# Gauge invariant quantum thermodynamics: consequences for the first law

Lucas Chibebe Céleri<EMAIL_ADDRESS>Institute of Physics, Federal
University of Goiás, 74.690-900 Goiânia, Goiás, Brazil  Łukasz Rudnicki
<EMAIL_ADDRESS>International Centre for Theory of Quantum
Technologies (ICTQT), University of Gdańsk, 80-308 Gdańsk, Poland

###### Abstract

Universality of classical thermodynamics rests on the central limit theorem,
due to which, measurements of thermal fluctuations are unable to reveal
detailed information regarding the microscopic structure of a macroscopic
body. When small systems are considered and fluctuations become important,
thermodynamic quantities can be understood in the context of classical
stochastic mechanics. A fundamental assumption behind thermodynamics is
therefore that of coarse-graning, which stems from a substantial lack of
control over all degrees of freedom. However, when quantum systems are
concerned, one claims a high level of control. As a consequence, information
theory plays a major role in the identification of thermodynamic functions.
Here, drawing from the concept of gauge symmetry, essential in all modern
physical theories, we put forward a new possible, intermediate route. Working
within the realm of quantum thermodynamics we explicitly construct physically
motivated gauge transformations which encode a gentle variant of coarse-
graining behind thermodynamics. As a consequence, we reinterpret quantum work
and heat, as well as the role of quantum coherence.

###### pacs:

03.67.Mn, 03.67.Lx, 42.50.Dv

Classical thermodynamics is based on a fundamental fact that we do not have
access to microscopic degrees of freedom of a macroscopic system. When we take
measurements, what we see is an average over space and time since our clocks
and rules are not able to follow the underlying complex dynamics of the
system. From this low ability of controlling the system at hand, also called
coarse-graining, thermodynamic variables emerge. This is the well known
thermodynamic limit Callen1985 . Leaving such limit and diving towards smaller
and smaller scales where fluctuations become important, we still expect the
laws of classical thermodynamics to hold on average, and statistical mechanics
can be employed in order to properly describe the system Sekimoto2010 ;
Seifert2012 ; Jarzynski1997 ; Crooks1999 .

Diving even deeper, we enter the realm of quantum mechanics, where
thermodynamics acquires a different flavor. Here the focus is often on systems
with a few degrees of freedom, subject to a high level of control. In an
extreme case, quantum state or process tomography can be performed, revealing
full information about density matrices or quantum gates. In this context
information theory starts playing a crucial role Horodecki2003 ;
Skrzypczyk2014 ; Bera2017 ; Goold2016  in the sense that thermodynamic
quantities like heat, work and entropy are operationally defined in terms of
informational ones Goold2016 .

Figure 1: Gauge theory of quantum thermodynamics. Quantum description of
systems and their dynamics, taken together with the limited access to the
details, characteristic to thermodynamics, do form an intermediate theory:
quantum thermodynamics subject to an emergent gauge symmetry.

Therefore, quantum and classical thermodynamics both rely on distinct
paradigms. While classical thermodynamics rests on the fundamental lack of
control over the system’s degrees of freedom, quantum thermodynamics requires
that the very details of the system’s evolution are at least accessible. For
example, limits to heat engines’ efficiency, tighter than the famous Carnot
bound, do rely on detailed information about quantum fluctuations during the
controlled Lindblad dynamics Pietzonka ; Miller .

Here we propose a new route towards quantum thermodynamics, which at the same
time aims to keep the spirit of classical thermodynamics. Therefore, our
approach situates itself in between the two paradigms —see Fig. 1. Our
approach is driven by the concept of gauge invariance, a key principle
underlying all modern physical theories Cabibbo2017 . In fact, even before the
development of quantum theory, symmetry considerations were increasingly seen
as a key agent in determining the form of the fundamental physical laws
Gross1996 . The basic idea behind this principle is that physical quantities
are invariant under certain symmetry transformations of the system. A well
known example is the Abelian $\mathcal{U}(1)$ group that describes the gauge
symmetry of electrodynamics Jackson2001 . Our aim is to build the symmetry
group behind quantum thermodynamics, such that the mean energy is invariant
under the group action.

There is, however, a crucial difference between the gauge symmetry of
classical and quantum field theories, and the one we identify behind
thermodynamics. In the former case the fundamental gauge transformations take
us from potentials to physically meaningful quantities such as scattering
amplitudes Cabibbo2017 . In our theory, the gauge takes us from information
theory to the thermodynamic quantities. In this sense, we rather speak about
an emergent gauge theory. Fig. 2 illustrates this difference.

Since we construct our theory with reference to mean energy, we infer
conclusions relevant for the first law of (quantum) thermodynamics. In
particular, we construct notions of invariant work and heat. Moreover, our
framework allows to clarify the role of quantum coherences, because the
resulting, gauge invariant notion of heat covers delocalized energy, thus
being present even in closed quantum systems. This result, even though not met
in a standard treatment Campisi2011 ; Landi2020 , is in fact quite intuitive,
since the generation of coherences in the energy eigenbasis implies that the
energy is not defined for a quantum system.

Thermodynamic gauge

As mentioned in the introduction, the gauge transformations that we introduce
are not of a fundamental kind. In field theories, the potentials do not have
physical meaning in the sense that they cannot be measured. In other words,
such types of the gauge are fundamental, coming from the redundancy in the
description of the relevant degrees of freedom. Considering thermodynamics,
carriers of quantum information play the same role as the potentials do, while
thermodynamic quantities are considered as the physically relevant ones. The
gauge, in this sense, is not fundamental since all thermodynamic quantities
emerge from a microscopic description by a process of coarse-graining. The
gauge emerges from this process. At the same time, the microscopic description
is physically meaningful, and experimentally accessible. Though, not from the
perspective of standard thermodynamics.

Figure 2: Emergent thermodynamic gauge. The panel on the left illustrates the
fundamental gauge behind field theories, both classical and quantum. The set
of potentials $A_{\mu}(x)$ does not have any physical meaning and the gauge
transformations takes us from this set to the scattering amplitudes that can
be experimentally investigated. The right panel represents the emergent
thermodynamic gauge $\mathcal{G}$ proposed in this work. Such transformations
take us from the set of information carriers $\rho_{t}$, that convey no
thermodynamic meaning, to the set of thermodynamically meaningful quantities.
The gauge emerges from the coarse-graining nature of thermodynamics, being, in
this sense, not fundamental as the one behind field theories.

We start by specifying the notation. Let $H_{t}$ be a (possibly time
dependent) Hamiltonian of our system, and let $\rho_{t}$ be a density matrix
describing it. The latter object, considered in its entity, is here ascribed
to the domain of information theory.

In a thermodynamic, coarse-grained description, we are only interested in a
particular set of quantities, such as the mean energy
$U\left[\rho_{t}\right]=\textrm{Tr}\rho_{t}H_{t}$. While the energy plays a
prominent role here, we expect that other quantities might also be of
relevance. We assume they all can be expressed as functionals
$F\left[\rho_{t}\right]$ depending on the density matrix. An example of such a
quantity, being of particular relevance for this contribution, is the work
associated with a given process that takes place during the time interval
$t\in[0,\tau]$, defined as Alicki1979

$W\left[\rho_{t}\right]=\int_{0}^{\tau}dt\textrm{Tr}\left[\rho_{t}\dot{H}_{t}\right],$
(1)

where the dot represents differentiation with respect to time. The functionals
$F\left[\rho_{t}\right]$, even though we do not write that explicitly, can as
well depend on time derivatives of the density matrix. The most obvious
example, relevant for the above context, is the usual notion of heat, being
defined as the difference between energy and work Alicki1979 ; Kieu2004 .

In fundamental physics, gauge is realized as an action of a gauge group.
Elements of this group act on variables describing a given theory
(potentials), such as the vector potential in electrodynamics. In our
parallel, the role of the potentials is played by carriers of quantum
information, namely, the density matrices $\rho_{t}$. Therefore, emergent
gauge transformations need to be represented as (some) unitary matrices
$V_{t}$ acting as follows $\rho_{t}\mapsto V_{t}\rho_{t}V_{t}^{\dagger}$. The
emergent gauge is allowed to be time dependent, the same way as the standard
one in field theories depends on space and time. As a consequence, the time
derivative of the density matrix transforms as $\dot{\rho}_{t}\mapsto
V_{t}\dot{\rho}_{t}V_{t}^{\dagger}+\dot{V}_{t}\rho_{t}V_{t}^{\dagger}+V_{t}\rho_{t}\dot{V}_{t}^{\dagger}$,
again resembling familiar transformation patterns.

Moving back and forth between the notions of fundamental and emergent gauges
we now go back to the former one, reminding the reader that only the
quantities which are gauge invariant are considered as being measurable and
physically meaningful. Within the framework of the emergent gauge in
thermodynamics, we adopt the same philosophy, namely, we require
thermodynamically meaningful quantities to be invariant with respect to the
proposed gauge. In other words, we take the full information content stored in
$\rho_{t}$ and extract just the _robust_ pieces of this information. Clearly,
not all unitary transformations can be allowed to render the emergent gauge
transformations, as otherwise the theory would trivialize too much. To select
a desired set of transformations we will build on the primary notion of our
framework, namely, the mean energy which has to be unambiguously gauge
invariant. We therefore define:

######  Definition 1 (Emergent thermodynamic gauge)

Unitary transformations are admissible gauge transformations if they preserve
the mean energy, independently of a particular state of the system, i.e.

