[1]Akshaya Jayashankar

[1]Department of Physics, Indian Institute of Technology Madras, Chennai,
600036, India

2]Department of Physics, Indian Institute of Technology Madras, Chennai,
600036, India

# Quantum Error Correction: Noise-adapted Techniques and Applications

<EMAIL_ADDRESS>Prabha Mandayam<EMAIL_ADDRESS>*  [

###### Abstract

The quantum computing devices of today have tens to hundreds of qubits that
are highly susceptible to noise due to unwanted interactions with their
environment. The theory of quantum error correction provides a scheme by which
the effects of such noise on quantum states can be mitigated, paving the way
for realising robust, scalable quantum computers. In this article we survey
the current landscape of quantum error correcting (QEC) codes, focusing on
recent theoretical advances in the domain of noise-adapted QEC, and
highlighting some key open questions. We also discuss the interesting
connections that have emerged between such adaptive QEC techniques and
fundamental physics, especially in the areas of many-body physics and
cosmology. We conclude with a brief review of the theory of quantum fault
tolerance which gives a quantitative estimate of the physical noise threshold
below which error-resilient quantum computation is possible.

###### keywords:

noise-adapted QEC, Petz map, amplitude damping, fault tolerance

##  1 Introduction

Quantum computing technologies have advanced by leaps and bounds over the last
decade. We are already witness to the first generation of quantum processors
successfully demonstrating that the theoretical promise of quantum
computational speedups can be realised in practice qSupremacy ;
boson_sampling2021 . There are however a few key challenges that must be
overcome in order to scale from the current generation of noisy intermediate-
scale quantum (NISQ) devices preskill2018_quantum , to robust universal
quantum computers. Error mitigation and fault tolerance is arguably the
biggest of these challenges, both from a theoretical and experimental
standpoint.

The theory of quantum error correction (QEC) lidar  lays down the basic
framework for dealing with noise affecting the quantum states of interest. The
early works of Shor Shor95 ; Calderbank_shor96  and Steane Steane97
demonstrated how quantum error correcting codes can be constructed for bit-
flip and phase-flip noise, by making use of entanglement to circumvent the
challenge posed by no-cloning theorem no_cloning82 . Subsequently, a general
theory of QEC was developed for arbitrary errors, in terms of algebraic
knill97  and information-theoretic conditions infoqec_96 . Today, the standard
approach to quantum error correction works by discretizing the errors
affecting the quantum state in terms of the _Pauli basis_ nielsen , with the
group structure of the Pauli operators naturally leading to the rich
mathematical framework of quantum stabilizer codes gottesman1997 . We refer to
a recent review terhal2015  that surveys the major developments in QEC, from
the stabilizer codes and CSS codes to the more recently proposed classes of
topological codes bombin2013  and surface codes raussendorf2007 .

In contrast to the _general-purpose_ QEC schemes described above, error
correction protocols tailored to specific noise models have been devised,
starting with a $4$-qubit code that protects against amplitude-damping noise
leung . Such adaptive QEC protocols have shown to offer the same degree of
protection while using fewer resources hui_prabha ; fletcher_AD , particularly
in the case of non-Pauli noise models. In this article, we focus on this
emerging area of _channel-adapted_ or _noise-adapted_ QEC and survey some of
the recent progress in this area. We will first review the early theoretical
progress on approximate QEC fletcher_rec ; beny2010 ; tyson2010 ; mandayam2012
, and introduce an important noise-adapted recovery map, namely, the Petz map
barnum ; Petz . We then discuss various recent approaches to constructing
noise-adapted quantum codes ak_cartan ; qvector ; cao2022quantum ;
reinforcement . Next we survey some interesting applications of noise-adapted
recovery maps, especially in the context of many-body quantum systems. We also
briefly touch upon the interesting role that approximate recovery maps are
coming to play in the AdS/CFT setting holography_qec .

While QEC focuses on dealing with the noise affecting the quantum states
assuming that all the gate operations are ideal, the theory of quantum fault
tolerance preskill1998  lays down the basic framework for dealing with faulty
gate operations in quantum computing devices. Fault-tolerant protocols build
upon the QEC framework and enable the construction of quantum circuits that
are truly noise-resilient, provided the noise is below a certain _threshold_
value knill2005 ; aliferis2006 . We devote the final section of our review to
discussing the latest developments in the theory of quantum fault tolerance,
focusing on fault-tolerant schemes that are based on noise-adapted codes ak_ft
.

