information, while the transmon qubits
can act as ancillas to aid universal control of the storage cavity (Fig. 1b,
c). Such an encoding scheme can make use of various bosonic QEC codes, which
are hardware-efficient compared to more standard qubit-based codes such as
surface codes. Moreover, bosonic QEC codes often have specific noise
resilience Guillaud2019 , and therefore can be concatenated with conventional
QEC codes to reduce the hardware overhead Guillaud2019 ; Noh2020 ; Terhal2020
. Recently there has been significant experimental progress in bosonic QEC.
QEC based on cat codes or binomial codes in superconducting cavities have
reached or approached the break-even point Ofek2016 ; Hu2019 , at which the
lifetime of the logical qubit exceeds that of the single best physical qubit
within the logical qubit. The encoding based on GKP codes has also been
demonstrated in trapped-ion mechanical oscillators Fluhmann2019 ; Neeve2020
and superconducting cavities Campagne-Ibarcq2019 . However, compared to
conventional one-qubit and two-qubit control, universal control of single and
multiple bosonic modes requires the introduction of nonlinearity and therefore
is more complex. This will be the main topic of this review.

This review is organized as follows. In Sec. II, we review the universal
control of a single bosonic mode with the aid of an ancilla qubit dispersively
coupled to it. Then we introduce, in Sec. III, the extension from the
universal unitary control to quantum feedback control and arbitrary quantum
channel construction for the bosonic mode by QND measurement of the ancilla.
In Sec. IV, it is shown that reservoir engineering and Hamiltonian engineering
can be promising strategies to realize universal quantum computation in some
unitarily evolving subspace of the bosonic mode. In Sec. V, the combination of
reservoir engineering and holonomic quantum control is introduced to realize
universal control of bosonic modes. Then in Sec. VI, we introduce the quantum
control schemes to entangle different bosonic modes for universal quantum
computation. In Sec. VII, we briefly summarize the review and outline some
future directions for quantum control of the bosonic modes. For convenience,
we take the reduced Plank constant as $\hbar=1$ throughout this review.

##  II Unitary quantum control

Quantum control of a single bosonic mode (typically a harmonic oscillator) can
be achieved in the coupled qubit-oscillator system with a qubit as an ancilla.
Many theoretical and experimental works were devoted to preparing arbitrary
oscillator states assisted by an ancilla qubit with Jaynes-Cummings (JC)
coupling Law1996 ; Brattke2001 ; Leibfried2003 ; Houch2007 ; Hofheinz2009 .
However, it is more challenging to achieve universal control of the
oscillator, which usually needs a multi-level ancilla Santos2005 , slow
adiabatic evolutions Strauch2012  or a large number of control operations
Mischuck2013 .

In circuit QED, the transmon (as an anharmonic oscillator) can act as an
ancilla to aid the control of cavity bosonic modes (as harmonic oscillators).
If the ancilla and a single oscillator are strongly off-resonant with the
detuning much larger than their coupling strength, we arrive at the dispersive
Hamiltonian Schuster2007 ; Kirchmair2013

$\displaystyle H_{0}=\sum_{j=0}^{\infty}(\Lambda_{j}-j\chi
a^{\dagger}a)|j\rangle\langle j|+\omega_{\rm C}a^{\dagger}a,$ (1)

where $|j\rangle$ labels the eigenstate of the ancilla, $\Lambda_{j}$ is the
eigenenergy, $\chi$ is the dispersive coupling strength, $\omega_{\rm C}$ is
the oscillator frequency, and $a$ ($a^{\dagger}$) is the annihilation
(creation) operator of the oscillator excitation. Note that here we have
neglected the weak anharmonicity of the cavity modes. The anharmonicity of the
ancilla ($\Lambda_{j}-\Lambda_{j-1}\neq\Lambda_{j+1}-\Lambda_{j}$) makes it
possible to selectively drive specific ancilla transitions, so the infinite-
dimensional ancilla can often be truncated to a finite-dimensional one. Below
the lowest ancilla eigenstates $\\{|0\rangle,|1\rangle,|2\rangle\cdots\\}$ are
denoted as $\\{|g\rangle,|e\rangle,|f\rangle\cdots\\}$, and the eigenenergy
difference between $|e\rangle$ and $|g\rangle$ is denoted as $\omega_{\rm
T}=\Lambda_{1}-\Lambda_{0}$.

The dispersive Hamiltonian can be interpreted from two different perspectives
(Fig. 2b). On the one hand, the oscillator frequency has a shift dependent on
the ancilla state. This ancilla-state-dependent shift of the cavity leads to
changes in the amplitude and phase of photons reflected from or transmitted
through the cavity and therefore enables a QND measurement on the ancilla
state Blais2004 ; Wallraff2004 . On the other hand, the ancilla transition
frequency has a shift proportional to the oscillator excitation number. In the
strongly dispersive regime of circuit QED, the ancilla frequency shift is much
larger than the cavity line width and ancilla line width, and therefore the
ancilla spectrum is split into a series of separately resolved peaks,
representing the distribution of photon numbers within the driven cavity
Schuster2007 . Moreover, for quantum control of the oscillator, such a
strongly-dispersive coupling regime makes it possible to selectively address
the ancilla if and only if the oscillator is in a specific number state, hence
providing new opportunities for universal control of the oscillator.

Typically we can achieve universal unitary control and quantum measurements of
the ancilla, but only limited unitary control on the oscillator, so the key
point is to use the ancilla to realize some other unitary control on the
oscillator to achieve universal control. Below we introduce two schemes: the
unitary control either separately acts on the ancilla or the oscillator and
then is combined, or acts on the both of them simultaneously (Fig. 2a).

Figure 2:  Universal unitary control of a harmonic oscillator via an ancilla
qubit. (a) Schematic drawing of the experimental circuit QED system. A
$\lambda/4$ coax-stub cavity resonator is coupled to a transmon and readout
resonator on a sapphire substrate. Input couplers close to the transmon and
cavity deliver the respective time-dependent microwave control fields
$\varepsilon_{\rm T}(t)$ and $\varepsilon_{\rm C}(t)$. (b) Schematic of
universal control of the qubit-oscillator system via displacements and SNAP
gates. A weak displacement operation (red dashed arrows) couples the states
$|g,n-1\rangle$ and $|g,n\rangle$ with strength $\sqrt{n}\varepsilon$ for all
$n$. The SNAP gate (blue solid arrows) can simultaneously accumulate different
geometric phases $\\{\varphi_{n}\\}$ to states $\\{|g,n\rangle\\}$. Here we
adopt the rotating frame associated with $\omega_{\rm C}a^{\dagger}a$ so that
the states $\\{|g,n\rangle\\}$ have the same energy. (c) Experimental
demonstration of the control strategy in (b) (separately acting on the
transmon and the cavity). Phasor representation, transmon spectrum, and Wigner
function are shown after each of the steps in the 1-photon Fock state creation
experiment. In the phasor representation, the arrow corresponds to the complex
amplitude $c_{n}$ of the initial cavity state
$|\psi\rangle=\sum_{n}c_{n}|n\rangle$ and the area of the circle is
proportional to $|c_{n}|^{2}$. The qubit spectrum refers to the ancilla
transmon transition frequency dependent on the number of photons in the
cavity. (d) Experimental demonstration of control strategy based on numerical
GRAPE algorithms (acting on both the transmon and the cavity simultaneously).
Lower panel: optimized transmon and oscillator control waveforms of length
approximately $2\pi/\chi$ to take the oscillator from vacuum to the 6-photon
Fock state. Solid (dotted) lines represent the in-phase (quadrature) field
component. Upper panel: oscillator photon-number population trajectory as a
function of time conditioned on the transmon in $|g\rangle$. A complex
trajectory occupying a wide range of photon numbers is taken to perform the
intended operation. Reprinted with permission from Krastanov2015 ; Heeres2015
; Heeres2017 .

###  II.1 Displacement operations and SNAP gates

The first scheme for universal control of the oscillator is to separately
apply unitary control on the subsystems (either the ancilla or the oscillator)
and then combine them Krastanov2015 ; Heeres2015 . The unitary control on the
ancilla may indirectly realize some unitary operations on the oscillator if we
make appropriate pre-section and post-selection of the ancilla state. Then
combining these indirect operations with the direct ones, we may realize
universal control of the oscillator.

One common kind of direct unitary transformation on the oscillator is the
displacement operation

$\displaystyle D(\alpha)=\exp(\alpha a^{\dagger}-\alpha^{*}a),$ (2)

which can be generated by a linear drive on the cavity $H_{\rm
C}=\epsilon_{\rm C}(t)e^{i\omega_{\rm C}t}a^{\dagger}+\rm{H.c.}$ with
$\alpha=-i\int\epsilon_{\rm C}(t)dt$ . However, the displacement operation
alone is not universal, i.e. it cannot generate arbitrary operations on the
oscillator. To see this, note that the displacement operation can only prepare
a coherent state from the vacuum, i.e.
$|\alpha\rangle=D(\alpha)|0\rangle=\exp(-|\alpha|^{2}/2)\sum_{n=0}^{\infty}(\alpha^{n}/\sqrt{n!})|n\rangle$,
while universal control requires that any given target state can be prepared
from any initial state including the vacuum state.

The dispersive coupling of the oscillator to the ancilla introduces a
nonlinear term for the oscillator Hamiltonian, and make it possible to realize
indirect control on the oscillator, such as the selective number-dependent
arbitrary phase (SNAP) gate,

$\displaystyle
S\left({\vec{\varphi}}\right)=\sum_{n=0}^{\infty}e^{i\varphi_{n}}|n\rangle\langle
n|,$ (3)

which imparts arbitrary phases
$\vec{\varphi}=\\{\varphi_{n}\\}_{n=0}^{\infty}$ to the different Fock states
of the oscillator. The original proposal to realize SNAP gates is to weakly
drive the ancilla with multiple frequency components, $H_{\rm T}=\epsilon_{\rm
T}(t)e^{i\omega_{\rm T}t}|g\rangle\langle e|+\text{H.c.}$ with $\epsilon_{\rm
T}(t)=\sum_{n}\Omega e^{i(\phi_{n}(t)-n\chi t)}$ and $\phi_{n}(t)$ being time-
dependent. If $\Omega\ll\chi$, the driving component with frequency
$\omega_{q}-n\chi$ induces a unitary evolution in the ancilla subspace
$\\{|g,n\rangle,|e,n\rangle\\}=\\{|g\rangle,|e\rangle\\}\otimes|n\rangle$ with
a negligible effect on the rest of the system, while the driving phases
{$\phi_{n}(t)$} depending on the oscillator excitation numbers can induce
different evolution paths in different ancilla subspaces, as shown in Fig. 2b.
Suppose the initial state of the whole system is a product state,
$|\psi(0)=|g\rangle\otimes\sum_{n=0}^{N}c_{n}|n\rangle=\sum_{n=0}^{N}c_{n}|g,n\rangle$
with $N$ being the truncated oscillator excitation number, then we may let the
ancilla undergo cyclic evolutions in each subspace
$\\{|g,n\rangle,|e,n\rangle\\}$ and return to $|g,n\rangle$ at time $\tau$. We
can tune $\phi_{n}(t)$ so that the final state accumulates different geometric
phases $\varphi_{n}$ for different $n$ Aharonov1987 , i.e.
$|\psi(\tau)\rangle=\sum_{n=0}^{N}c_{n}e^{i\varphi_{n}}|g,n\rangle$. For
example, we may set $\phi_{n}(t)=0$ for $t\in[0,\tau/2)$ and
$\phi_{n}(t)=\varphi_{n}$ for $t\in[\tau/2,\tau]$ with $\tau=\pi/\Omega$ being
the Rabi period, and the unitary propagator on the whole system [in the
interaction picture associated with the dispersive Hamiltonian in Eq. (1)] is

$\displaystyle U(\tau,0)=|g\rangle\langle g|\otimes
S(\vec{\varphi})+|e\rangle\langle e|\otimes S(-\vec{\varphi}),$ (4)

which implies that the unitary gate on the oscillator is $S(\vec{\varphi})$
[$S(-\vec{\varphi})$] if the initial ancilla state is $|g\rangle$
($|e\rangle$).