$U\left[V_{t}\rho_{t}V_{t}^{\dagger}\right]=U\left[\rho_{t}\right],$ (2)

for all $\rho_{t}$, where $U\left[\rho_{t}\right]=\mathrm{Tr}\rho_{t}H_{t}$.

By virtue of the above definition we are in position to explicitly construct
the gauge. The invariance of the mean energy of any state requires that
$\textrm{Tr}\left[\rho_{t}\left(V_{t}^{\dagger}H_{t}V_{t}-H_{t}\right)\right]=0$
independently of $\rho_{t}$, which simply means that the unitary operations
need to commute with the Hamiltonian, i.e.
$\left[V_{t},H_{t}\right]=0=\left[V_{t}^{\dagger},H_{t}\right]$. Therefore,
thermodynamic gauge transformations are simply unitary channels which commute
with the Hamiltonian. It is important to observe that our emergent gauge is
not equivalent to the well known thermal operations employed in the resource
theory of thermodynamics as free transformations Brandao2013 ; Chitambar2019 .
The emergent gauge rather forms a very particular subclass of the latter
operations which, importantly, are not attributed to additional physical
interpretation.

While exploring the structure of the gauge, we first assume that we work with
a $d$-dimensional system. Then, $V_{t}\in\mathcal{U}\left(d\right)$, with
$\mathcal{U}\left(d\right)$ representing the set of $d\times d$ unitary
matrices. In fact, unitaries which commute with $H_{t}$, by definition form a
subgroup of $\mathcal{U}\left(d\right)$. In this way we again get a very close
contact with the fundamental gauge in field theories, as this subgroup can be
called the _emergent gauge group_.

Now, since the Hamiltonian is Hermitian, we can conveniently parametrize all
the members of the emergent gauge group. To this end we express the
Hamiltonian as

$H_{t}=u_{t}\left(\bigoplus_{k=1}^{p}\varepsilon_{k}(t){\rm
1l}_{n_{k}}\right)u_{t}^{\dagger}\equiv u_{t}h_{t}u_{t}^{\dagger},$ (3)

where $p$ is the number of its distinct eigenvalues, $n_{k}$ is a multiplicity
of each $\varepsilon_{k}(t)$, such that $\sum_{k=1}^{p}n_{k}=d$, and
$u_{t}\in\mathcal{U}\left(d\right)$. For the sake of clarity, we assume that
the degeneracy of the Hamiltonian does not change in time, however, below and
whenever appropriate, we pinpoint the consequences of an alternative scenario.
Detailed mathematical account of the latter is left for future work.

Every complex $d\times d$ matrix $M$ (not necessarily Hermitian or unitary)
commutes with (3) if and only if it is of the form

$M=u_{t}\left(\bigoplus_{k=1}^{p}M_{k}\right)u_{t}^{\dagger},$ (4)

where each $M_{k}$ is an $n_{k}\times n_{k}$ block with arbitrary matrix
elements. The constraint $\left[V_{t},H_{t}\right]=0$ therefore enforces

$V_{t}=u_{t}\left(\bigoplus_{k=1}^{p}v_{k}\right)u_{t}^{\dagger},$ (5)

with $v_{k}\in\mathcal{U}\left(n_{k}\right)$. From a topological point of
view, the emergent gauge group is isomorphic to

$\mathcal{G}=\mathcal{U}\left(n_{1}\right)\times\mathcal{U}\left(n_{2}\right)\times\ldots\times\mathcal{U}\left(n_{p}\right).$
(6)

As long as the degeneracy of $H_{t}$, i.e. the multiplicities of all its
eigenvalues do not change in time, the above structure remains intact. If the
degeneracy of $H_{t}$ changes, it just means that the topological structure in
Eq. (6) does change as well.

On the group $\mathcal{G}$ there is an invariant Haar measure induced from
that on $\mathcal{U}\left(d\right)$. Therefore, the group averaging allows one
to assign to each relevant quantity its gauge invariant version:

######  Definition 2 (Gauge invariant quantities)

Given a functional $F\left[\rho_{t}\right]$, its counterpart invariant with
respect to the emergent gauge is

$F_{\mathrm{inv}}\left[\rho_{t}\right]=\int
d\mathcal{G}F\left[V_{t}\rho_{t}V_{t}^{\dagger}\right],$ (7)

where $V_{t}$ is given in (5) and $d\mathcal{G}$ is the Haar measure on the
group $\mathcal{G}$ defined in Eq. (6).

It is important to remember that the Haar measure is normalized, i.e. $\int
d\mathcal{G}=1$. When the degeneracy of $H_{t}$ is not fixed, this normalized
measure just needs to be replaced by an appropriate (still normalized)
functional measure $\mathcal{D}\mathcal{G}_{t}$ where $\mathcal{G}_{t}$ is a
counterpart of (6), with all $\\{n_{k}\\}$ and $p$ being time-dependent.

Before proceeding to a particularly relevant case of work and heat, let us
point out that there is a natural class of quantities which are gauge
invariant "by construction". These are all quantities solely defined on the
equilibrium manifold, or even on a family of instantaneous equilibrium
operators $\rho_{t}^{\beta}\sim e^{-\beta H_{t}}$, with $\beta$ being the
inverse temperature. Also, other states which solely depend on the
Hamiltonian, e.g. the microcanonical state, are preserved.

Gauge invariant work and the role of coherence

It is an easy task to recognize that the average work defined in (1)
transforms as $W[\rho_{t}]\mapsto
W[\rho_{t}]+\int_{0}^{\tau}dt\textrm{Tr}\Omega_{t}$, where
$\Omega_{t}=\left[H_{t},\rho_{t}\right]V_{t}^{\dagger}\dot{V}_{t}$, being not
invariant with respect to the emergent gauge. This transformation rule has a
deep physical meaning, as the work is gauge invariant only when
$\left[H_{t},\rho_{t}\right]$ for all $t$, which means that the process is
adiabatic.

However, the general formalism presented above allows us to provide the notion
of work which is invariant with respect to the emergent gauge.

######  Theorem 1 (Gauge invariant work)

Let $h_{t}=u_{t}^{\dagger}H_{t}u_{t}$ be the diagonal form of the Hamiltonian
$H_{t}$, with $u_{t}$ unitary. Then the notion of work which is invariant with
respect to the emergent thermodynamic gauge is given by

$W_{\mathrm{inv}}[\rho_{t}]=\int_{0}^{\tau}dt\mathrm{Tr}\left[\rho_{t}u_{t}\dot{h}_{t}u_{t}^{\dagger}\right].$
(8)

It is straightforward to recognize that this assertion is true, namely, that
the right hand side is invariant. Nevertheless, we make a formal proof in the
Appendix. Importantly, the above result does also hold if the degeneracy of
$H_{t}$ changes in time, as potential contribution coming from that fact is of
zero measure with respect to the time integral. It is a bit more involved to
show that the invariant notion of work is the same as would be given by Eq.
(7). This fact is also demonstrated in the Appendix, with the help of the
group averaging techniques.

Taking the invariant notion of work together with primordial invariance of the
energy, the gauge invariant notion of heat $Q_{\mathrm{inv}}$ immediately
follows from energy conservation. The explicit form of the heat is given
below, where the role of quantum coherence is discussed in light of our
findings.

We represent the invariant heat as consisting of two independent
contributions, $Q_{\mathrm{inv}}=Q_{u}+Q_{c}$, where the usual heat reads
Kieu2004

$Q_{u}=\int_{0}^{\tau}dt\textrm{Tr}\left[\dot{\rho}_{t}H_{t}\right]\equiv\int_{0}^{\tau}dt\textrm{Tr}\left[\dot{\rho}_{t}u_{t}h_{t}u_{t}^{\dagger}\right],$
(9)

while the coherent heat is now being defined as

$Q_{c}=\int_{0}^{\tau}dt\textrm{Tr}\left[\rho_{t}\dot{u}_{t}h_{t}u_{t}^{\dagger}+\rho_{t}u_{t}h_{t}\dot{u}_{t}^{\dagger}\right].$
(10)

The quantity $Q_{u}$ has very frequently appeared in the literature as a
natural definition of heat Alicki1979 ; Kieu2004 ; Binder2018 ; Landi2020 .
The additional term $Q_{c}$, the presence of which distinguishes the invariant
heat with respect to the usual one, very interestingly, contains only
contributions from the coherences. This fact can be seen after writing it in
the energy eigenbasis.

Let $\left\\{\left|a_{k}\left(t\right)\right\rangle\right\\}$ and
$\left\\{\varepsilon_{k}\right\\}$ be the set of time-dependent energy
eigenstates and eigenvalues, respectively. For simplicity, we for a moment
assume that the Hamiltonian is non-degenerate, i.e. all $n_{k}=1$.
Qualitatively the same results can however be obtained without this
assumption. Since we are talking about work, heat and thermodynamics in
general, the energy eigenbasis seems to be the most appropriate one. At any
instant of time, the density matrix of the system can be represented in this
basis as
$\rho_{t}=\sum_{jl}c_{jl}\left(t\right)\left|a_{j}\left(t\right)\right\rangle\left\langle
a_{l}\left(t\right)\right|$.