The original SNAP gate based on the geometric phases can be simplified by
first decomposing the above propagator [Eq.(4)] as
$U(\tau,0)=U(\tau,\tau/2)U(\tau/2,0)$ with

$\displaystyle U(\tau/2,0)=(|g\rangle\langle e|+|e\rangle\langle
g|)\otimes\mathbb{I},$ (5a) $\displaystyle U(\tau,\tau/2)=|g\rangle\langle
e|\otimes S(\vec{\varphi})+|e\rangle\langle g|\otimes S(-\vec{\varphi}),$ (5b)

where $\mathbb{I}$ is the identity operator for the oscillator. Note that the
first half evolution $U(\tau/2,0)$ causes a flip of the ancilla state while
leaving the oscillator state unchanged, and the second half evolution
$U(\tau,\tau/2)$ causes a further flip of the ancilla state and produces the
SNAP gate on the oscillator at the same time (Fig. 2b). So we may simplify the
SNAP gate by applying only the drive during the second half period, which we
may call the simplified SNAP gate. Moreover, if the simplified SNAP gate is
not completed, we have

$\displaystyle U(\tau/2+\Delta t,\tau/2)$
$\displaystyle=\cos\theta(|g\rangle\langle g|+|e\rangle\langle
e|)\otimes\mathbb{I}$ $\displaystyle-i\sin\theta\left[|g\rangle\langle
e|\otimes S(\vec{\varphi})+|e\rangle\langle g|\otimes
S(-\vec{\varphi})\right],$ (6)

where $\Delta t\in[0,\tau/2]$ and $\theta=\Omega\Delta t$. In this case, the
pre-selection and post-selection of the ancilla state induces either the
identity operation or SNAP gate on the oscillator. For example,
$P_{e}U(\tau/2+\Delta t,\tau/2)P_{e}=\mathbb{I}$ and $P_{g}U(\tau/2+\Delta
t,\tau/2)P_{e}=S(\vec{\varphi})$ with $P_{m}=|m\rangle\langle m|$ ($m=g,e$).
Note that in the above discussions, we consider the limiting case
$\Omega/\chi\rightarrow 0$, while in practice $\Omega/\chi$ is finite and
causes deviations from the ideal SNAP gates Krastanov2015 ; Heeres2015 .
Nevertheless, it is possible to minimize such gate errors due to finite
$\Omega/\chi$ by optimizing the detunings and pulse shapes of the multi-
frequency drive on the ancilla Wang2020 . Besides the resonant driving
approach for SNAP gates, a photon-number dependent Hamiltonian of the
oscillator can also be engineered by off-resonantly driving the ancilla with
multiple frequencies Wang2020 .

It has been demonstrated that universal control of an oscillator can be
achieved by combining the displacement operations $D(\alpha)$ and the SNAP
gates $S(\vec{\varphi})$ Krastanov2015 , since the generators of $D(\alpha)$
and $S(\vec{\varphi})$ and the commutators between these generators generate
the full Lie algebra $\mathfrak{u}(N)$ for any truncated oscillator space
$\\{|0\rangle,\cdots,|N-1\rangle\\}$ Lloyd1999 . As an example, we show in
Fig. 2c that a Fock state $|1\rangle$ of the oscillator can be created by
applying the operation $D(\beta_{2})S(\vec{\varphi})D(\beta_{1})$, where
$\vec{\varphi}$ is fixed to be $(\pi,0,0,\cdots)$ while the displacement
parameters $\beta_{1}$, $\beta_{2}$ are obtained by numerical optimization. A
systematic method was presented in Krastanov2015  to construct an arbitrary
unitary operation in any truncated oscillator space. With this method, the
number of operations to prepare the oscillator Fock state $|n\rangle$ can be
significantly decreased from $\mathcal{O}(n)$ to $\mathcal{O}(\sqrt{n})$.
Recently a more efficient scheme by parameter optimization has been proposed
to implement a broad range of cavity control with only 3 to 4 SNAP gates
Fosel2020 . Nevertheless, it is still an open problem to find the optimal way
for decomposing an arbitrary target unitary into displacement operations and
SNAP gates.

###  II.2 Universal control by numerical optimization algorithms

The previous analytic approach based on displacement operations and SNAP gates
implicitly assumes that the cavity drive [$\epsilon_{\rm C}(t)$] and transmon
drive [$\epsilon_{\rm T}(t)$] are never applied simultaneously, which makes
the evolution more tractable. However, to find more efficient control schemes,
it is better to include the possibility of simultaneously driving both the
ancilla and the oscillator. The arbitrary control field can take the form

$\displaystyle H_{\rm CT}=\epsilon_{\rm C}(t)e^{i\omega_{\rm
C}t}a+\epsilon_{\rm T}(t)e^{i\omega_{\rm T}t}|g\rangle\langle e|+\rm{H.c.},$
(7)

where $\epsilon_{\rm C}(t)$, $\epsilon_{\rm T}(t)$ are arbitrary complex-
valued functions of time. The exact form of the control field can be obtained
by numerical optimization algorithms Heeres2017 , such as the Gradient Ascent
Pulse Engineering (GRAPE) method Khaneja2005 ; deFouquieres2011 . The basic
procedure of the GRAPE method is as follows: (1) specify the target unitary
$U$ and the evolution time $\tau$; (2) discretize the total time $\tau$ into
$M$ equal steps of duration $\Delta t=\tau/M$, and during each step the
control amplitudes are constant; (3) make an initial guess of the control
amplitudes, then calculate the fidelity between the implemented unitary and
target unitary, and also the gradient of the fidelity with respect to each
variation of the control amplitude in each time step; (4) adapt the control
amplitudes according to the fidelity gradient, and repeat step (3) until a
local maximum of the gate fidelity is achieved.

Both $\epsilon_{\rm C}(t)$ and $\epsilon_{\rm T}(t)$ can be optimized with
GRAPE to achieve universal control of the cavity. The numerical optimized
pulses thus obtained have been extensively used in experiments to control
superconducting cavity modes Ofek2016 ; Axline2018 ; Rosenblum2018a ; Hu2019 .
As an example, we show in Fig. 2d the control amplitudes $\epsilon_{\rm
C}(t)$, $\epsilon_{\rm T}(t)$ to prepare the cavity state from the vacuum
state $|0\rangle$ to the Fock state $|6\rangle$. With this approach, Heeres et
al. Heeres2017  have also realized a universal set of gates on the logical
qubit based on error-correcting cat codes in a cavity. Compared with the SNAP
gate that takes a rather long time $2\pi/\Omega$ due to $\Omega/\chi\ll 1$,
the logical gates based on GRAPE algorithm take a much shorter time
$2\pi/\chi$.

###  II.3 Other approaches

Besides the above schemes, there are various other approaches to control
harmonic oscillators via ancilla-induced nonlinearity. One approach is called
photon blockade control, in which the frequency of $\epsilon_{\rm T}(t)$ is
set as $\omega_{\rm T}-N\chi$ to drive resonantly the transition
$|g,N\rangle\leftrightarrow|e,N\rangle$, therefore blockading the population
transfer between the cavity subspace
$\\{|0\rangle,|1\rangle,\cdots,|N-1\rangle\\}$ and the rest of the cavity
Hilbert space Bretheau2015 . Then universal control of the $N$-level qudit can
be realized by optimizing $\epsilon_{\rm C}(t)$ with GRAPE Schirmer2001 ,
which has been experimentally demonstrated in Chakram2020 . In another
approach, using a single transmon as the central processor, universal quantum
operations have been realized between arbitrary eigenmodes of a linear array
of coupled superconducting resonators, realizing a random access quantum
information processor Naik2017 .

It should be mentioned that the weak point of the transmon as an ancilla is
its small dispersive coupling strength with the cavity modes. This limits the
control fidelity of both SNAP gates and blockade control. Such limitations may
be overcome in the future by using better ancillas. For example, with an
ancilla oscillator, a superconducting oscillator can have stimulated
nonlinearity by a three-wave interaction, enabling control of the single-
photon manifold at rates faster than the dispersive protocols Vrajitoarea2020
. Other possible better ancillas include the C-shunt flux qubit with large
anharmonicity Abdurakhimov2019  and the fluxonium with millisecond coherence
time Manucharyan2009 ; Somoroff2021 .

##  III Quantum feedback control

In the last section, the system (an oscillator and an ancilla) as a whole are
assumed to be a closed system and therefore can be sufficiently described by
unitary dynamics. However, the inevitable coupling of the system to the
environment typically induces non-unitary evolutions of the system, which can
be fully characterized by completely positive and trace preserving (CPTP) maps
Nielsen2000 ; Wilde2013  (also called quantum operations or quantum channels).
Hence, it is important to systematically extend the quantum control techniques
from a closed system to an open quantum system. In this section, we will show
that an arbitrary CPTP map of the system can be constructed by coupling the
system to an ancilla qubit with QND readout and quantum feedback control.

Feedback control, where information about the system state is fed back to the
controller for correction, is widely used in classical control theory.
However, its extension to the quantum world is nontrivial Wiseman2010 , since
a quantum measurement of the system will inevitably affect the quantum state
of the system. Quantum feedback control generally falls into two categories:
measurement-based feedback control Wiseman1993  and coherent feedback control
Lloyd2000 . Below we will show that the measurement-based approach can be used
to construct arbitrary CPTP maps and realize robust quantum operations.

Figure 3:  Arbitrary CPTP map construction with quantum feedback control. (a)
Binary tree representation with depth $L=3$. The Kraus operators $K_{b^{(L)}}$
are associated with the leaves of the binary tree, $b^{(L)}\in\\{0,1\\}^{L}$.
The system-ancilla joint unitary to apply in $l$th round $U_{b^{(l)}}$ depends
on the previous ancilla readout record $b^{(l)}=(b_{1}b_{2}\cdots
b_{l})\in\\{0,1\\}^{l}$ associated with a node of the binary tree. (b)
Schematic setup of a circuit QED system used for constructing an arbitrary
quantum channel. (c) Quantum circuit for arbitrary channel construction. The
dimension of the system $d$ can be arbitrary and the circuit depth depends
only on the Kraus rank of the target channel. (d) The quantum circuit to
implement an arbitrary Kraus rank-2 channel with the circuit QED system.
Reprinted with permission from Shen2017 .

###  III.1 Arbitrary CPTP map construction

A CPTP map can be described by the Kraus representation Wilde2013

$\displaystyle\mathcal{\varepsilon}(\rho)=\sum_{i=1}^{N}K_{i}\rho
K_{i}^{\dagger},$ (8)

where $\rho$ is the density matrix of the system we consider and
$\\{K_{i}\\}_{i=1}^{N}$ is the set of Kraus operators satisfying
$\sum_{i=1}^{N}K_{i}^{\dagger}K_{i}=\mathbb{I}$ to preserve the trace of
$\rho$. The Kraus representation is not unique, since a new set of Kraus
operators $\\{F_{i}\\}_{i=1}^{N}$ can be constructed with any $N\times N$
unitary matrix $U$, $F_{i}=\sum_{j}U_{ij}K_{j}$, characterizing the same CPTP
map. The minimum number of Kraus operators is called the Kraus rank of the
CPTP map, and is no larger than $d^{2}$ with $d$ being the Hilbert space
dimension of the system.