In the Appendix we show that $Q_{c}$ depends only on the differences
$\varepsilon_{j}\left(t\right)-\varepsilon_{k}\left(t\right)$ and,
consequently, all the terms containing the diagonal coefficients of the
density matrix, $c_{ii}(t)$, do vanish. This is a universal result stating
that the development of internal coherences are part of the heat.

It is interesting to observe that work and (total) heat are independent of the
coherences in the energy eigenbasis. The average total energy can be written
as
$U_{t}\left[\rho_{t}\right]=\sum_{k}c_{kk}\left(t\right)\varepsilon_{k}\left(t\right)$,
while the invariant work takes the form
$W_{\mathrm{inv}}=\int_{0}^{\tau}\differential
t\sum_{k}c_{kk}\left(t\right)\dot{\varepsilon}_{k}\left(t\right)$ and the heat
is written as
$Q_{\mathrm{inv}}=\sum_{k}\dot{c}_{kk}\left(t\right)\varepsilon_{k}\left(t\right)$.
Such results are intuitively expected and are fully compatible with our
classical notions of work and heat, in agreement with the strategy adopted
while proposing the gauge invariant theory of quantum thermodynamics. While
the standard quantum definitions of the work and the heat involve terms with
coherences, our gauge invariant approach provides fully coherence-independent
thermodynamic quantities.

Examples

For the sake of illustration, we consider three examples. In the first two,
usual notions of work and heat are recovered (which means that for the models
considered they coincide with the gauge invariant ones), while the
contributions from coherences becomes clear due to the third example.

Let us first discuss a single qubit, described by the Hamiltonian
$H_{t}=\alpha_{t}\sigma_{z}$, undergoing a dissipative process described by
generalized amplitude damping or phase damping channels Nielsen2011 . These
processes can be described by Lindblad master equations of the general form
$\dot{\rho}_{t}=-i\commutator{H_{t}}{\rho_{t}}+\mathcal{D}[\rho_{t}]$, with
$\mathcal{D}$ being the non-unitary part of the dynamics. Since the
Hamiltonian is already diagonal in the energy eigenbasis, we take $u_{t}={\rm
1l}$, implying that $\dot{h}_{t}=\dot{\alpha}_{t}\sigma_{z}$. This leads to
the following result for the invariant work

$W_{\mathrm{inv}}=\int_{0}^{\tau}\differential
t\dot{\alpha}_{t}\expectationvalue{\frac{\partial
H_{t}}{\partial\alpha_{t}}},$ (11)

for both channels. Of course, $W_{\mathrm{inv}}=0$ if $\alpha_{t}$ is time
independent. Let us now move to the invariant heat. Since $u_{t}$ is time
independent, we have $Q_{\mathrm{inv}}=Q_{u}$, resulting in

$Q_{\mathrm{inv}}=\int_{0}^{\tau}\differential
t\tr{\mathcal{D}[\rho_{t}]H_{t}}.$ (12)

As in detail described in the Appendix, for the case of the dephasing channel
we obtain $Q_{\mathrm{inv}}^{\mathrm{deph}}=0$. On the other hand, in the case
of the generalized amplitude damping the invariant heat takes the form
$Q_{\mathrm{inv}}^{\mathrm{gad}}=\int_{0}^{\tau}\differential t\Phi_{E}^{t}$,
with $\Phi_{E}^{t}$ being the instantaneous energy flux. Since the dephasing
channel only involves exchange of phase information between the system and the
environment, while the amplitude damping is characterized by energy exchange,
the above results agree with what is expected from classical thermodynamics.
It is therefore important to stress that our gauge invariant approach agrees
with the standard one in cases which are consistently described from a point
of view of classical thermodynamics. Thus, deviations between the two
approaches appear together with quantum effects. This can be seen in the third
example below, where the role of coherences is being exposed.

Let us now consider an externally driven single qubit whose Hamiltonian can be
written as $H_{t}=\sigma_{z}+\gamma_{t}\sigma_{x}$, where $\gamma_{t}$ is a
time-dependent driven frequency. In this case we will have coherences being
generated during the evolution. Fig. 3 shows the invariant heat and work,
along with the relative entropy of coherence Baumgratz2014  (see Appendix for
details).

Figure 3: Energies and coherence. The red dashed line represents the invariant
work $W_{\mathrm{inv}}$ while the solid black line is the coherence and the
dotted blue line is the invariant heat. We considered $\gamma_{t}=\cos(t)$ and
the ground state as the initial one. It is clear that the heat dynamics is
perfectly correlated with the coherence.

Note that this example is in deep contrast with the usual understanding met in
quantum thermodynamics, where it is assumed that a unitary evolution does not
involve heat Alicki1979 ; Kieu2004 ; Binder2018 ; Landi2020 . Here, heat is
linked with the delocalization of energy, which explains why it oscillates for
a closed evolution where coherences change in time.

Statistical work

An important issue when discussing quantum thermodynamics (or classical
thermodynamics of small systems) is the notion of statistical work.
Classically, when fluctuations become important we can use statistical
mechanics in order to properly describe thermodynamics Sekimoto2010 ;
Seifert2012 ; Jarzynski1997 ; Crooks1999 . A crucial assumption behind such
theory is that the system can be constantly and perfectly monitored, which is
clearly not true in quantum mechanics. Therefore, ambiguities in defining work
arise when we consider the thermodynamic properties of quantum systems. The
present article also deals with gauge invariant notion of average work, but
nothing has so far been said about the statistical work.

A widely accepted definition of statistical work is the two-point measurement
Talkner2007 ; Campisi2011 . An energy measurement $\Pi_{0}^{n}$, where we
define the projectors
$\Pi_{t}^{l}=u_{t}\left(\bigoplus_{k=1}^{p}\delta_{kl}{\rm
1l}_{n_{k}}\right)u_{t}^{\dagger}$ consistently with Eq. (3), is performed
before the work process, resulting in the energy $\varepsilon_{n}(0)$. After
the process, a second measurement $\Pi^{m}_{\tau}$ results in the energy
$\varepsilon_{m}(\tau)$. The stochastic work is then defined as
$W_{m,n}=\varepsilon_{m}(\tau)-\varepsilon_{n}(0)$. This is a random variable
associated with the probability distribution
$p_{m,n}=\mathrm{Tr}\left[\Pi^{m}_{\tau}U_{\tau}\Pi^{n}_{0}\rho_{0}\Pi^{n}_{0}U^{\dagger}_{\tau}\right]$,
where $U_{\tau}$ is such that
$\rho_{\tau}=U_{\tau}\rho_{0}U_{\tau}^{\dagger}$. This protocol has been
considered both theoretically Esposito2009 ; Campisi2011 ; Landi2020  and
experimentally Batalhao2014 ; Zhang2015 ; Cerisola2017 ; Gomez2020 ; Zanin2019
; Ribeiro2020 ; Peterson2016  in distinct contexts within quantum
thermodynamics. The probability distribution $p_{m,n}$ associated with the
work is a central quantity in the discussed scheme.

We observe that the emergent gauge transformations of $\rho_{t}$ induce the
following transformation of the process $U_{\tau}\mapsto
V_{\tau}U_{\tau}V_{0}^{\dagger}$. Since due to (5) we have
$V_{t}\Pi^{m}_{t}V_{t}^{\dagger}=\Pi^{m}_{t}$ for every $t$, we can
immediately see that $p_{m,n}$ is invariant with respect to the emergent
gauge. Therefore, our concept of emergent gauge invariance is fully compatible
with the two-point measurement and, thus, with the (classical and quantum)
fluctuations theorems.

Discussion

In the classical world the first law of thermodynamics tells us how to sharply
split the internal energy of a system into work and heat. The extension of
this result to the quantum realm typically relies on operational grounds.
Therefore, work is associated with a given unitary process while heat is
usually related with the irreversible flux of energy. In the current
contribution we take a distinct route, proposing a new theory of quantum
thermodynamics based on the principle of gauge invariance. We also address the
consequences of such theory to the first law of thermodynamics. The fact that
gauge theories are the most successful ones in describing nature (standard
model Weinberg2005 , general relativity Wald1994 , quantum matter Wen2004 ,
etc.) provides us a fresh ground on which quantum thermodynamics may be
considered.

Concerning the first law, the usual notions of heat and work are based on
Clausius idea. Heat is associated with the change in the thermodynamic entropy
while work is linked with the changes in some externally controlled parameter.
Such notions cannot be simply extended to the quantum world, thus leading to
several definitions of work and heat that are, in general, nonequivalent, but
operationally well defined Talkner2016 . Here enters information theory where
heat is usually associated with the change in the von Neumann entropy, while
work preserves the informational content of the system. The theory proposed
here lies in between such perspectives. We adopt the coarse-graining spirit of
classical thermodynamics in order to build the gauge theory of quantum
thermodynamics. This theory leads to gauge invariant definitions of
thermodynamic quantities, in particular, heat and work that are not solely
related with the change in the information entropy. Work is associated with
the engenvalues of the Hamiltonian while heat is connected with the change in
the eigenbasis of the Hamiltonian, thus being deeply linked with energy
delocalization. This implies that we can have heat even in a closed quantum
system, which is in sharp contrast with the usual notions of quantum heat
Alicki1979 ; Kieu2004 ; Binder2018 ; Landi2020 . In the adopted perspective
coherences in the energy eigenbasis indicate that energy is not defined and
its spread over the system space is interpreted as heat.