####  III.1.1 Construction of CPTP maps with arbitrary Kraus rank

For the construction of arbitrary CPTP maps, Lloyd and Viola first showed that
it is sufficient to repeatedly apply Kraus rank-2 channels in an adaptive
fashion Lloyd2001 , but they did not consider efficient construction with a
low-depth quantum circuit. Recently Shen et al. have extended the binary-tree
construction for arbitrary positive operator-valued measure (POVM)
Andersson2008  to an efficient protocol for CPTP map construction Shen2017 .
In this protocol, a CPTP map with Kraus rank-$N$ can be constructed with an
ancilla qubit by the lowest possible circuit depth $L=\,\log_{2}N\\!$, where
each round of operation consists of one joint unitary of system and ancilla
and one QND measurement on the ancilla qubit. Below we will briefly introduce
such a binary-tree construction for CPTP maps.

Let us first consider the construction of a rank-2 CPTP map with Kraus
operators $\\{K_{0},K_{1}\\}$, which can be achieved by only one round of
operation: (1) initialize the ancilla qubit in $|0\rangle$ (the qubit state
basis being $\\{|0\rangle,|1\rangle\\}$); (2) perform a joint unitary
operation $U\in SU(2d)$ with $d$ being the dimension of the system; (3)
discard or trace over the ancilla qubit. The key point is to design $U$ so
that its $d\times d$ submatrices satisfy $\langle 0|U|0\rangle=K_{0}$,
$\langle 1|U|0\rangle=K_{1}$.

The quantum circuit to implement a rank-$N$ CPTP map with Kraus operators
$\\{K_{0},\cdots,K_{N}\\}$ consists of $L=\,\log_{2}N\\!$ rounds of operations
(Fig. 3a). Each round of operation includes: (1) initialization of the ancilla
qubit in $|0\rangle$; (2) joint unitary gate over the system and ancilla
(conditional on the measurement outcomes from previous rounds), (3) QND
readout of the ancilla, and (4) storage of the classical measurement outcome
for later use. The $l$th round unitary gate $U_{b^{(l)}}$ is represented by
the node of the binary tree $b^{(l)}=(b_{1}b_{2}\cdots b_{l})\in\\{0,1\\}^{l}$
with $l=0,\cdots,L-1$, while the Kraus operators $K_{b^{(L)}}$ are associated
with the leaves of the binary tree $b^{(L)}\in\\{0,1\\}^{L}$ (Fig. 3b). A
systematic way is presented in Shen2017  to design the nodes $U_{b^{(l)}}$ so
that the leaves of the binary tree are exactly the desired Kraus operators,
$K_{b^{(L)}}=K_{i}$ for $i=(b_{1}b_{2}\cdots b_{L})_{2}+1\leq N$ and
$K_{i>N}=0$ [$(\cdots)_{2}$ denotes a binary number]. Arbitrary quantum
channels can also be constructed in the quantum circuit model including
controlled-not (CNOT), single-qubit gates and partial trace operations on the
qubits and any ancilla, and with free single-qubit gates the minimum number of
CNOT gates has been found in Iten2017 .

####  III.1.2 Physical implementation with circuit QED

Circuit QED in the dispersive regime is a promising platform to implement the
arbitrary CPTP map construction. The transmon qubit acts as the ancilla (the
transmon state $|g/e\rangle$ corresponds to the ancilla state $|0/1\rangle$ in
the last subsection), and a $d$-dimensional subspace (e.g., the lowest $d$
Fock states) of the storage cavity with high-quality-factor (high-$Q$) acts as
the qudit. The QND readout of the transmon qubit can be realized by coupling a
readout cavity with low-$Q$ to the transmon. Then the readout result is fed
back to a controller that induces an effector to implement the conditional
control on the qudit (Fig. 3c).

Similar to the SNAP gates, we can implement the following entangling unitary
gate for the whole system including the transmon and the cavity,

$\displaystyle U_{\rm
ent}(\vec{\theta})=\prod_{n=0}^{d}\exp(-iY_{n}\theta_{n}/2),$ (9)

where $\vec{\theta}=(\theta_{0},\cdots,\theta_{d})$, and
$Y_{n}=-i|g,n\rangle\langle e,n|+{\rm H.c.}$ is the the Pauli-$Y$ operator for
the two-dimensional subspace $\\{|g,n\rangle,|e,n\rangle\\}$. The drive on the
transmon for the above gate is $H_{\rm
ent}=\sum_{n}\Omega_{n}e^{-i(\omega_{\rm T}-n\chi)t}|g\rangle\langle
e|+\text{H.c.}$, where the driving amplitude $\Omega_{n}$ and the gate time
$\tau$ should satisfy $\theta_{n}=2\Omega_{n}\tau$. This entangling gate
produces a CPTP map with Kraus operators $\\{S_{g},S_{e}\\}$ with $S_{g}={\rm
diag}(\cos\theta_{1},\cdots,\cos\theta_{d})$ and $S_{e}={\rm
diag}(\sin\theta_{1},\cdots,\sin\theta_{d})$. If we precede $U_{\rm ent}$ with
a unitary $V^{\dagger}$ acting on the qudit alone and perform an conditional
unitary $W=|g\rangle\langle g|\otimes W_{g}+|e\rangle\langle e|\otimes W_{e}$
after $U_{\rm ent}$, the entangling gate becomes $U_{\rm ent}^{\prime}=WU_{\rm
ent}V^{\dagger}$ (Fig. 3d), which is known as the “cosine-sine” decomposition
Shende2006  that can decompose an arbitrary unitary into CNOT and single-qubit
gates. The Kraus operators corresponding to $U_{\rm ent}^{\prime}$ are
$\langle g|U_{\rm ent}^{\prime}|g\rangle=W_{g}S_{g}V^{\dagger}$, $\langle
e|U_{\rm ent}^{\prime}|g\rangle=W_{e}S_{e}V^{\dagger}$, which are singular
value decomposition of any operator for the qubit Nielsen2000  and therefore
can simulate any rank-2 CPTP map. Likewise we can use such entangling gates to
simulate the CPTP map with any Kraus rank.

Recently there have been several experiments for quantum channel simulations
in various platforms, including trapped ions Fluhmann2019 , nuclear mangnetic
resonance (NMR) system Xin2017  and IBM’s cloud computer Wei2018 . In
particular, using a scheme similar to the above one, Hu et al. Hu2018b  first
realized arbitrary quantum channel simulation for a single photonic qubit in
circuit QED. Although this experiment only simulates quantum channels with
Kraus rank-2 for a 2-level qubit with one round of adaptive control, a recent
experiment has extended the capability to simulate arbitrary rank-16 channels
for a $4$-level qudit with 4 rounds of adaptive control Cai2020 . For the
platforms other than circuit QED, the real-time adaptive control is often the
main limitation: for trapped ions, it is quite challenging to avoid the recoil
problem when performing adaptive measurement for trapped ions; for NMR
systems, single-shot readout is not available, so many ancillas must be used
to simulate the adaptiveness; the IBM’s cloud computer does not allow real-
time adaptive control.

The ability to construct an arbitrary CPTP map may have various applications,
such as QEC and quantum state initialization/stabilization. For example, the
simulated quantum channel enabling QEC can help achieve the Heisenberg limit
in quantum metrology Reiter2017 ; Zhou2018 , and dissipative quantum circuits
consisting of sequences of quantum channels subject to specific constraints
can lead to finite-time robust state stabilization Johnson2017 .

###  III.2 Robust quantum operations with adaptive control

Figure 4:  Error-corrected (PI) SNAP gate in circuit QED. (a) PI principle of
error-corrected SNAP gate demonstrated by the ancilla transition graph. The
ancilla transition from the ground state $|g\rangle$ to the second excited
state $|f\rangle$ (green arrows) implements the SNAP gate $S(\vec{\varphi})$
on the logical system (boxes), while the reverse ancilla transition from
$|f\rangle$ to $|g\rangle$ implements the inverse SNAP gate
$S(-\vec{\varphi})$. The ancilla transition from any state to itself (red
closed loops) produces the identity operation on the logical system. Without
ancilla errors, all closed loops in the ancilla transition graph produce the
identity operation on the logical system, satisfying the PI condition. With
the $\chi$-matching condition in Eq. (10), the dominant ancilla relaxation
error from $|f\rangle$ to $|e\rangle$ (blue arrows) produces an identity
operation on the logical system, still ensuring the PI condition. The ancilla
relaxation from $|e\rangle$ to $|g\rangle$ breaks the PI condition but is a
second-order error. (b) The SNAP operation for implementing a logical rotation
$S(\varphi)=e^{-iZ_{L}\varphi}$ with $Z_{L}=|0_{L}\rangle\langle
0_{L}|-|1_{L}\rangle\langle 1_{L}|$ for the binomial code
$\\{||0_{L}\rangle=(|0\rangle+|4\rangle)/\sqrt{2},|1_{L}\rangle=|2\rangle\\}$.
The control consists of applying a Raman drive detuned from the
$|g\rangle\leftrightarrow|e\rangle$ transition (blue arrow) as well as a comb
of control drives (green arrows), detuned in the opposite sense from the
$|e\rangle\leftrightarrow|f\rangle$ transition and separated in frequency by
twice the ancilla-cavity dispersive shift $2\chi$. The measured Wigner
tomograms of the cavity state, postselected on the final ancilla state
following a $|g\rangle\leftrightarrow|f\rangle$ swap (dashed arrows), are
shown to the right. (c) Error-corrected SNAP gate performance from randomized
benchmarking (RB) and interleaved randomized benchmarking (IRB). The effective
gate error probability can be learnt by fitting both the RB and IRB results to
an exponential model (dotted lines). The error probability without interleaved
logical gates is $\gamma_{\rm RB}$ = 2.5%$\pm$0.1% (black), while the error
probability associated with the error-corrected SC (non-error-corrected SNC)
operation is $\gamma_{\rm IRB}-\gamma_{\rm IRB}$ = 2.4%$\pm$0.1%
(4.6%$\pm$0.1%) from the red (blue) curve. (d) Robustness of the error-
corrected SNAP gate with added ancilla dephasing ($|f\rangle\langle
f|-|g\rangle\langle g|$) and relaxation noise rates ($|e\rangle\langle f|$).
In both cases, SC (red markers) is significantly less likely than SNC (blue
markers) to translate ancilla errors induced by the added noise into logical
errors. The dotted lines are derived from a full quantum simulation using
independently measured system parameters. Reprinted with permission from
Reinhold2019 .

The measurement-based adaptive control can also help achieve robust quantum
operations, such as FT quantum measurements and FT quantum gates. Below we
show the recent theoretical and experimental advances of FT operations enabled
by adaptive control in circuit QED.

In Sec. II, we have shown that universal control of a bosonic mode can be
achieved with the aid of an ideal ancilla. In addition, the ancilla can
measure the even-odd excitation number parity of the bosonic mode (as an error
syndrome detecting single excitation loss). This can be done by first
preparing the ancilla in state $(|g\rangle+|e\rangle)/\sqrt{2}$, evolving with
the dispersive Hamiltonian [Eq. (1)] for a time $\pi/\chi$, and finally
performing Ramsey interferometry on the ancilla to determine its phase.
However, ancilla systems are typically more vulnerable to environmental noise,
e.g. the transmon coherence time ($\sim\mu$s) is much shorter than the cavity
mode coherence time ($\sim$ms), so the ancilla errors (e.g. relaxation error
$|g\rangle\langle e|$ and dephasing error $|e\rangle\langle
e|-|g\rangle\langle g|$) during the operation time can propagate to the
oscillator and corrupt the encoded information irreversibly. This drawback
calls for new operation schemes that are FT to these ancilla errors.

A recent experiment shows that the parity measurement of a cavity mode in
circuit QED can be made FT to the ancilla transmon errors by using three
transmon levels and adaptive control Rosenblum2018b . The three-level transmon
($|g\rangle,|e\rangle,|f\rangle$) is coupled to a cavity mode with

$\displaystyle H_{\rm dis}=-\chi a^{\dagger}a(|e\rangle\langle
e|+|f\rangle\langle f|).$ (10)

Note that the dispersive coupling strength is the same for the transmon in
$|e\rangle$ or $|f\rangle$ ($\chi$-matching condition), which can be realized
with an engineered side-band drive Rosenblum2018b . The dispersive Hamiltonian
commutes with the dominant ancilla relaxation error ($|e\rangle\langle f|$)
and also any ancilla dephasing error ($c_{g}|g\rangle\langle
g+c_{e}|e\rangle\langle e|+c_{f}|f\rangle\langle f|$ with
$c_{g},c_{e},c_{f}\in\mathbb{C}$). Such an ancilla error during the
measurement is equivalent to an ancilla error at the end, so although the
measurement fails if the error happens, the cavity logical state is still well
protected and the measurement errors can be overcome by majority voting.

With the same three-level ancilla satisfying the $\chi$-matching condition,
the SNAP gates in Sec. II.1 can be made FT to the dominant ancilla relaxation
error and any dephasing error by adaptive control Ma2019 ; Reinhold2019 . Such
a SNAP gate is implemented by applying the Hamiltonian that drives the
$|g\rangle\leftrightarrow|f\rangle$ transition instead of the
$|g\rangle\leftrightarrow|e\rangle$ transition, with the effective Hamiltonian
in the interaction picture as

$\displaystyle H_{\rm int}=\Omega\left[|g\rangle\langle f|\otimes
S(-\vec{\varphi})+|f\rangle\langle g|\otimes S(\vec{\varphi})\right].$ (11)

Without any ancilla error, the logical gate on the cavity with the ancilla
going from $|g\rangle$ to $|f\rangle$ is the ideal SNAP gate
$S(\vec{\varphi})$. With a single ancilla relaxation error $|e\rangle\langle
f|$ during the control, the ancilla ends in $|e\rangle$ and the final logical
operation is still $S(\vec{\varphi})$. With a single ancilla dephasing error
(e.g. $|f\rangle\langle f|-|g\rangle\langle g|$) and a projective measurement
of the ancilla after the gate, the ancilla may end in $|f\rangle$ with the
logical gate still being $S(\vec{\varphi})$, or end in $|g\rangle$ with the
logical gate being the identity operation (Fig. 4a, b). Thus the control
protocol can be repeated if the ancilla is measured in $|g\rangle$ until the
SNAP gate succeeds. Such error-corrected SNAP gates have recently been
experimentally realized Reinhold2019  with a reduction of the logical gate
error by a factor of two in the presence of naturally occurring decoherence, a
sixfold suppression of the gate error with increased transmon relaxation rates
and a fourfold suppression with increased transmon dephasing rates (Fig. 4c,
d).

Recent theory shows that the error-corrected SNAP gate belongs to a general
class of FT gates on a logical system protected against Markovian ancilla
errors, called path-independent (PI) quantum gates Ma2019 . The PI principle
requires that for given initial and final ancilla states, the logical system
undergoes a unitary gate independent of the specific ancilla path induced by
control drives and ancilla error events. With a certain initial ancilla state,
the desired quantum gate on the logical system is successfully implemented for
some final ancilla states, while the other final ancilla states herald a
failure of the attempted operation, but the logical system still undergoes a
deterministic unitary evolution without loss of coherence. So the PI gate on
the central system can be repeated until it succeeds. A special class of the
PI gates is the error-transparent (ET) gates for a QEC code, theoretically
proposed in Vy2013 ; Kapit2018  and experimentally demonstrated Ma2019b
against a specific system error.

The FT measurement and PI gates belong to an interesting class of CPTP maps,
called quantum instruments Shen2017 . For quantum instruments, both the
classical measurement outcomes and the post-measurement states of the quantum
system are tracked, with the corresponding CPTP map

$\displaystyle\mathcal{\varepsilon}_{\rm
QI}(\rho)=\sum_{\mu=1}^{M}\mathcal{\varepsilon}_{\mu}(\rho)\otimes|\mu\rangle\langle\mu|,$
(12)

where $\\{|\mu\rangle\langle\mu|\\}_{\mu=1}^{M}$ is a set of $M$ orthogonal
projections of the measurement device, and
$\\{\mathcal{\varepsilon}_{\mu}\\}_{\mu=1}^{M}$ is a set of completely
positive and trace nonincreasing maps while
$\sum_{\mu=1}^{M}\mathcal{\varepsilon}_{\mu}(\rho)$ preserves the trace. For
the FT parity measurement, $\\{\mathcal{\varepsilon}_{\mu}\\}$ contains either
the parity measurement channels or the identity channel, while for PI gates
$\\{\mathcal{\varepsilon}_{\mu}\\}$ is a set of unitary channels.

##  IV Driven-dissipative control

Figure 5:  Formation and control of a stabilized manifold by reservoir
engineering. (a) Confinement of a quantum state belonging to a large Hilbert
space into a two-dimensional quantum manifold spanned by
$\\{|C_{\alpha}^{+}\rangle,|C_{\alpha}^{-}\rangle\\}$. The cube represent a
multi-dimensional Hilbert space, while the sphere represents the manifold of
states. Stabilizing forces (orange arrows) direct all states toward the inner
sphere without inducing any rotation in this subspace. (b) Conceptual
representation of quantum Zeno dynamics in the stabilized manifold. The dark
blue circle represents a cross section of the Bloch sphere of a two-state
manifold in (a). The quantum Zeno dynamics corresponds to the motion along the
circle. The trajectory induced by a drive in the large Hilbert space has a
component both along the circle and out of it. The nonlinear dissipation and
drive (orange arrows) cancels the movement outside the circle while leaving
only the rotation on the circle. (c) Schematics of the experimental device.
The quantum manifold is stabilized within the Hilbert space of the fundamental
mode of an aluminium post storage cavity (cyan). This resonator is coupled to
two Josephson junctions on sapphire (yellow for the reservoir and crimson for
the transmon qubit), which are read out by stripline resonators (gray). Three
couplers (brown) bring microwave drives into the system and carry signals out
of it. (d) Sequence of different drives in (c) for quantum Zeno dynamics. To
make resonant the conversion between one reservoir photon and two storage
photons, two pumps on the reservoir with frequencies $2f_{\rm S}-f_{\rm R}$
and $f_{\rm R}$ are used to create pairs of storage photons. An additional
linear displacement drive on the storage cavity induces the quantum Zeno
dynamics. The drive on the transmon qubit is to initialize the storage cavity
in the stabilized manifold and read-out the parity of the storage cavity. (e)
Evolution of the measured parity of the storage cavity as a function of time.
The initial cat states are even cat state $|C_{\alpha}^{+}\rangle$ with
$|\alpha|^{2}=2,3,5$ (circles, squares, diamonds). The storage drive is either
off (black markers) or on (colored markers) with various strengths given in
units of a chosen base strength $\epsilon_{0}$. Reprinted with permission from
Touzard2018 .

The inevitable coupling of a quantum system to the reservoir generally
deteriorates the coherence and coherent control of the system. However, in
some cases, the system can be driven into a unitarily evolving steady
subspace, which can encode and process the quantum information while being
largely immune to environmental noise. This can be achieved by either
reservoir engineering (designing both the system Hamiltonian and the coupling
to the reservoir) or Hamiltonian engineering (designing only the system
Hamiltonian), which are both called driven-dissipative control in this paper.
In this section, we will discuss the formation and control of stabilized
manifolds of Schrödinger cat states in cavity bosonic modes with both
approaches.

###  IV.1 Reservoir engineering

Reservoir engineering is a powerful technique to realize steady state or
subspace in condensed matter physics and quantum information processing
Poyatos1996 ; Diehl2008 ; Verstraete2009 , since the steady state is often an
exotic phase of matter that is difficult to stabilize in nature Diehl2008 ,
while the steady subspace may be used to store, protect and process quantum
information Verstraete2009 . In particular, when the quantum system is coupled
to a Markovian reservoir, the time evolution of the system is governed by the
Lindblad master equation Gardiner2000 ,

$\displaystyle\dot{\rho}=$
$\displaystyle\mathcal{L}\rho=-i[H,\rho]+\sum_{l}\mathcal{D}[F_{l}]\rho,$ (13)

where the Liouvillian $\mathcal{L}$ is a superoperator on the system, $H$ is
the Hamiltonian of the system including the driving term,
$\mathcal{D}[F_{l}]\rho=2F_{l}\rho
F_{l}^{\dagger}-F_{l}^{\dagger}F_{l}\rho-\rho F_{l}^{\dagger}F_{l}$ is the
Lindbladian dissipator with $F_{l}$ being the dissipation-inducing jump
operator that can depend on a parameter. The Markovian reservoir engineering
refers to the design of the system Hamiltonian $H$ and the jump operators
$\\{F_{l}\\}$, so that a stabilized manifold consisting of multiple steady
states Albert2014 ; Albert2016b ; Mirrahimi2014  is formed to encode quantum
information and even allow QEC.

####  IV.1.1 Stabilized manifold with quantum information

Single-mode two-photon process. Consider that a single cavity mode is driven
by an external field such that it can only absorb photons in pairs, and the
energy decay of the mode also happens in pairs of photons, then the
Lindbladian master equation describing such a two-photon driven-dissipative
process is

$\displaystyle\dot{\rho}=$ $\displaystyle[\epsilon_{2}a^{\dagger
2}-\epsilon_{2}^{*}a^{2},\rho]+\mathcal{D}[\sqrt{\kappa_{2}}a^{2}]\rho$
$\displaystyle=$
$\displaystyle\mathcal{D}[\sqrt{\kappa_{2}}(a^{2}-\alpha^{2})]\rho,$ (14)

where $\epsilon_{2}$ and $\kappa_{2}$ are the driving amplitude and decay
rate, respectively. The second line of the above equation shows that the
driven-dissipative dynamics can be described by a single Lindbladian
dissipator $\mathcal{D}[\sqrt{\kappa_{2}}(a^{2}-\alpha^{2})]$ with
$\alpha=\sqrt{2\epsilon_{2}/\kappa_{2}}$. The stabilized manifold is
determined by $\mathcal{D}[\sqrt{\kappa_{2}}(a^{2}-\alpha^{2})]\rho=0$, and
any state satisfying $a^{2}|\xi\rangle=\alpha^{2}|\xi\rangle$ or
$a|\xi\rangle=\pm\alpha|\xi\rangle$ is in this manifold. Such a stabilized
manifold also forms a decoherence-free subspace Lidar2013 . The stabilized
manifold for two-photon process is the two-dimensional Hilbert space spanned
by two coherent states $\\{|\alpha\rangle,|-\alpha\rangle\\}$ (Fig. 5a). For
any initial state $\rho(0)$, the cavity mode asymptotically converges to some
pure or mixed state $\rho(\infty)$ in such a stabilized manifold. For example,
if the initial state is the vacuum state $|0\rangle$ or the single-photon Fock
state $|1\rangle$, the asymptotic state is the pure even
($|C_{\alpha}^{+}\rangle$) or odd ($|C_{\alpha}^{-}\rangle$) Schrödinger cat
state with

$\displaystyle|C_{\alpha}^{\pm}\rangle=\mathcal{N}_{2}(|\alpha\rangle\pm|-\alpha\rangle),$
(15)

where $\mathcal{N}_{2}$ is a normalization constant.