Regarding the notion of statistical work (and heat), the most considered
definition is based on the two-point measurement. Although widely accepted,
such scheme does not take into account the coherences, which are destroyed by
the measurements. Fully quantum fluctuation theorems were developed in order
to take into account this contribution Jevtic2015 ; Alhambra2016 ; Aberg2018 ;
Santos2019 ; Francica2019 ; Kwon2019 . A naturally arising question is to
scrutinize or generalize/modify the fluctuation theorems from the perspective
of the invariant work and heat presented in this work. The implications of
such definitions for the irreversible entropy production will certainly deepen
our understanding of quantum coherence in non-equilibrium thermodynamics.

Another question that can be addressed under this theory is the local
thermalization of closed quantum many-body systems and how this is linked with
the eigenstate thermalization hypothesis Deutsch2018  and irreversible entropy
production Landi2020 .

Perhaps the most important and broad message of the current work is relevant
for all aspects of quantum technologies associated with operational principles
behind potential quantum engines, batteries or refrigerators. Our approach
might in the future allow to delineate the boundaries between controlled
quantum systems and their thermodynamic behaviour, forming a bridge between
classical and quantum thermodynamics, which starts in the latter paradigm and
expands towards the former one.

Last but not least, since our theory is based on gauge invariance, it may also
pave investigations of quantum thermodynamics in the domains of quantum field
theory and general relativity.

Acknowledgments The authors are in debit to Gabriel Landi for his critical
reading of the manuscript. LCC would like to acknowledge the financial support
from funding agencies CNPq, FAPEG and the Brazilian National Institute of
Science and Technology of Quantum Information (INCT-IQ). This study was
financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior – Brasil (CAPES) – Finance Code 001. ŁR acknowledges support by the
Foundation for Polish Science (IRAP project, ICTQT, contract no. 2018/MAB/5,
co-financed by EU within Smart Growth Operational Programme).

##  Appendix A Gauge invariance of work

We first present a formal proof that $W_{\mathrm{inv}}$ defined in Eq. (8) in
the main text is invariant with respect to the emergent gauge transformations.
To this end we study

$W_{\mathrm{inv}}[V_{t}\rho_{t}V_{t}^{\dagger}]=\int_{0}^{\tau}dt\mathrm{Tr}\left[V_{t}\rho_{t}V_{t}^{\dagger}u_{t}\dot{h}_{t}u_{t}^{\dagger}\right].$
(13)

Substituting the explicit form of $V_{t}$ and $h_{t}$ we find

$\mathrm{Tr}\left[V_{t}\rho_{t}V_{t}^{\dagger}u_{t}\dot{h}_{t}u_{t}^{\dagger}\right]=\mathrm{Tr}\left[\rho_{t}u_{t}\left(\bigoplus_{k=1}^{p}v_{k}^{\dagger}\dot{\varepsilon}_{k}(t)v_{k}\right)u_{t}^{\dagger}\right].$
(14)

Since
$v_{k}^{\dagger}\dot{\varepsilon}_{k}(t)v_{k}=\dot{\varepsilon}_{k}(t){\rm
1l}_{n_{k}}$, we get the desired result.

In the next step we pass to the second assertion, namely, we study the
invariant work from the perspective of group averaging. To this end, we plug
(1) in the Definition 2, getting

$\int
d\mathcal{G}W\left[V_{t}\rho_{t}V_{t}^{\dagger}\right]=\int_{0}^{\tau}dt\textrm{Tr}\left[\rho_{t}u_{t}f_{t}u_{t}^{\dagger}\right],$
(15)

where

$f_{t}=\int
d\mathcal{G}\left(\bigoplus_{k=1}^{p}v_{k}^{\dagger}\right)u_{t}^{\dagger}\dot{H}_{t}u_{t}\left(\bigoplus_{k=1}^{p}v_{k}\right).$
(16)

Further we expand

$\displaystyle u_{t}^{\dagger}\dot{H}_{t}u_{t}$ $\displaystyle=$
$\displaystyle\dot{h}_{t}+h_{t}\dot{u}_{t}^{\dagger}u_{t}+u_{t}^{\dagger}\dot{u}_{t}h_{t}$
$\displaystyle=$
$\displaystyle\dot{h}_{t}+\left[h_{t},\dot{u}_{t}^{\dagger}u_{t}\right],$

where the last line follows from unitarity of $u_{t}$, which implies
$u_{t}^{\dagger}\dot{u}_{t}=-\dot{u}_{t}^{\dagger}u_{t}$. Therefore, using the
arguments spelled out previously, we find

$f_{t}=\dot{h}_{t}+\left[h_{t},b_{t}\right],$ (17)

where

$b_{t}=\int
d\mathcal{G}\left(\bigoplus_{k=1}^{p}v_{k}^{\dagger}\right)\dot{u}_{t}^{\dagger}u_{t}\left(\bigoplus_{k=1}^{p}v_{k}\right).$
(18)

Due to the fundamental property of unitary group averaging we know that
$b_{t}=\bigoplus_{k=1}^{p}b_{k}\left(t\right){\rm 1l}_{n_{k}}$, for some
functions $b_{k}(t)$. Therefore, we find that $[h_{t},b_{t}]=0$, so the group
averaged work is the same as the invariant work proposed in Theorem 1.

Finally, let us consider the heat and its connections to quantum coherence. In
the energy eigenbasis the coherent heat Eq. (10) reads

$\displaystyle
Q_{c}=\int_{0}^{\tau}dt\sum_{jl}c_{jl}(t)\left[\varepsilon_{j}\left(t\right)\left\langle
a_{l}\left(t\right)\left|\dot{a}_{j}\left(t\right)\right\rangle\right.\right.$
(19)
$\displaystyle\left.+\varepsilon_{l}\left(t\right)\left\langle\dot{a}_{l}\left(t\right)\left|a_{j}\left(t\right)\right\rangle\right.\right].$
(20)

Since for all times we have

$\left\langle
a_{l}\left(t\right)\left|a_{j}\left(t\right)\right\rangle\right.=\delta_{lj},$
(21)

we also consequently get

$\left\langle
a_{l}\left(t\right)\left|\dot{a}_{j}\left(t\right)\right\rangle\right.+\left\langle\dot{a}_{l}\left(t\right)\left|a_{j}\left(t\right)\right\rangle\right.=0.$
(22)

Therefore, we obtain the final result

$Q_{c}=\int_{0}^{\tau}dt\sum_{jl}c_{jl}\left(t\right)\left\langle\dot{a}_{l}\left(t\right)\left|a_{j}\left(t\right)\right\rangle\right.\left(\varepsilon_{l}\left(t\right)-\varepsilon_{j}\left(t\right)\right).$
(23)

Clearly, only off-diagonal terms of the density matrix (coherences) contribute
to this part of heat.

##  Appendix B Examples

Let us consider the driven single qubit under the action of distinct
decoherence channels. The Hamiltonian of the system is simply
$H_{t}=\alpha_{t}\sigma_{z}$. As discussed in the text, the invariant work is
the same for both channels, but heat behaves differently. The dephasing
channel is

$\mathcal{D}[\rho]=-\frac{\Gamma_{\mathrm{dec}}}{2}\commutator{\sigma_{z}}{\commutator{\sigma_{z}}{\rho}},$
(24)

with $\sigma_{s}$ being the $s$-th Pauli matrix while $\Gamma_{\mathrm{dec}}$
represents the decoherence rate. The generalized amplitude damping is
described by

$\displaystyle\mathcal{D}[\rho]$ $\displaystyle=$
$\displaystyle\Gamma_{a}(\bar{n}+1)\left[a\rho
a^{\dagger}-\frac{1}{2}\left\\{a^{\dagger}a,\rho\right\\}\right]$ (25)
$\displaystyle+$ $\displaystyle\Gamma_{a}\bar{n}\left[a^{\dagger}\rho
a-\frac{1}{2}\left\\{aa^{\dagger},\rho\right\\}\right],$

where $a$ ($a^{\dagger}$) is the annihilation (creation) operator of the bath
excitations and $\bar{n}=\left(e^{\beta\omega}-1\right)^{-1}$ is the mean
excitation number of the bath mode with frequency $\omega$. $\beta$ stands for
the inverse temperature while $\Gamma_{a}$ is the decoherence rate. A direct
calculation leads us to
$Q_{\mathrm{inv}}^{\mathrm{gad}}=\int_{0}^{\tau}\differential t\Phi_{E}^{t}$,
with $\Phi_{E}^{t}$ being the instantaneous energy flux from the system to the
bath.

Let us now move to the third example discussed in the main text. The dynamics
of a single qubit governed by the Hamiltonian

$H_{t}=\sigma_{z}+\gamma_{t}\sigma_{x}.$ (26)

This Hamiltonian can be put into a diagonal form
$h_{t}=u_{t}^{\dagger}H_{t}u_{t}=\mathrm{diag}(-\lambda_{t},\lambda_{t})$,
with $\lambda_{t}^{2}=1+\gamma_{t}^{2}$. Here $u_{t}$ is the matrix whose
columns are the normalized eigenvectors of $H_{t}$

$u_{t}=\begin{bmatrix}\frac{1-\lambda_{t}}{n_{-}}&\frac{1+\lambda_{t}}{n_{+}}\\\
\frac{\gamma_{t}}{n_{+}}&\frac{\gamma_{t}}{n_{-}},\end{bmatrix}$ (27)

with $n_{\pm}=2\lambda_{t}(\lambda_{t}\pm 1)$. Denoting
$\left\\{\ket{a_{1}},\ket{a_{2}}\right\\}$ the eigenstates of the Hamiltonian,
one can show that

$\innerproduct{\dot{a}_{2}}{a_{1}}=-\innerproduct{\dot{a}_{1}}{a_{2}}=\frac{\dot{\gamma}_{t}}{2\lambda_{t}^{2}}\hskip
5.69046pt\mbox{and}\hskip 5.69046pt\innerproduct{\dot{a}_{i}}{a_{i}}=0.$ (28)

The invariant heat can be written as

$Q_{\mathrm{inv}}=2\int_{0}^{\tau}\differential
t\mbox{Re}[c_{12}(t)]\frac{\dot{\gamma}_{t}}{\lambda_{t}}.$ (29)

Note that the only contribution that we have here comes from the coherences of
the density operator in the energy eigenbasis. However, it is important to
stress that, if the engenbasis is constant, the coherences does not matter and
the heat is zero. The important thing here is the change in the eigenbasis of
the Hamiltonian.

The invariant work takes the form

$W_{\mathrm{inv}}=\int_{0}^{\tau}\differential
t[c_{22}(t)-c_{11}(t)]\dot{\lambda}_{t}.$ (30)

We shall numerically solve the evolution equation for the density operator
(for notation brevity we omit time dependence of $c_{ij}$ coefficients)

$\derivative{\rho_{t}}{t}=-i\commutator{H_{t}}{\rho_{t}}=-i\sum_{i,j}\commutator{H_{t}}{c_{ij}\outerproduct{a_{i}}{a_{j}}}.$
(31)

Since

$\derivative{\rho_{t}}{t}=\sum_{i,j}\left[\dot{c}_{ij}\outerproduct{a_{i}}{a_{j}}+c_{ij}\outerproduct{\dot{a}_{i}}{a_{j}}+c_{ij}\outerproduct{a_{i}}{\dot{a}_{j}}\right],$
(32)

one can in a straightforward way show that the coefficients $c_{ij}$ fulfill
the following set of differential equations

$\displaystyle\dot{c}_{11}$ $\displaystyle=$
$\displaystyle-\frac{\dot{\gamma}_{t}}{2\lambda_{t}^{2}}\left(c_{12}+c_{21}\right),$
$\displaystyle\dot{c}_{12}$ $\displaystyle=$ $\displaystyle
2i\lambda_{t}c_{12}+\frac{\dot{\gamma}_{t}}{2\lambda_{t}^{2}}\left(c_{11}-c_{22}\right),$
$\displaystyle\dot{c}_{21}$ $\displaystyle=$
$\displaystyle-2i\lambda_{t}c_{21}-\frac{\dot{\gamma}_{t}}{2\lambda_{t}^{2}}\left(c_{22}-c_{11}\right),$
$\displaystyle\dot{c}_{22}$ $\displaystyle=$
$\displaystyle\frac{\dot{\gamma}_{t}}{2\lambda_{t}^{2}}\left(c_{12}+c_{21}\right).$
(33)

In order to study the role of coherences, we consider the relative entropy of
coherence, which is defined as Baumgratz2014

$C=S(\rho_{\mathrm{diag}})-S(\rho),$ (34)

where $\rho_{\mathrm{diag}}$ is the density operator with the off-diagonal (in
the the energy eigenbasis) elements removed, while $S$ stands for the von
Neumann entropy.

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# Quantum Error Detection Without Using Ancilla Qubits

Nicolas J. Guerrero  David E. Weeks<EMAIL_ADDRESS>Air Force Institute
of Technology

###### Abstract

In this paper, we describe and experimentally demonstrate an error detection
scheme that does not employ ancilla qubits or mid-circuit measurements. This
is achieved by expanding the Hilbert space where a single logical qubit is
encoded using several physical qubits. For example, one possible two qubit
encoding identifies $|0\rangle_{L}=|01\rangle$ and $|1\rangle_{L}=|10\rangle$.
If during the final measurement a $|11\rangle$ or $|00\rangle$ is observed an
error is declared and the run is not included in subsequent analysis. We
provide codewords for a simple bit-flip encoding, a way to encode the states,
a way to implement logical $U_{3}$ and logical $C_{x}$ gates, and a
description of which errors can be detected. We then run
Greenberger–Horne–Zeilinger circuits on the transmon based IBM quantum
computers, with an input space of $N\in\\{2,3,4,5\\}$ logical qubits and
$Q\in\\{1,2,3,4,5\\}$ physical qubits per logical qubit. The results are
compared relative to $Q=1$ with and without error detection and we find a
significant improvement for $Q\in\\{2,3,4\\}$.

††preprint: APS/123-QED

##  I  Introduction

In an ideal world, physical qubits of quantum computers would be isolated from
outside influence to permit useful and reliable quantum computation. Of
course, it is well understood that such perfect isolation is likely an
impossibility, and that there are many things interacting with quantum
computers which prevent perfect qubits. These include, but are not limited to
quantum vacuum fluctuations [1, 2], cosmic radiation [3, 4, 5], material
defects [6, 7, 8], and many others.

The most promising idea to mitigate these errors is currently quantum error
correction (QEC) and quantum error detection (QED). The first error correction
code employed nine physical qubits per logical qubit and six ancilla qubits
for syndrome extraction [9]. More efficient codes were soon produced such as
codes on seven qubits [10] and five qubits [11], as well as other error
correction schemes such as surface codes [12] and continuous QEC [13, 14].
These codes have been studied extensively through simulations [15, 16, 17] as
well as more recently through experimentation [18, 19, 20, 21, 22].

In a similar manner, QED was first described using four physical qubits per
logical qubit and two ancilla qubits for syndrome extraction [23]. Other types
of detection process exist such as QED in surface codes [24] and hardware
based sensors [25]. QED has been experimentally verified for small numbers of
logical qubits [24, 26] and continues to be a useful companion to QEC. In this
paper, we will introduce and experimentally verify a new type of quantum error
detection which differs significantly from both QEC and QED. The experimental
verification is conducted on the transmon based IBM quantum computers [27].

###  I.1 No ancilla error detection

No ancilla error detection (NAED) is a comparable process to both QEC and QED.
In all three, $Q$ physical qubits represent a single logical qubit with states
$|0\rangle_{L}$ and $|1\rangle_{L}$. NAED differs from QEC and the standard
QED in the rather obvious way that it does not use ancilla qubits. This has
two benefits over previous schemes: the resources required are generally less
than QEC and QED and mid-circuit measurements are not a requirement for NAED.
The only measurements are made at the end of the circuit, thereby bypassing
any noise that might otherwise be incurred by measuring ancilla qubits.

An implementation of NAED begins by considering a single logical qubit
$|\psi\rangle_{L}=\alpha|0\rangle_{L}+\beta|1\rangle_{L}$ over $Q$ physical
qubits. Define the set
$H^{+}=\\{\alpha|0\rangle_{L}+\beta|1\rangle_{L}:|\alpha|^{2}+|\beta|^{2}=1\\}$
and define the set $H^{-}=\mathcal{H}_{Q}-H^{+}$ where we have used
$\mathcal{H}_{Q}$ to be the Hilbert space over the $Q$ physical qubits. The
basic premise behind NAED is that a detectable error on any of the physical
qubits will move a state $|\psi\rangle$ from $H^{+}$ to $H^{-}$. If the final
state is measured and found to be in $H^{-}$, an error is declared. If not,
then we measured something in $H^{+}$ and can identify that the logical qubit
was measured in the $|0\rangle_{L}$ or $|1\rangle_{L}$ state. We stress that
this scheme can produce false positives but no false negatives.

In this paper, we will describe a procedure to implement a simple NAED code.
This includes the operators required to encode the initial states as well as
methods to create logical gates out of the standard base set of $U_{3}$ and
$C_{x}$ gates. These logical gates are then used to produce logical
Greenberger–Horne–Zeilinger (GHZ) states with differing amounts logical qubits
and physical qubits per logical qubit. In general, constructing logical gates
remains an active area of research for all error mitigation schemes [28, 29,
30, 31].

The unencoded GHZ circuit can be thought of as generalized bell circuit and
produces the state

$GHZ(N,1)=\frac{|00...0\rangle+|11...1\rangle}{\sqrt{2}}$ (1)

Here, there are $N$ $0$s and $N$ $1$s in each ket and the $1$ in $GHZ(N,1)$
represents the fact that there is no encoding with one physical qubit per
logical qubit. We then run these circuits and compare the results with and
without error detection.