The logical qubit can be encoded into the even-odd Schrödinger cat states
$\\{|C_{\alpha}^{+}\rangle,|C_{\alpha}^{-}\rangle\\}$ (with large $\alpha$ so
that $|\langle C_{\alpha}^{+}|C_{\alpha}^{-}\rangle|\approx 0$) Mirrahimi2014
. A qubit encoded in such a way is called the dissipative-cat qubit. For such
a logical qubit, the dephasing error
$\mathcal{D}[\sqrt{\kappa_{\phi}}a^{\dagger}a]$ can be largely suppressed when
$\kappa_{\phi}\ll\kappa_{2}$, while the single photon loss error
$\mathcal{D}[\sqrt{\kappa_{1}}a]$ causes a bit-flip error and therefore cannot
be suppressed by the two-photon process. Experimentally Leghtas et al.
Leghtas2015  first successfully confined the quantum states of a
superconducting cavity to the stabilized manifold spanned by the even-odd cat
states.

Single-mode four-photon process. The four-photon process is described by
letting both the absorption from the driving field and the energy decay into
the bath happen through quadruples of photons,

$\displaystyle\dot{\rho}=$ $\displaystyle[\epsilon_{4}a^{\dagger
4}-\epsilon_{4}^{*}a^{4},\rho]+\mathcal{D}[\sqrt{\kappa_{4}}a^{4}]\rho$
$\displaystyle=$
$\displaystyle\mathcal{D}[\sqrt{\kappa_{4}}(a^{4}-\beta^{4})]\rho.$ (16)

The stabilized manifold is the four-dimensional Hilbert space spanned by
$\\{|\pm\beta\rangle,|\pm i\beta\rangle\\}$ with
$\beta=(2\epsilon_{4}/\kappa_{4})^{1/4}$. When the cavity mode starts at
initial Fock states $|0\rangle,|1\rangle,|2\rangle,|3\rangle$, it
asymptotically converges to the pure states

$\displaystyle|C_{\beta}^{(\rm
0mod4)}\rangle=\mathcal{N}_{4}(|C_{\beta}^{+}\rangle+|C_{i\beta}^{+}\rangle),$
$\displaystyle|C_{\beta}^{(\rm
1mod4)}\rangle=\mathcal{N}_{4}(|C_{\beta}^{-}\rangle-i|C_{i\beta}^{-}\rangle),$
$\displaystyle|C_{\beta}^{(\rm
2mod4)}\rangle=\mathcal{N}_{4}(|C_{\beta}^{+}\rangle-|C_{i\beta}^{+}\rangle),$
$\displaystyle|C_{\beta}^{(\rm
3mod4)}\rangle=\mathcal{N}_{4}(|C_{\beta}^{-}\rangle+i|C_{i\beta}^{-}\rangle),$
(17)

which form the four-component subspace of the Schrödinger cat states.

To suppress the single photo loss error, which is usually the dominant error
channel of the cavity modes, we can use the encoding scheme that can track the
single-photon jump event and perform QEC. This can be achieved by encoding the
qubit into the logical subspace spanned by the two cat states
$\\{|C_{\alpha}^{\rm 0mod4}\rangle,|C_{\alpha}^{\rm 2mod4}\rangle\\}$ with
even photon number parity. Then a single photon loss changes the photon number
parity from even to odd. The photon number parity of the cavity mode can be
monitored in a QND manner by a Ramsey experiment on an ancilla transmon qubit
dispersively coupled to the cavity.

Single-mode $d$-photon process. The two-photon and four-photon processes can
be generalized to $d$-photon processes ($d=2,4,6,\cdots$ being an even
integer) with

$\displaystyle\dot{\rho}=$ $\displaystyle[\epsilon_{d}a^{\dagger
d}-\epsilon_{d}^{*}a^{d},\rho]+\mathcal{D}[\sqrt{\kappa_{d}}a^{d}]\rho$
$\displaystyle=$
$\displaystyle\mathcal{D}[\sqrt{\kappa_{d}}(a^{d}-\gamma^{d})]\rho,$ (18)

with $\gamma=(2\epsilon_{d}/\kappa_{d})^{1/d}$. The stabilized manifold is the
$d$-dimensional Hilbert spanned by $\\{|\gamma\lambda_{\nu}\rangle\\}$ with
$\lambda_{\nu}=\exp(2i\pi\nu/d)$ ($\nu=0,1,\cdots,d-1$), which are $d$
coherent states lying equidistantly in the phase space. The asymptotic states
or cat code are $d$ different superpositions of such $d$ coherent states
$\\{|C_{\gamma}^{\mu{\rm mod}d}\rangle\\}$ ($\mu=0,1,\cdots,d-1$) with

$\displaystyle|C_{\gamma}^{\mu{\rm
mod}d}\rangle=\mathcal{N}_{d}\sum_{\nu=0}^{2d-1}\lambda_{\nu}^{-\mu}|\gamma\lambda_{\nu}\rangle,$
(19)

which is a superposition of $\mu$mod$d$ Fock states. The $d$-dimensional
Hilbert cat space can be divided into $d/2$ subspaces labeled by
$s=0,1,\cdots,d/2-1$, where the $s$-subspace is spanned by two states
$\\{|C_{\gamma}^{s{\rm mod}d}\rangle,|C_{\gamma}^{(s+d/2){\rm
mod}d}\rangle\\}$ and may encode a logical qubit Li2017 ; Bergmann2016 . After
losing $k$ photons, the $s$ subspace is mapped to the $s-k$ subspace. Hence we
can distinguish up to $d/2-1$ photon losses without destroying the encoded
logical states by projectively measuring the excitation number mod $d/2$
(called the “$\mathbb{Z}_{d/2}$ measurement”). We can also encode a qudit into
the $d/2$-dimensional subspace $\\{|C_{\gamma}^{0{\rm
mod}d}\rangle,|C_{\gamma}^{2{\rm mod}d}\rangle,\cdots,|C_{\gamma}^{(d-2){\rm
mod}d}\rangle\\}$ that can correct a single photon loss error.

Multimode processes. The driven-dissipative processes can be extended from a
single cavity mode to a two modes with operators $\\{a,b\\}$ Albert2019 .
Suppose that both modes simultaneously absorb energy from the driving field
and release energy to the bath through pairs of photons,

$\displaystyle\dot{\rho}=$ $\displaystyle[\epsilon_{\rm p4}b^{\dagger
2}a^{\dagger 2}-\epsilon_{\rm
p4}^{*}a^{2}b^{2},\rho]+\mathcal{D}[\sqrt{\kappa_{\rm p4}}a^{2}b^{2}]\rho$
$\displaystyle=$ $\displaystyle\mathcal{D}[\kappa_{\rm
p4}(a^{2}b^{2}-\delta^{4})]\rho.$ (20)

The stabilized manifold is spanned by the pair-coherent/Barut-Girardello
states Perelomov1986 . Quantum information encoded in a subspace of such a
manifold is immune to the dephasing errors in both modes. Most interestingly,
arbitrary photon loss errors in either mode can be corrected by continuously
monitoring the photon number difference between the two modes. The two-mode
generalization above can also be extended to the multimode case, with the
additional advantage of being able to correct for higher-weight products of
losses or for photon losses and gains at the same time Albert2019 .

####  IV.1.2 Quantum gates by quantum Zeno dynamics

We have shown that the logical qubit encoded in the stabilized manifold can be
dynamically protected from the photo loss and dephasing errors and therefore
act a good quantum memory. It is also possible to perform universal gates on
such a logical qubit. The arbitrary rotations around $x$-axis of a single
qubit and the two-qubit entangling gate can be generated by quantum Zeno
dynamics.

When a quantum system is frequently measured to determine whether it is in the
initial state, the system will always stay in the initial state, which is
called the quantum Zeno effect Misra1977 . But if frequent measurements are
performed to see if it is in a multi-dimensional subspace, the system is not
freezed but evolves according to an effective Hamiltonian obtained by
projecting the initial Hamiltonian into the measurement subspace. Such
dynamics are called Quantum Zeno dynamics Facchi2002 . The driven-dissipative
processes act as a continuous measurement on the quantum system to see if it
is in the multi-dimensional stabilized manifold, so if we apply another
driving Hamiltonian $H$, the effective driving Hamiltonian is $H_{\rm
eff}=P_{C}HP_{C}$ with $P_{C}$ being the projector onto the stabilized
manifold (Fig. 5b).

For the two-photon process with the logical qubit
$\\{|C_{\alpha}^{+}\rangle,|C_{\alpha}^{-}\rangle\\}$, we may apply a linear
drive on the oscillator, $H_{x}=\epsilon_{x}(a+a^{\dagger})$. The two-photon
process acts as a continuous measurement which projects the driving
Hamiltonian onto an effective $x$-axis rotation Hamiltonian in the qubit
space,

$\displaystyle P_{C}H_{x}P_{C}=\Omega_{x}X,$ (21)

where $P_{C}=|C_{\alpha}^{+}\rangle\langle
C_{\alpha}^{+}|+|C_{\alpha}^{-}\rangle\langle C_{\alpha}^{-}|$,
$X=|C_{\alpha}^{+}\rangle\langle C_{\alpha}^{-}|+|C_{\alpha}^{-}\rangle\langle
C_{\alpha}^{+}|$ and $\Omega_{x}=\epsilon_{x}(\alpha+\alpha^{*})$. One can see
that a population transfer between the even cat state $|C_{\alpha}^{+}\rangle$
and the odd cat state $|C_{\alpha}^{-}\rangle$ is enabled by a resonant
single-photon drive on the system (Fig. 5d-f). Recently Touzard et al.
Touzard2018  have experimentally observed such coherent oscillations between
the even and odd cat states by tuning the desired dissipation rate (two-photon
loss rate $\kappa_{2}$) to be 2 orders of magnitude larger than the undesired
dissipation rate (single-photon loss rate $\kappa_{1}$) (Fig. 5c).

For the four-photon process with the logical qubit $\\{|C_{\alpha}^{\rm
0mod4}\rangle,|C_{\alpha}^{\rm 2mod4}\rangle\\}$, the population transfer
between two logical states needs a two-photon drive
$H_{x2}=\epsilon_{x2}(a^{2}+a^{\dagger 2})$ with the projected Hamiltonian in
the stabilized manifold as

$\displaystyle P_{C^{\prime}}H_{x2}P_{C^{\prime}}=\Omega_{x2}(X_{02}+X_{13}),$
(22)

where $P_{C^{\prime}}=\sum_{i=0}^{3}|C_{\alpha}^{i\rm mod4}\rangle\langle
C_{\alpha}^{i\rm mod4}|$, $X_{ij}=|C_{\alpha}^{i\rm mod4}\rangle\langle
C_{\alpha}^{j\rm mod4}|+|C_{\alpha}^{j\rm mod4}\rangle\langle C_{\alpha}^{i\rm
mod4}|$ and $\Omega_{x2}=\epsilon_{x2}(\beta^{2}+\beta^{*2})$. The above
effective Hamiltonian have two driving components: one acting on the qubit
subspace to drive the Rabi oscillation between $|C_{\alpha}^{\rm
0mod4}\rangle$ and $|C_{\alpha}^{\rm 2mod4}\rangle\\}$, and the other one
acting on the remaining subspace to drive the Rabi oscillation between
$|C_{\alpha}^{\rm 1mod4}\rangle$ and $|C_{\alpha}^{\rm 3mod4}\rangle\\}$ with
the same driving amplitude. Such a gate has the additional advantage of being
error-transparent to single-photon loss error, since in the stabilized
manifold the single photon loss operator commutes with the effective
Hamiltonian and therefore can be detected/corrected at the end of the gate
without compromising the encoded quantum information Vy2013 ; Kapit2018 . For
a general $d$-photon processes to a qubit, the $x$-axis rotation Hamiltonian
is $H_{x,d}=\epsilon_{x,d}(a^{d}+a^{\dagger d})$.