##  II Bit-flip error detection

The bit-flip encoding (based on classical Hamming Codes [32]) is the simplest
NAED code. In general, a bit-flip encoding on $Q$ physical qubits can be
defined by a set $S\subseteq\\{0,1,...,Q-1\\}$. We then define the two
integers

$x=\left\\{\begin{array}[]{ll}0&\quad S=\emptyset\\\ \sum_{i\in
S}2^{i}&\text{otherwise}\end{array}\right.$ (2)

$y=2^{Q}-1-x$ (3)

The code words are then defined as $|0\rangle_{L}=|x\rangle$ and
$|1\rangle_{L}=|y\rangle$ where $x$ and $y$ are written in binary. From this,
it is apparent that for $Q$ physical qubits, there are $2^{Q}$ encodings that
we might utilize. For example, a simple set of codewords for $Q=3$ physical
qubits is $|0\rangle_{L}=|001\rangle$ and $|1\rangle_{L}=|110\rangle$ as
determined by the set $S=\\{0\\}$.

A word on the notation used throughout this paper: let $\sigma_{x}$ be the
standard $2\times 2$ Pauli $x$ matrix. We will use the notation $\sigma_{i}$
to represent the $2^{Q}\times 2^{Q}$ matrix

$\sigma_{i}=I_{2^{i}}\otimes\sigma_{x}\otimes I_{2^{Q-i-1}}$ (4)

where $I_{n}$ is the $n\times n$ identity matrix. For example, for $Q=3$ we
have

$\sigma_{0}=\sigma_{x}\otimes I_{4}$ (5)

$\sigma_{1}=I_{2}\otimes\sigma_{x}\otimes I_{2}$ (6)

$\sigma_{2}=I_{4}\otimes\sigma_{x}$ (7)

Note that qubits are counted starting from $0$ rather than $1$.

For any two physical qubits in the circuit, the gate $C_{x}(i,j)$ is defined
to be a controlled-not gate with control qubit $i$ and target qubit $j$.
Additionally, qubits other than $i$ and $j$ are assumed to be operated on by
identity gates. The gate $C_{x}(i,j)$ represents a $2^{n}\times 2^{n}$ matrix,
rather than a $4\times 4$ matrix, where $n$ will be obvious from context. As
an example, for $n=3$ physical qubits

$C_{x}(1,2)=I_{2}\otimes C_{x}$ (8)

$C_{x}(0,2)=(I_{2}\otimes SWAP)(C_{x}\otimes I_{2})(I_{2}\otimes SWAP)$ (9)

$C_{x}(1,0)=(SWAP\cdot C_{x}\cdot SWAP)\otimes I_{2}$ (10)

Here, both $C_{x}$ and $SWAP$ are the standard $4\times 4$ controlled-not and
swap gates respectively. Finally, when using product notation for matrix
multiplication, we will use the convention

$\prod_{n=1}^{N}A_{n}=A_{N}A_{N-1}\cdots A_{2}A_{1}$ (11)

###  II.1 Encoding, Logical Gates, and Detectable Errors

In order use the bit-flip encoding, two things are required: a way to encode
the initial $|0\rangle_{L}$ state and a method to construct logical gates that
operate on $|0\rangle_{L}$ and $|1\rangle_{L}$. To encode $|0\rangle_{L}$ for
a given set $S$ (with $Q$ physical qubits starting in the state
$|00...0\rangle$) we simply need to operate on each qubit $q_{i}$ with a
$\sigma_{x}$ if $i\in S$. This can be written mathematically as

$M_{S}=\prod_{i\in S}\sigma_{i}$ (12)

where $M_{S}$ is identified as the encoding matrix.

The construction of logical gates for the bit-flip encoding is slightly more
complicated. By logical gates, we mean that a logical $R$ gate (denoted
$L_{S}(R)$) has the property $L_{S}(R)|\psi\rangle_{L}=|\phi\rangle_{L}$ when
$R|\psi\rangle=|\phi\rangle$. Here, the subscript $L$ denotes the logical
version of non-encoded qubit states $|\psi\rangle$ and $|\phi\rangle$ and $R$
is an arbitrary physical gate. For $Q$ physical qubits and $S=\emptyset$ we
have

$L_{\emptyset}(U_{3})=\left[\prod_{i=1}^{Q-1}C_{x}(0,Q-i)\right]U_{3}\otimes
I_{2^{Q-1}}\left[\prod_{i=1}^{Q-1}C_{x}(0,i)\right]$ (13)

Note that this logical gate uses $2(Q-1)$ physical $C_{x}$ gates. For a
general set $S$ we have

$L_{S}(U_{3})=\left\\{\begin{array}[]{ll}L_{\emptyset}(\sigma_{x}U_{3}\sigma_{x})&\quad
0\in S\\\ L_{\emptyset}(U_{3})&\quad 0\notin S\end{array}\right.$ (14)

Finally, to define the logical $C_{x}$ gate we must introduce some notation
describing the physical qubits of each logical qubit: let
$\\{r_{0},r_{1},...,r_{Q-1}\\}$ be the $Q$ physical qubits that correspond to
the logical control qubit and let $\\{p_{0},p_{1},...,p_{Q-1}\\}$ be the $Q$
physical qubits that correspond to the logical qubit being operated on. Then
for an arbitrary $S$ the logical $C_{x}$ gate is given by

$L_{S}(C_{x})=\prod_{i=0}^{Q-1}C_{x}(r_{i},p_{i})(I_{2^{Q}}\otimes M_{S})$
(15)

Here, $M_{S}$ is the encoding matrix defined in equation 12. Proofs of
equations 14 and 15 are provided in the appendix.

An example of bit-flip encoding and the associated logical gates is given by
encoding a bell state with $Q=2$ and $S=\\{1\\}$. In this example $x=2$ and
$y=1$ with $|0\rangle_{L}=|10\rangle$ and$|1\rangle_{L}=|01\rangle$. The
circuit includes an encoding step, a logical Hadamard $L_{\\{1\\}}(H)$, and a
logical controlled-not $L_{\\{1\\}}(C_{X})$ as shown in FIG 1. The final state
will be given by
$|\psi\rangle=1/\sqrt{2}(|00\rangle_{L}+|11\rangle_{L})=1/\sqrt{2}(|0101\rangle+|1010\rangle)$.

After measurement, there are 16 possibilities: $4$ states in $H^{+}$ and $12$
states in $H^{-}$. If we measure $|0101\rangle$ then we obtain
$|00\rangle_{L}$. In a similar fashion $|1010\rangle$ yields $|11\rangle_{L}$,
$|0110\rangle$ yields $|01\rangle_{L}$, and $|1001\rangle$ yields
$|10\rangle_{L}$. If we measure any other of the 12 combinations of $0$s and
$1$s, then we are in $H^{-}$ and an error is declared.

Figure 1: This circuit encodes a bell state for $Q=2$ physical qubits per
logical qubit with codewords $|0\rangle_{L}=|01\rangle$ and
$|1\rangle_{L}=|10\rangle$. The first set of gates to the left of the first
barrier transforms $|0000\rangle$ to $|0101\rangle=|00\rangle_{L}$. The second
set of gates is the logical Hadamard gate $L_{\\{1\\}}(H)$. The third set of
gates is the logical $C_{x}$ gate $L_{\\{1\\}}(C_{X})$. At the end of this
circuit, the qubits will be in the linear combination
$|\psi\rangle=1/\sqrt{2}(|00\rangle_{L}+|11\rangle_{L})=1/\sqrt{2}(|0101\rangle+|1010\rangle)$.

It is possible to describe the errors that we have a chance to detect while
using the bit-flip code. If we model these errors by $2\times 2$ unitary
gates, then any error not of the form

$P(\theta,\phi)=e^{i\phi}\left(\begin{array}[]{cc}1&0\\\ 0&e^{i\theta}\\\
\end{array}\right)$ (16)

for real $\phi$ and $\theta\neq 2\pi k$, will force any state $|\psi\rangle\in
H^{+}$ into some state $|\psi^{{}^{\prime}}\rangle\in H^{-}$. These detectable
errors include $\sigma_{x}$, $\sigma_{y}$, and any other non-phase error that
might occur.

##  III An Application of the Bit-Flip Encoding

The logical $U_{3}$ and $C_{x}$ gates are used to construct $GHZ(N,Q)$
circuits where $N$ is the number of logical qubits and $Q$ is the number of
physical qubits per logical qubit. Gates corresponding to this circuit are
given by

$GHZ(N,1)=\left[\prod_{i=0}^{N-1}C_{x}(i,i+1)\right](H\otimes I_{2^{N-1}})$
(17)

which uses $N-1$ physical $C_{x}$ gates and a single Hadamard gate. Note that
this is not the only way to create the $GHZ(N,1)$ state [33].