In addition to the $x$-axis single-qubit gates, the two-qubit entangling gates
can be realized by applying appropriate driving fields. To complete the set of
universal gate, we may turn off the driven-dissipative control and apply a
Kerr Hamiltonian to implement single-qubit $\pi/2$-rotation around the $z$
axis Mirrahimi2014 . The universal control of the qubits encoded in single
modes can be extended to those encoded in multiple modes Albert2019 . For
example, an arbitrary $x$-axis rotation of the qubit encoded in double modes
can be realized by the drive $H_{\rm p}=\epsilon_{x,\rm
p}(ab+b^{\dagger}a^{\dagger})$.

###  IV.2 Hamiltonian engineering

Figure 6:  Formation and control of a stabilized manifold by Hamiltonian
engineering. (a) Illustration of the eigenspectrum of a Kerr-nonlinear
resonator with a squeezing drive. The even-odd cat states
$|C_{\alpha}^{\pm}\rangle$ are two eigenstates with a large energy gap from
the other eigenstates. (b) Photograph of the nonlinear resonator (purple
frame) inside the copper section of the readout cavity. Also represented are
the $x$-axis rotation drive ($\omega_{a}$) and the squeezing-generation drive
($2\omega_{a}$). Here $\omega_{a}$ denotes the resonator frequency and is
equivalent to $\omega_{\rm C}$ in Eq. (1). (c) Schematic of the nonlinear
resonator with pad offset $\delta$ to set the dispersive coupling to the
readout cavity and spiral symbol representing the nonlinear inductor (SNAIL
element). (d) Scanning electron micrograph of the SNAIL element consisting of
four Josephson junctions in a loop threaded by an external magnetic flux. (e)
Pulse sequence for initialization (to $|C_{\alpha}^{+}\rangle$), Rabi
oscillation and readout of the Kerr-cat qubit. Here $\omega_{s}=2\omega_{a}$,
and $\omega_{b}$ is the frequency of the readout cavity. (f) Dependence of the
Rabi frequency on $\sqrt{\varepsilon_{2}}$. (g) Dependence of the
experimentally measured Rabi oscillations on evolution time $\Delta t$ and on
the phase of the Rabi drive arg($\varepsilon_{2}$). (h) Cuts of (g) for the
three Rabi-drive phases indicated by dashed lines. (i) Simulated Wigner
function of the oscillator density matrix corresponding to the symbols in the
bottom panel of (h). Reprinted with permission from Grimm2020 .

Apart from reservoir engineering, it is also possible to form and process a
stabilized manifold by only Hamiltonian engineering. The stabilized manifold
can be chosen to be a degenerate eigenspace of the system with a designed
Hamiltonian, which is typically decoupled to the remaining eigenspace by a
large energy gap and therefore can be protected from specific system errors
(Fig. 6a).

####  IV.2.1 Formation of the Kerr-cat qubit

Consider the Hamiltonian of a Kerr-nonlinear resonator under the application
of a single-mode squeezing drive Puri2017 ; Goto2016 , written in a frame
rotating at the resonator frequency $\omega_{\rm C}$,

$\displaystyle{H}_{\rm Kerr}$
$\displaystyle=-Ka^{{\dagger}2}a^{2}+(\epsilon_{2}a^{{\dagger}2}+\epsilon_{2}^{*}a^{2}),$
$\displaystyle=-K\left(a^{{\dagger}2}-\alpha^{*2}\right)\left(a^{2}-\alpha^{2}\right)+\frac{|\epsilon_{2}|^{2}}{K}.$
(23)

Here $K$ is the strength of the nonlinearity and
$\alpha=\sqrt{\epsilon_{2}/K}$. The second line makes it clear that the even-
and odd-parity cat states $|C_{\alpha}^{\pm}\rangle$ are the degenerate
eigenstates of this Hamiltonian Puri2017 ; Goto2016 . This cat subspace is
separated from the rest of Hilbert space by a gap $\omega_{\rm gap}\propto
4K|\alpha|^{2}$ Puri2017 . A qubit encoded in such a way is called the Kerr-
cat qubit. Observe that as the strength of the two-photon drive decreases,
that is $|\epsilon_{2}|\rightarrow 0$ and hence $|\alpha|\rightarrow 0$, the
states $|C_{\alpha}^{\pm}\rangle$ continuously approach the vacuum and single-
photon Fock state, respectively. In fact, in this limit the Kerr-cat qubit is
essentially the well-known transmon which encodes a “Fock qubit” in the two
photon-number states: vacuum and single-photon Fock state. It follows that, an
initially undriven Kerr-nonlinear resonator ($\equiv$ Fock-qubit) prepared in
vacuum or single-photon Fock state will respectively evolve to the states
$|C_{\alpha}^{+}\rangle$ or $|C_{\alpha}^{-}\rangle$ as the amplitude of the
squeezing drive is increased adiabatically. For the adiabatic condition to be
satisfied, the rate of change of the two-photon drive must be slower than the
minimum energy gap, $\dot{|\epsilon_{2}}(t)|/|\epsilon_{2}(t)|\ll 2K$. So
typically a large Kerr-nonlinearity results in faster cat state. Nevertheless,
it is possible to apply counter-adiabatic two-photon drive to go faster than
the adiabatic condition would allow Puri2017 .

Like the case of a dissipative-cat qubit [Eq. (IV.1.1)], the probability of a
bit-flip error (e.g., due to frequency fluctuations
$\mathcal{D}[\sqrt{\kappa_{\phi}}\hat{a}^{\dagger}\hat{a}]$) is exponentially
suppressed compared to a phase-flip error (for example due to single photon
loss $\mathcal{D}[\sqrt{\kappa}\hat{a}]$) in the Kerr-cat qubit as well. While
the dissipative-cat qubit is protected against bit-flip errors by a
decoherence-free subspace enabled by engineered dissipation Lescanne2020 ;
Guillaud2019 ; Chamberland2020 , the Kerr-cat qubit is protected from such
errors by the underlying eigenspace structure of the two-photon driven Kerr-
nonlinear resonator Puri2019 ; Puri2019b ; Grimm2020 . Interestingly, the
Kerr- and dissipative-cat qubit realizations are completely compatible with
each other and have complementary properties Puri2017 ; Puri2019 ; Puri2019b .
The inherent nonlinearity of the Kerr-cat mode provides the ability to
implement fast, high-fidelity gates. It also naturally provides the ability to
parametrically engineer two-photon dissipation, which can be subsequently used
for autonomous correction of possible leakage errors Puri2017 ; Puri2019 ;
Puri2019b . Recently, the adiabatic preparation of Kerr-cat was experimentally
demonstrated Grimm2020  and the asymmetry in the bit- and phase-flip errors
was also confirmed. Figure 6b-d shows the device of the superconducting setup
for realization of the Kerr-cat qubit Grimm2020 .

####  IV.2.2 Quantum gates for the Kerr-cat qubit

Selective control of the dynamics of the Kerr-cat qubit in the two-dimensional
subspace $\\{|C_{\alpha}^{+}\rangle,C_{\alpha}^{-}\rangle\\}$ is possible
because of the energy gap $\omega_{\mathrm{gap}}$ separating the qubit
subspace from the rest of the Hilbert space.

Consider a coherent microwave tone applied to the resonator at the resonator’s
resonance frequency $\omega_{\mathrm{\rm C}}$. In the rotating frame, the
resulting Hamiltonian is ${H}_{1}={H}_{\rm
Kerr}+\varepsilon_{x}a^{\dagger}+\varepsilon_{x}^{*}a$. The ${a}^{\dagger}$
term can cause transitions outside the cat-subspace. However, these
transitions are off-resonant and in the limit
$|\varepsilon_{x}|\ll\omega_{\mathrm{gap}}$, leakage out of the qubit subspace
can be neglected. Similar to Eq. (21), the effective Hamiltonian in the qubit
subspace is

$\displaystyle{P}_{C}{H}_{1}{P}_{C}=\Omega_{x}{X}-\Omega_{y}{Y}$ (24)

where ${P}_{C}$, $X$ are the same as those defined in Eq. (21),
$Y=i|C_{\alpha}^{-}\rangle\langle
C_{\alpha}^{+}|-i|C_{\alpha}^{+}\rangle\langle C_{\alpha}^{-}|$,
$\Omega_{x}=\alpha(\varepsilon_{x}+\varepsilon_{x}^{*})\left(r^{-1}+r\right)/2$,
$\Omega_{\mathrm{y}}=i\alpha(\varepsilon_{x}^{*}-\varepsilon_{x})\left(r^{-1}-r\right)/2$
and $r={\sqrt{1-e^{-2\alpha^{2}}}}/\sqrt{1+e^{-2\alpha^{2}}}$. Consequently, a
resonant coherent microwave drive applied in phase with the squeezing drive
($\varepsilon_{x}=\varepsilon_{x}^{*}$) causes Rabi-oscillations around the
$x$-axis and hence implements a $X(\theta)=\exp(i\theta X/2)$ operation
Puri2017 ; Puri2019 ; Puri2019b , where $\theta=\Omega_{x}T$ with $T$ being
the evolution time. Since $r-r^{-1}\sim 2e^{-2\alpha^{2}}$ in the limit of
large $\alpha$, the Rabi oscillations around $y$-axis is exponentially
suppressed with $\alpha^{2}$. The Rabi oscillations of the Kerr-cat qubit were
demonstrated in a recent experiment Grimm2020 , as shown in Fig. 6e-i. Readout
of the Kerr-cat qubit can be realized by coupling the Kerr-cat cavity to a
line resonator with a beam-splitter interaction followed by a homodyne
measurement of the line resonator Grimm2020 . Furthermore, it follows from Eq.
(24) that a resonant beam-splitter interaction, generated parametrically
between two driven nonlinear resonators, leads to an Ising coupling and
realizes a $XX(\theta)=\exp{(i\theta X_{1}X_{2}})$ gate Puri2017 ; Puri2019 ;
Puri2019b .

The Kerr-cat qubit has an asymmetric noise channel such that $Y$ and ${Z}$
errors are exponentially suppressed. This asymmetry illustrates that the qubit
couples to the environment predominantly along the $x$-axis, while coupling
along the $y$ and $z$-axis is suppressed. This coupling asymmetry, also
evident from Eq. (24), results from the Hilbert-space structure of the
Hamiltonian of the driven Kerr-nonlinear resonator. Consequently, in order to
allow coupling to the $z$-axis, it becomes necessary to turn-off the two-
photon pump. When this pump is turned off, the cat-qubit freely evolves under
the Kerr-nonlinearity and a $Z(\pi/2)$ gate is realized after a duration
$\pi/2K$ Yurke1988 ; Kirchmair2013 ; Grimm2020 . After finishing the
operation, the two-photon pump can be turned on again in order to recover the
Kerr-cat qubit. It is important to note that unlike the $X(\theta)$ and
$XX(\theta)$ gates, a $Z(\pi/2)$ rotation propagates a $X$ error as a linear
combination of $X$ and $Y$ errors and consequently destroys the underlying
asymmetric noise structure of the qubit Puri2019b .