In order to turn the $GHZ(N,1)$ circuit into the $GHZ(N,Q)$ circuit (using
bit-flip encoding) we simply replace all physical gates in the $GHZ(N,1)$
circuit with their logical equivalents from equations 14 and 15. Of course, we
also have to decide which set $S$ we will use for the encoding. In order to
balance the number of $0$s and $1$s that make up $|0\rangle_{L}$ and
$|1\rangle_{L}$, we will use $S=\left\\{0,1,...,\lceil Q/2\rceil\right\\}$.
The last thing we need to do is slightly simplify the resulting circuit before
implementation. For example, for the circuit $GHZ(2,2)$ we have the the
circuit given in FIG 1, however the first $C_{x}$ gate as well as the two
$\sigma_{x}$ gates on $q_{3}$ are redundant. Removing these gives us the
circuit in FIG 2, and a similar simplification will be used for all
experimental runs.

Figure 2: The simplified $GHZ(2,2)$ circuit. This circuit is identical to the
circuit in FIG 1 except that the first $C_{x}$ gate and bottom $\sigma_{x}$
gates have been removed as redundant. This does not change the overall state
$|\psi\rangle=1/\sqrt{2}(|0101\rangle+|1010\rangle)$ that this circuit
produces.

###  III.1 Experimental Design, Results, and Discussion

In order to measure how well our NAED works, we will employ a modified version
of the total variation distance [34] between probability measure denoted
$\tau(A,B)$ for finite probability distribution functions (PDFs) $A$ and $B$.
Define the similarity between two finite PDFs to be

$0\leq\mu(A,B)=100(1-\tau(A,B))$

$=100-50\sum_{i=1}^{|A|}|A_{i}-B_{i}|\leq 100$ (18)

The similarity is equal to $0$ if two PDFs are completely dissimilar and $100$
if they are identical. For our experiments we will use the IBMQ quantum
computers and compare the experimentally determined PDF for the $GHZ(N,Q)$
circuit with theoretical expected PDF. For the simple $GHZ(N,Q)$ circuit, this
theoretical PDF is easily computed and will always be an equally split between
two states in the full $\mathcal{H}_{NQ}$ circuit space.

All experiments were performed on the $27$ qubit processor ibmq_motreal [27]
with a quantum volume [35] of $128$. For each
$(N,Q)\in\\{2,3,4,5\\}\times\\{1,2,3,4,5\\}$, the $GHZ(N,Q)$ circuit was
submitted at optimization level $1$ for $2^{13}$ shots. Two similarity
measures are computed using equation 18. The first retains all of the
experimental results and represents the full similarity measure of the encoded
circuit, $\mu_{Full}$. The second omits all experimental results for which an
error is detected and is the NAED result, denoted $\mu_{NAED}$. This process
was repeated between $220$ and $230$ times for each circuit, and averaged to
yield $\mu_{Full}$ and $\mu_{NAED}$ for each $(N,Q)$ pair.

Figure 3: The similarity measure $\mu_{Full}$ of the $GHZ(N,Q)$ circuits over
the input space $(N,Q)\in\\{2,3,4,5\\}\times\\{1,2,3,4,5\\}$. Not
surprisingly, the best results occur at $GHZ(2,1)$ with a similarity measure
of $90.8$. The similarity decreases as both $N$ and $Q$ increase, with the
worst similarity of $0.4$ for $N=Q=5$.

Values of $\mu_{Full}$ are shown in FIG 3 where the best run overall is the
$GHZ(2,1)$ circuit with a similarity measure of $90.8$. The similarity then
decreases as both $N$ and $Q$ increase. Values of $\mu_{NAED}$ are shown in
FIG 4 and

Figure 4: The similarity measure $\mu_{NAED}$ of the $GHZ(N,Q)$ circuits over
the input space $(N,Q)\in\\{2,3,4,5\\}\times\\{1,2,3,4,5\\}$. The highest
similarity is now $97.2$ for $GHZ(2,2)$ with the circuit from FIG 2. while the
greatest increase in similarity from the unencoded circuit occurs between
$GHZ(5,1)$ and $GHZ(5,2)$.

demonstrate that NAED is a viable option for improving quantum computation. As
seen in FIG 4, values of $\mu_{NAED}$ for $GHZ(N,Q)$ where $2\leq Q\leq 4$ are
all greater than $\mu_{NAED}$ for the unencoded $GHZ(N,1)$ circuit. It is not
until $Q=5$ that NAED produced lower values of $\mu_{NAED}$ than $Q=1$ where
no error detection is performed.

Figure 5: The percentage of runs retained for each $GHZ(N,Q)$ circuit over the
input space $(N,Q)\in\\{2,3,4,5\\}\times\\{1,2,3,4,5\\}$ after error detection
has been performed. For $Q=1$, there are no runs removed. The next highest
percentage is for the $GHZ(2,2)$ circuit (given by FIG 2) at $76.4\%$ kept.
From here, the percentage retained decreases as both $N$ and $Q$ increase.

Also seen in FIG 4 is evidence that even values of $Q$ outperform odd values
of $Q$. For example, the similarity of $GHZ(N,4)$ is greater than the
similarity of $GHZ(N,3)$ for all $N$. This behavior is caused by an assymmetry
in the number of $0$’s and $1$’s that make up the codewords for odd values of
$Q$. Since super conducting qubits will naturally decohere towards the
$|0\rangle$ state [36, 37], this leads to an asymmetry in measured states for
odd $Q$.

A primary cost of NAED is the number of runs that are discarded when an error
is declared. The total number of runs that are not included in the calculation
of $\mu_{NAED}$ increases with $N$ and $Q$ as seen in FIG 5 where only a few
percent of the runs are retained for larger $N$ and $Q$. It is interesting to
note the values of $\mu_{Full}$ in FIG 3 and percentage retained in FIG 5
approach each other in value as both $N$ and $Q$ increase. This observation is
used to show that the ratio of false positives to the total number of
measurements decreases as $N$ and $Q$ increase. To illustrate why this is the
case, define the following

  * •

$T=$ the total number of measurements

  * •

$r_{0}=$ the total number of $|00...0\rangle_{L}$ measured

  * •

$r_{1}=$ the total number of $|11...1\rangle_{L}$ measured

  * •

$r_{a}=$ the total number of logical states other than $|00...0\rangle_{L}$ or
$|11...1\rangle_{L}$ measured

  * •

$r_{b}=$ the total number of states which don’t fall into any logical state
measured

Here, $r_{b}$ are errors that NAED would catch while $r_{a}$ are errors that
NAED would not catch. For example, with $N=3$ and $Q=2$ the state
$|010110\rangle=|001\rangle_{L}$ is an invalid $GHZ$ state but will not be
caught by error detection. Also note that $T=r_{0}+r_{1}+r_{a}+r_{b}$. Then
the percentage of measurements kept is given by

$P_{Kept}=1-\frac{r_{b}}{T}$ (19)

Without detection, $\mu_{Full}$ can be written as

$\mu_{Full}=1-\frac{1}{2}\left(\left|\frac{1}{2}-\frac{r_{0}}{T}\right|+\left|\frac{1}{2}-\frac{r_{1}}{T}\right|+\frac{r_{a}}{T}+\frac{r_{b}}{T}\right)$
(20)

The absolute values may be ignored since the chance that
$\frac{r_{0}}{T}>\frac{1}{2}$ or $\frac{r_{1}}{T}>\frac{1}{2}$ is negligible
for higher $N$ and $Q$. Then equation 20 becomes

$\mu_{Full}=1-\frac{1}{2T}(T-r_{0}-r_{1}+r_{a}+r_{b})$

$=1-\frac{1}{2T}(T-r_{0}-r_{1}-r_{a}-r_{b}+2r_{a}+2r_{b})$

$=1-\frac{r_{a}}{T}-\frac{r_{b}}{T}=P_{Kept}-\frac{r_{a}}{T}$ (21)

As $N$ and $Q$ increase, the value of $r_{a}/T$ in equation 21 goes to zero.

##  IV Conclusion

These experiments have shown that NAED is a viable option for increasing the
fidelity of quantum circuits. At its best, it was able to improve the
similarity measure of a $GHZ(5,1)$ circuit from $\mu_{Full}73.9$ to
$\mu_{NAED}92.5$ using a $GHZ(5,2)$ circuit. Additionally, we have shown that
the ratio of false positives to the total number of measurements decreases as
the number of logical qubits and the number of physical qubits per logical
qubits increase. Current efforts are focused on expanding NAED to other
encoding algorithms. As a final comment, NAED may provide higher fidelity
quantum communication applications.

###### Acknowledgements.

The authors would like to thank the Air Force Research Laboratory for
providing access to the IBMQ quantum computer as well as Dr. L. D. Merkle for
many useful discussions. The views expressed are those of the authors, and do
not reflect the official policy or position of IBM or the IBM Quantum team.