Remarkably, recent theory shows that it is possible to realize a two-qubit,
controlled-Z (CZ) gate without turning off the two-photon drive and thereby
preserving the structure of the noise bias (termed as CX gate in Puri2019b
due to the different bases adopted there). The ability to implement a bias-
preserving CZ gate makes the Kerr-cat qubits desirable for efficient quantum
error correction. Moreover, the CZ gate can be implemented with parametric
drives and four-wave mixing via the inherent Kerr-nonlinearity in the cat-
qubit mode, which is very convenient to realize as no additional coupling
elements are required.

##  V Holonomic quantum control

In the last section, we have shown that through Markovian reservoir
engineering, the Lindbladian dynamics can be designed to support a mult-
dimensional stabilized manifold or decoherence-free subspace to encode the
quantum information without suffering dissipation, and it is also possible to
realize univeral control of the states in the stabilized manifold with the aid
of the quantum Zeno dynamics. In this section, we will show that the universal
control of the states in the stabilized manifold can be achieved by an
alternative method - holonomic quantum control.

In holonomic quantum computation (HQC) Lidar2013 ; Zanardi1999 ; Pachos1999 ,
the qubit states undergo adiabatic closed-loop parallel transport in parameter
spaces, acquiring Abelian Berry phases Berry1984  or non-Abelian adiabatic
quantum holonomies Wilczek1984  to achieve noise-resistant universal
computation. Recently Albert et al. Albert2016a ; Albert2016b  introduced the
idea of HQC to Markovian reservoir engineering and found that universal
computation of a quantum system consisting of superpositions of well-separated
coherent states of single or multiple harmonic oscillators can be achieved by
three families of adiabatic holonomic gates, including the loop gates,
collision gates for single oscillator mode and controlled-phase gates for
multiple harmonic oscillators. Below we will briefly introduce the first two
family of gates.

Consider the following Lindbladian for a single oscillator,

$\displaystyle\dot{\rho}=\mathcal{D}\left[\kappa\prod_{\nu=0}^{d-1}\left(a-\alpha_{\nu}(t)\right)\right]\rho.$
(25)

which is a generalization of Eq.(IV.1.1) that supports the stabilized manifold
spanned by a set of coherent states
$\\{|\alpha_{0}(t)\rangle,\cdots,|\alpha_{d-1}(t)\rangle\\}$. Note that
different from the constant parameters $\\{\alpha_{\nu}\\}$ in the last
section, $\\{\alpha_{\nu}(t)\\}$ here is time-dependent and can be tuned as
external parameters. Then by adiabatically changing $\\{\alpha_{\nu}(t)\\}$
through closed paths in phase space, the stable coherent states
$\\{|\alpha_{\nu}(t)\rangle\\}$ also undergo the same adiabatic evolutions.

Figure 7:  Holonomic gates for the logical subspace spanned by
$\\{|C_{\alpha}^{+}\rangle,|C_{\alpha}^{-}\rangle\\}$. (a) Wigner function
sketch of the state before (top) and after (bottom) a loop gate acting on
$|-\alpha\rangle$, depicting the path of $|-\alpha\rangle$ during the gate
(blue) and a shift in the fringes between. (b) Phase space diagram for the
loop gate with $X=\langle a+a^{\dagger}\rangle/2$ and $P=-i\langle
a-a^{\dagger}\rangle/2$. The parameter $\alpha_{1}(t)$ is varied along a
closed path (blue) of area $A$, and the state $|-\alpha\rangle$ gains a phase
$\theta=2A$ relative to $|\alpha\rangle$. (c) Effective Bloch sphere of the
cat qubit $|\pm\alpha\rangle$ depicting the rotation caused by the loop gate.
The black arrow depicts the initial state while the red arrow is the state
after application of the gate. The dotted blue arrow does not represent the
path traveled since the states leave the logical space during the gate.
(d)-(f) Analogous descriptions of the collision gate, which consists of
reducing $\alpha$ to 0, driving back to $\alpha e^{i\phi}$, and rotating back
to $\alpha$. Reprinted with permission from Albert2016a . .

###  V.1 Loop gates

One type of the holonomic control is the loop gate, which can accumulate
tunable relative Berry phases over superpositions of stabilized coherent
states. First consider the simple case with $d=2$ (e.g., the single-mode two-
photon process), the steady state space is
$\\{|\alpha_{0}(t)\rangle,|\alpha_{1}(t)\rangle\\}$ with
$\alpha_{0}(0)=-\alpha_{1}(0)=\alpha$ (Fig. 7a). This stabilized manifold
holds the even-odd cat qubit
$\\{|C_{\alpha}^{+}\rangle,|C_{\alpha}^{-}\rangle\\}$. Suppose that
$\alpha_{1}(t)$ undergoes an adiabatic variation through a closed path while
$\alpha_{0}(t)$ is kept constant and well separated from $\alpha_{1}(t)$ (Fig.
7a, b), the state $|\alpha_{1}(t)\rangle$ will accumulate a Berry phase
$\theta=2A$ with $A$ being the area enclosed by the closed path. Such an
operation is called a loop gate with implemented unitary

$\displaystyle U_{\rm loop}$
$\displaystyle=e^{i\theta/2}\exp[-i\theta(|\alpha\rangle\langle\alpha|-|-\alpha\rangle\langle-\alpha|)/2]$
$\displaystyle=e^{i\theta/2}\exp[-i\theta(|C_{\alpha}^{+}\rangle\langle
C_{\alpha}^{-}|+|C_{\alpha}^{-}\rangle\langle C_{\alpha}^{+}|)/2],$ (26)

which performs an $x$-axis rotation for the even-odd cat qubit (Fig. 7c). For
the general case with an arbitrary $d$, the loop gates consist of an adiabatic
evolution of $\alpha_{\nu}(t)$ around a closed path isolated from all the
other $\alpha_{\nu^{\prime}}(t)$ ($\nu^{\prime}\neq\nu$).

###  V.2 Collision gates

The other type of the holonomic control is the collision gate, which can
coherently convert the population of a stabilized coherent state to another.
To get the idea of collision gates, notice that there are two distinct
parameter regimes for the even-odd cat cat states: $\alpha\gg 1$ and
$\alpha\ll 1$. In the regime $\alpha\gg 1$, the cat states
$|C_{\alpha}^{\pm}\rangle$ are well separated and nearly orthogonal. However,
in the regime $\alpha\ll 1$, the cat states are reduced to the Fock states
with $|C_{\alpha}^{+}\rangle\rightarrow|0\rangle$ and
$|C_{\alpha}^{-}\rangle\rightarrow|1\rangle$, so a bosonic rotation
$R_{\phi}=\exp(i\phi a^{\dagger}a)$ will make the two Fock states $|0\rangle$
and $|1\rangle$ accumulate a relative phase $\phi$. If we start with the even-
odd cat qubit $|C_{\alpha}^{\pm}\rangle$ with large $\alpha$, first reduce
$\alpha$ to $0$, then apply bosonic rotation $R_{\phi}$ and final drive back
from $0$ to $\alpha$, the net result is that $|C_{\alpha}^{+}\rangle$ and
$|C_{\alpha}^{-}\rangle$ accumulate a relative phase $\phi$ with the
implemented unitary

$\displaystyle U_{\rm coll}$
$\displaystyle=e^{i\phi/2}\exp[-i\phi(|C_{\alpha}^{+}\rangle\langle
C_{\alpha}^{+}|-|C_{\alpha}^{-}\rangle\langle C_{\alpha}^{-}|)/2]$
$\displaystyle=e^{i\phi/2}\exp[-i\phi(|\alpha\rangle\langle-\alpha|+|\alpha\rangle\langle-\alpha|)/2],$
(27)

which performs a $z$-axis rotation for the even-odd cat qubit (Fig. 7f).
Denote the nonunitary driving from $0$ to $\alpha e^{i\phi}$ as $S_{\phi}$,
then the collision gate can also be represented as
$S_{0}R_{\phi}S_{0}^{-1}=R_{\phi}(R_{\phi}^{\dagger}S_{0}R_{\phi})S_{0}^{-1}=R_{\phi}S_{\phi}S_{0}^{-1}$.
So an equivalent construction of the collision gate is reducing $\alpha$ to 0,
driving back to $\alpha e^{i\phi}$ and rotating back to $\alpha$ (Fig. 7d, e).
The generalization of the collision gate to an arbitrary $d$ is
straightforward: start with the
$\\{|\alpha\lambda_{\nu}\rangle\\}_{\nu=0}^{d-1}$ configuration with
$\lambda_{\nu}=e^{i2\pi\nu/d}$ and large enough $\alpha$, tune $\alpha$ to
zero (or close to zero), pump back to a different phase $\alpha e^{i\phi}$,
and rotate back to the initial configuration. In the cat state basis with
$\alpha\ll 1$, $|C_{\alpha}^{\mu{\rm mod}d}\rangle\rightarrow|\mu\rangle$ will
gain a phase proportional to its mean photon number Albert2016a .

Besides the adiabatic HQC approaches above, it is also possible to implement
nonadiabatic HQC based on shortcuts-to-adiabatic (STA) dynamics Guery2019 .
Recent theory shows that in the ultrastrong and deep-strong coupling regimes
of the Rabi model Kockum2019 , STA methods can generate arbitrary nonclassical
bosonic states and induce fast nonadiabatic gates in tens of nanoseconds
Chen2021a ; Chen2021b .

##  VI Multi-mode quantum control

Figure 8:  Design of eSWAP gate with circuit QED. (a) Schematic drawing of the
three-dimensional circuit QED system used to realize the quantum Fredkin gate
and eSWAP operations between two bosonic modes Alice and Bob. (b)
Decomposition of the Fredkin gate into two 50:50 beam-splitters and a
controlled phase shift (CPS). The CPS for one cavity mode is described by the
unitary $U_{\rm CPS}=|g\rangle\langle g|\otimes I_{\rm C}+|e\rangle\langle
e|\otimes e^{i\pi a^{\dagger}a}$, which can be realized through the dispersive
coupling between the cavity mode and an ancilla transmon. (c) Quantum circuit
to realize the eSWAP unitary between two bosonic modes controlled by an
ancilla transmon. Reprinted with permission from Gao2019 . Figure 9:  Modular
architecture and teleported CNOT gate with circuit QED. (a) Schematic drawing
of the modular quantum architecture for quantum computation. Modules are
represented as nodes of a quantum network, and coupling between modules is
generated through reconfigurable communication channels that can be enabled
(dark purple lines) or disabled (light purple lines). (b) Each module acts a
small quantum processor that is capable of high-fidelity operations among data
qubits (magenta) and communication qubits (cyan). Here two modules are shown
with data qubit D1, D2 and communication qubits C1, C2. (c) Quantum circuit
for teleported CNOT circuit between D1 and D2. (d) Experimental realization of
modular structure in circuit QED. The data qubit is realized by a high-$Q$
cavity (magenta), a communication qubit by a transmon qubit (cyan) and a
low-$Q$ readout resonator (black), and the communication channel by another
cavity with a bus mode. Reprinted with permission from Chou2018 .

In all the sections above, we have concentrated on the quantum control of
single bosonic mode, although sometimes we briefly mention the extension to
the multi-mode control. In this section, we will focus on the quantum control
of multiple bosonic modes, mainly how to entangle two bosonic modes, which is
a prerequisite for universal quantum computation. Recent experimental advances
in circuit QED to entangle cavity modes include preparation of two-mode cat
state Wang2016 , two-mode W state Chakram2020  and three-mode W state
Chakram2020 , on-demand state transfer and entanglement generation Axline2018
; Campagne-Ibarcq2018 ; Kurpiers2018 ; Zhong2018 ; Leung2018 , and the
realization of CNOT gate Rosenblum2018a , controlled-Z (CZ) gate Xu2020 ,
exponential-SWAP (eSWAP) gate Gao2019  and teleported CNOT gate Chou2018 .
Below we will introduce the eSWAP gate and teleported CNOT gate.

###  VI.1 Exponential-SWAP gate

The eSWAP gate can coherently transfer the states between two bosonic modes,
regardless of the choice of encoding Lau2016 . To illustrate the eSWAP gate,
we first introduce the unitary SWAP operation $S_{ij}$ between any two bosonic
modes $a_{i}$, $a_{j}$, defined as $S_{ij}a_{i}S_{ij}^{\dagger}=a_{j}$ and
$S_{ij}a_{j}S_{ij}^{\dagger}=a_{i}$ (the same relation for $a_{i}^{\dagger}$,
$a_{j}^{\dagger}$). Applying the SWAP operation twice results in
$S_{ij}^{2}a_{i/j}S_{ij}^{\dagger 2}=a_{i/j}$, which is the identity operation
$I_{ij}$ for the two bosonic modes. The eSWAP gate is defined as the unitary
propagator induced by a Hamiltonian in the form of the SWAP operation

$\displaystyle U_{ij}(\theta)=\exp(i\theta S_{ij})=\cos\theta
I_{ij}+i\sin\theta S_{ij},$ (28)

which represents a superposition of the identity and the SWAP operation with
the superposition coefficient tunable by the rotation angle $\theta$. At
$\theta=\pi/2,\pi/4$, the eSWAP gate is reduced to the SWAP gate and
$\sqrt{\rm SWAP}$ gate, respectively. One powerful feature of the eSWAP gate
is that it can entangle two bosonic modes for any bosonic code. To see this,
suppose the qubit code for the bosonic mode is
$|0_{L}/1_{L}\rangle_{i}=f_{0/1}(a_{i}^{\dagger})|0\rangle$ with
$f_{0/1}(a_{i}^{\dagger})$ being a function of $a_{i}^{\dagger}$ and
$|0\rangle_{i}$ the vacuum state for the $i$th bosonic mode, and the initial
state of the $i$th and $j$th bosonic modes is
$|0_{L}\rangle_{i}|1_{L}\rangle_{j}$, then applying the eSWAP gate results in
$U_{ij}(\theta)|0_{L}\rangle_{i}|1_{L}\rangle_{j}=\cos\theta|0_{L}\rangle_{i}|1_{L}\rangle_{j}+i\sin\theta|1_{L}\rangle_{i}|0_{L}\rangle_{j}$
[Note that
$S_{ij}f_{0/1}(a_{i}^{\dagger})S_{ij}^{\dagger}=f_{0/1}(a_{j}^{\dagger})$ and
$S_{ij}|0\rangle_{i}|0\rangle_{j}=|0\rangle_{i}|0\rangle_{j}$].

To implement the eSWAP operations, one can first use an ancilla qubit with
states $\\{|g\rangle,|e\rangle\\}$ coupled to the two bosonic modes to realize
the controlled-SWAP or Fredkin gate $C_{ij}=|g\rangle\langle g|\otimes
I_{ij}+|e\rangle\langle e|\otimes S_{ij}$. The Fredkin gate can be decomposed
as (Fig. 8b)

$\displaystyle
C_{ij}=e^{-\frac{\pi}{4}(a_{i}a_{j}^{\dagger}-a_{j}a_{i}^{\dagger})}e^{i\frac{\pi}{2}|e\rangle\langle
e|a_{i}^{\dagger}a_{i}}e^{\frac{\pi}{4}(a_{i}a_{j}^{\dagger}-a_{j}a_{i}^{\dagger})},$
(29)

where the first and last unitaries are 50:50 beam splitters and the middle one
is the controlled-phase shift (CPS) of one bosonic mode conditioned on the
ancilla state. The CPS operation can be achieved by a dispersive coupling
between the ancilla and the bosonic mode [Eq. (1)]. Then the eSWAP gate can be
realized as (Fig. 8c)

$\displaystyle
U_{ij}(\theta)|+\rangle|\psi\rangle_{ij}=C_{ij}X_{\theta}C_{ij}|+\rangle|\psi\rangle_{ij},$
(30)

where $|+\rangle=(|g\rangle+|e\rangle)/\sqrt{2}$,
$X_{\theta}=e^{i\theta(|g\rangle\langle e|-|e\rangle\langle g|)}$ and
$|\psi\rangle_{ij}$ is the wavefunction for the two bosonic modes.

Recently Gao et al. Gao2019  have experimentally implemented the eSWAP
operations in three-dimensional (3D) circuit QED system (Fig. 8a) and
demonstrated high-quality deterministic entanglement between two cavity modes
with several different encodings including the Fock- and coherent-state coding
schemes. As opposed to the eSWAP gate, a traditional CNOT gate between the
multiphoton qubits in two cavities has also be realized by the mediation of a
driven ancilla transmon, with the driving pulse obtained from GRAPE optimal
control algorithm Rosenblum2018a . Moreover, a geometric method has been
utilized for realizing CZ gates between two logical qubits encoded in two
cavities Xu2020 .

###  VI.2 Teleported CNOT gate

A promising strategy toward scalable quantum computation is to adopt a quantum
modular architecture (Fig. 9a), which is a distributed network of modules that
communicate with one another through quantum and classical channels Kimble2008
; Monroe2014 . Each module is composed of two functional subsystems (Fig. 9b):
the data qubits that store and process quantum information and the
communication qubits that mediate interactions between different modules. The
intra-modular operations between the data and communication qubits are
performed independently in each module so that the crosstalk and residual
interactions between different modules are minimized even for a scaled-up
system, while the inter-modular operations between the data qubits are enabled
by distributing entanglement between communication qubits.

Due to the isolation between different modules, the multi-qubit operations
between modules cannot depend on direct interactions but instead utilize
quantum teleported gates Eisert2000 ; Duan2004 ; Gottesman1999 ; Jiang2007
that are enabled by entanglement sharing, local operations and classical
communications. Consider two modules with the data qubits (D1 and D2) and
communication qubits (C1 and C2), the teleported CNOT gate between D1 and D2
can be implemented by the following steps (Fig. 9c): (1) generation of
entanglement in the communication qubits C1 and C2, (2) local operations
performed within each module entangle the data and communication qubits, (3)
measurement of C1 in the Pauli-$Z$ basis and C2 in the Pauli-$X$ basis and (4)
classical communication and feedforward operations.

Recently Chou et al. Chou2018  have experimentally realized such a teleported
CNOT gate in a deterministic way in cicuit QED. The experimental architecture
consists of two modules (Fig. 9d). Each module consists of a high-$Q$ 3D
electromagnetic cavity as the data qubit, a transmon qubit as the
communication qubit and a Purcell-filtered, low-$Q$ stripline resonator for
readout of the transmon qubit. The local operations on the data cavity mode in
each module were realized by the optimal control pulses obtained by GRAPE
method (see Sec. IIB). The communication channel was realized by an additional
cavity mode that functions as a quantum bus coupling to both communication
qubits in the two modules. With the first-level bosonic binomial quantum code
Michael2016 , the teleported CONT gate was implemented deterministically with
the process fidelity reaching 79%.

##  VII Summary and outlook

Encoding quantum information in bosonic modes is a hardware-efficient approach
to quantum computation, and universal quantum control of the bosonic modes is
a crucial step towards this goal. Here we have given an extensive account of
the recent advances in universal control of the bosonic modes. Although the
approaches in this review were initially developed in the context of circuit
QED, they can be extended to various other platforms, such as cavity QED
Mabuchi2002 , trapped ions Leibfried2003 , nanophotonics Tiecke2014  and
Rydberg atoms Signoles2014  in the strongly dispersive regime Schuster2007 .

We have shown that universal control of a single bosonic mode can be achieved
with the aid of an ancilla qubit. The SNAP gates of a harmonic oscillator
(cavity resonator) can be implemented by indirect control of a dispersively
coupled ancilla (transmon qubit), and the SNAP gates combined with
displacement operation are sufficient for universal control. We can even
construct arbitrary quantum channels for the oscillator by QND readout of the
ancilla and quantum feedback control. However, it is still an open problem to
find the optimal control of the qubit-oscillator system with minimized
expenditure of energy and resources Glaser2015 . Another problem with this
qubit-oscillator system is that the ancilla qubit usually suffer relaxation
and dephasing errors during the quantum gates and these ancilla errors may
propagate to the logical qubits in the oscillator and corrupt the quantum
information. We have shown recent theoretical and experimental advances in
this respect, including the theoretical discovery of PI gates Ma2019  and
experimental realization of FT parity measurement Rosenblum2018b  and PI SNAP
gates Reinhold2019  in circuit QED.

Universal quantum control can also be achieved in some noise-resilient
subspace of the bosonic modes. With the aid of reservoir engineering or
Hamiltonian engineering, the bosonic modes may support some multi-dimensional
decoherence-free subspace to encode quantum information. Applying appropriate
drive can implement the desired unitary on this stabilized manifold allowed by
quantum Zeno dynamics. Universal control in such stabilized manifold can also
be achieved by holonomic quantum control, where the external parameters are
tuned so that the stable states undergo some adiabatic evolutions. Recent
experimental advances include the formation of stabilized manifold in two-
photon process Leghtas2015 ; Grimm2020 , Rabi population oscillations in such
a manifold Touzard2018 ; Grimm2020  and the formation and control of a Kerr-
cat qubit Grimm2020 . However, it is still challenging to experimentally
generate desired engineered dissipation that is much stronger than the
undesired dissipations. Moreover, it remain unsolved to systematically extract
high-order nonlinear Hamiltonian of the oscillator, in order to support high-
dimensional steady state subspaces.

Apart from universal control of single bosonic modes, coupling different
bosonic modes is also needed for universal quantum computation. We have
introduced two approaches to entangling two bosonic modes with recent
experimental realizations: the eSWAP gate independent of the bosonic encoding
Gao2019  and the teleported CNOT gate for a modular architecture Chou2018 . It
is interesting to further design some robust generalization of eSWAP gates
that are FT to the ancilla errors and bosonic loss errors. Moreover, the
teleported CNOT gate has only been realized for adjacent modules, and it will
be the next milestone to demonstrate the non-local teleported gates using
spatially separate modules.

##  VIII Acknowledgements

We acknowledge support from the ARO (W911NF-18-1-0020, W911NF-18-1-0212), ARO
MURI (W911NF-16-1-0349), AFOSR MURI (FA9550-19-1-0399), NSF (EFMA-1640959,
OMA-1936118, EEC-1941583), NTT Research, the Packard Foundation (2013-39273),
and the Startup Foundation of Institute of Semiconductors, Chinese Academy of
Sciences (No. E0SEBB11).

##  IX Author contributions

Liang Jiang, Wenlong Ma and S. M. Girvin organized the manuscript. Wenlong Ma
and Liang Jiang carried out the literature search and wrote most parts of the
manuscript. Shruti Puri wrote the part in Sec. IV-B. S. M. Girvin, Michel H.
Devoret and Robert J. Schoelkopf contributed to the manuscript revision. All
authors contributed to the discussion.

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