*

##  Appendix A Proofs of Logical Gates

In order to prove the validity of the logical gates described in equations
14-15, we will use the following four identities:

$(I_{2}\otimes\sigma_{x})C_{x}=C_{x}(I_{2}\otimes\sigma_{x})$ (22)

$(\sigma_{x}\otimes I_{2})C_{x}(\sigma_{x}\otimes
I_{2})=(I_{2}\otimes\sigma_{x})C_{x}$ (23)

$\alpha|0\rangle_{L}+\beta|1\rangle_{L}=M_{S}(\alpha|00...0\rangle+\beta|11...1\rangle)$
(24)

where $M_{S}$ is the encoding matrix given in equation 12. The first two of
these equations are easily checked while the third equation follows from the
definition of $M_{S}$. For the rest of these proofs, we will drop the $S$
subscript and simply refer to the encoding matrix as $M$. After noting that
$M=M^{\dagger}=M^{-1}$, this also gives us

$M(\alpha|0\rangle_{L}+\beta|1\rangle_{L})=\alpha|00...0\rangle+\beta|11...1\rangle$
(25)

###  A.1 Logical $U_{3}$ gate

We will start with the $S=\emptyset$ case (with codewords
$|0\rangle_{L}=|00...0\rangle$ and $|1\rangle_{L}=|11...1\rangle$) and from
there prove the general case. To prove that $L_{\emptyset}(U_{3})$ is a
logical $U_{3}$ gate, we must show that if
$U_{3}(\alpha|0\rangle+\beta|1\rangle)=\tau|0\rangle+\delta|1\rangle$ then
$L_{\emptyset}(U_{3})(\alpha|0\rangle_{L}+\beta|1\rangle_{L})=\tau|0\rangle_{L}+\delta|1\rangle_{L}$.
With the first part of $L_{\emptyset}(U_{3})$ we have

$|\psi_{1}\rangle=\left[\prod_{i=1}^{Q-1}C_{x}(0,i)\right](\alpha|0\rangle_{L}+\beta|1\rangle_{L})$

$=\alpha|00...0\rangle+\beta|10...0\rangle=(\alpha|0\rangle+\beta|1\rangle)\otimes|00...0\rangle$
(26)

where the state $|00...0\rangle$ in the resulting expression has $Q-1$ zeros.
Then applying the $U_{3}$ gate gives us

$|\psi_{2}\rangle=U_{3}\otimes I_{2^{q-1}}|\psi_{1}\rangle=U_{3}\otimes
I_{2^{q-1}}(\alpha|0\rangle+\beta|1\rangle)\otimes|00...0\rangle$
$=(\tau|0\rangle+\delta|1\rangle)\otimes|00...0\rangle$ (27)

The final part of $L_{\emptyset}(U_{3})$ gives us

$|\psi_{3}\rangle=\left[\prod_{i=1}^{Q-1}C_{x}(0,Q-i)\right]|\psi_{2}\rangle$

$=\left[\prod_{i=1}^{Q-1}C_{x}(0,Q-i)\right](\tau|0\rangle+\delta|1\rangle)\otimes|00...0\rangle$

$=\tau|00...0\rangle_{L}+\delta|11...1\rangle_{L}=\tau|0\rangle_{L}+\delta|1\rangle_{L}$
(28)

as desired.

For the general case, if $0\not\in S$ then

$L_{S}(U_{3})(\alpha|0\rangle_{L}+\beta|1\rangle_{L})=L_{\emptyset}(U_{3})(\alpha|0\rangle_{L}+\beta|1\rangle_{L})$
$=L_{\emptyset}(U_{3})MM(\alpha|0\rangle_{L}+\beta|1\rangle_{L})=L_{\emptyset}(U_{3})M(\alpha|00...0\rangle+\beta|11...1\rangle)$
(29)

We also know that $[L_{\emptyset}(U_{3}),M]=\hat{0}$ since every $C_{x}$ in
$L_{\emptyset}(U_{3})$ is controlled by $q_{0}$, the physical $U_{3}$ gate in
$L_{\emptyset}(U_{3})$ acts on $q_{0}$, and $0\not\in S$ which allows us to
use equation 22. Thus, equation 29 becomes

$=L_{\emptyset}(U)M(\alpha|00...0\rangle+\beta|11...1\rangle)$

$=ML_{\emptyset}(U)(\alpha|00...0\rangle+\beta|11...1\rangle)$

$=M(\tau|00...0\rangle+\delta|11...1\rangle)=\tau|0\rangle_{L}+\delta|1\rangle_{L}$
(30)

For the case where $0\in S$, let $M^{{}^{\prime}}=\sigma_{1}M$. Then using the
fact $\sigma_{i}=\sigma_{i}^{-1}$, we have

$L_{\emptyset}(\sigma_{x}U_{3}\sigma_{x})M=\sigma_{0}\sigma_{0}L_{\emptyset}(\sigma_{x}U_{3}\sigma_{x})\sigma_{0}M^{{}^{\prime}}$
(31)

Then using equation 23 $2(Q-1)$ times (one for each physical $C_{x}$ in
$L_{\emptyset}(\sigma_{x}U\sigma_{x})$) and simplifying using equation 22 we
get

$=\sigma_{0}M^{{}^{\prime}}M^{{}^{\prime}}L_{\emptyset}(U_{3})M^{{}^{\prime}}=ML_{\emptyset}(U_{3})$
(32)

Using this relation we may now conclude

$L_{S}(U_{3})(\alpha|0\rangle_{L}+\beta|1\rangle_{L})=L_{\emptyset}(\sigma_{x}U\sigma_{x})(\alpha|0\rangle_{L}+\beta|1\rangle_{L})$

$=L_{\emptyset}(\sigma_{x}U\sigma_{x})MM(\alpha|0\rangle_{L}+\beta|1\rangle_{L})=ML_{\emptyset}(U)(\alpha|00...0\rangle+\beta
0|11...1\rangle)$

$=M(\tau|00...0\rangle+\delta|11...1\rangle)=\tau|0\rangle_{L}+\delta|1\rangle_{L}$
(33)

###  A.2 Logical $C_{x}$ gate

In a similar manner to the previous proof, We will start with the
$S=\emptyset$ case and from there prove the general case. To prove that
$L_{\emptyset}(C_{x})$ is a logical $C_{x}$ gate, we must show that
$L_{\emptyset}(C_{x})(\alpha|00\rangle_{L}+\beta|01\rangle_{L}+\tau|10\rangle_{L}+\delta|11\rangle_{L})=\alpha|00\rangle_{L}+\beta|01\rangle_{L}+\delta|10\rangle_{L}+\tau|11\rangle_{L}$.
As in the main paper, let $\\{r_{0},r_{1},...,r_{Q-1}\\}$ be the $Q$ physical
qubits the make up the logical control qubit and let
$\\{p_{0},p_{1},...,p_{Q-1}\\}$ be the $Q$ physical qubits that make up the
logical target bit. For ease of notation, the full ket corresponding to these
$2Q$ physical qubits shall be written as
$|r_{0}r_{1}...r_{Q-1};p_{0}p_{1}...p_{Q-1}\rangle$, note the semi-colon used
to distinguish between both sets of qubits. We then have

$C_{x}(r_{0},p_{0})(\alpha|00\rangle_{L}+\beta|01\rangle_{L}+\tau|10\rangle_{L}+\delta|11\rangle_{L})$

$=C_{x}(r_{0},p_{0})(\alpha|00...0;00...0\rangle+\beta|00...0;11...1\rangle$
$+\tau|11...1;00...0\rangle+\delta|11...1;11...1\rangle)$

$=\alpha|00...0;00...0\rangle+\beta|00...0;11...1\rangle$

$+\tau|11...1;10...0\rangle+\delta|11...1;01...1\rangle$ (34)

Repeating this process for the other $Q-1$ physical $C_{x}$ gates in
$L_{\emptyset}(C_{x})$ gives us the desired result. To generalize to all $S$,
note that by equations 22 and 23 we have

$L_{S}(C_{x})=L_{\emptyset}(C_{x})(I_{2^{Q}}\otimes M)=(I_{2^{Q}}\otimes
M)L_{\emptyset}(C_{x})$

$=(M\otimes I_{2^{Q}})L_{\emptyset}(C_{x})(M\otimes I_{2^{Q}})$

$=(M\otimes MM)L_{\emptyset}(C_{x})(M\otimes I_{2^{Q}})=(M\otimes
M)L_{\emptyset}(C_{x})(M\otimes M)$ (35)

Using this equivalent definition for $L_{S}(C_{x})$, we get

$L_{S}(C_{x})(\alpha|00\rangle_{L}+\beta|01\rangle_{L}+\delta|10\rangle_{L}+\tau|11\rangle_{L})$

$=(M\otimes M)L_{\emptyset}(C_{x})(M\otimes
M)(\alpha|00\rangle_{L}+\beta|01\rangle_{L}+\delta|10\rangle_{L}+\tau|11\rangle_{L})$

$=(M\otimes
M)L_{\emptyset}(C_{x})(\alpha|00...000...0\rangle+\beta|00...011...1\rangle)$
$+\delta|11...100...0\rangle+\tau|11...111...1\rangle)$ (36)

by equation 25. But this is precisely the relationship we just showed (the
$S=\emptyset$ encoding). Thus, it simplifies to

$=(M\otimes M)(\alpha|00...000...0\rangle+\beta|00...011...1\rangle)$
$+\tau|11...100...0\rangle+\delta|11...111...1\rangle)$

$=\alpha|00\rangle_{L}+\beta|01\rangle_{L}+\tau|10\rangle_{L}+\delta|11\rangle_{L}$
(37)

as desired.

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