The rest of this review is organized as follows. In Sec. 2, we introduce the
basic mathematical formalism of perfect and approximate QEC. In Sec. 3 we
formally introduce the idea of noise-adapted codes and recovery maps. We then
proceed to discuss various analytical and numerical approaches to constructing
such noise-adapted recovery maps and noise-adapted quantum codes. In Sec. 4,
we survey some interesting applications of adaptive QEC protocols. In Sec. 5
we discuss how quantum fault tolerance can be achieved using noise-adapted QEC
protocols. We conclude with a brief summary and future outlook in Sec. 6.

##  2 Preliminaries

Figure 1:  Unitary evolution of a closed system (left) vs non-unitary dynamics
of an open quantum system (right) thesis .

We begin with a brief review of the mathematical description of decoherence in
quantum systems, and refer to some of the well-known textbooks nielsen ; lidar
for further details. Noise in quantum systems is described by the mathematical
framework of quantum operations, often referred to as _quantum channels_ in
the context of quantum computation nielsen . A quantum channel is a completely
positive trace-preserving (CPTP) map describing the non-unitary evolution of
an open quantum system interacting with its environment, as shown in Fig. 1.
The action of such a map $\mathcal{E}$ on the state $\rho$ of a quantum system
can be described via a set of _Kraus operators_ $\\{E_{i}\\}$, as,
$\mathcal{E}(\rho)=\sum_{i}E_{i}\rho E_{i}^{\dagger}$, satisfying the
normalization condition, $sum_{i}E_{i}^{\dagger}E_{i}=I$. These Kraus
operators essentially represent the different _error operators_ associated
with the noise channel $\mathcal{E}$.

A classical example of quantum channel arising out of a system-environment
interaction is the _amplitude-damping channel_. This is a channel that models
dissipation in a two-dimensional quantum system (a qubit!), such as a two-
level atom interacting with an optical mode in a cavity via the Jaynes-
Cummings Hamiltonian, as shown in Fig. 2. The action of the amplitude-dampng
channel on the qubit system is described by a a pair of Kraus operators,
$E_{0}$ and $E_{1}$ which are given by nielsen ,

$E_{0}=|0\rangle\langle 0|+\sqrt{1-\gamma}\,|1\rangle\langle 1|,\;\quad
E_{1}=\sqrt{\gamma}\,|0\rangle\langle 1|.$ (1)

Here, $\gamma$ is the probability of decay of the excited state $|1\rangle$ to
the ground state $|0\rangle$. Amplitude-damping noise is an important model of
_non-Pauli_ noise, since its Kraus operators are not simply proportional to
Pauli operators. This is in contrast to other well known noise models such as
bit-flip, phase-flip and depolarizing noise which are all described by Kraus
operators that belong to the set $\\{I,X,Y,Z\\}$ of single-qubit Pauli
operators.

Figure 2:  An atom in a cavity undergoing spontaneous emission cao .

###  2.1 Quantum Error Correction

Quantum error correction involves protecting the information in say, a
$d_{0}$-dimensional Hilbert space $\mathcal{H}_{0}$, by _encoding_ it in a
$d_{0}$-dimensional subspace $\mathcal{C}$ of a larger Hilbert space
$\mathcal{H}$. Subsequently, a _recovery_ map $\mathcal{R}$ is applied in
order to reverse the effects of errors, followed by _decoding_ , which brings
the information back into the original Hilbert space $\mathcal{H}_{0}$. The
encoding $\mathcal{W}$ is an isometry whose action can be described as,
$\mathcal{W}:\mathcal{B}(\mathcal{H}_{0})\rightarrow\mathcal{B}(\mathcal{C})\subseteq\mathcal{B}(\mathcal{H})$,
where $\mathcal{B}(.)$ denotes the set of bounded linear operators on the
respective Hilbert spaces. The subspace $\mathcal{C}$ is called the
_codespace_ or simply, a quantum code nielsen ; lidar .

Figure 3: Perfect QEC: codespace gets mapped to mutually orthogonal subspaces
by the errors $\\{E_{i}\\}$ thesis .

Ideally, to protect against the error $\\{E_{i}\\}$ caused by a quantum
channel $\mathcal{E}$, the codespace must be chosen such that it gets mapped
onto mutually orthogonal subspaces under the action of the errors, as shown in
Fig. 3. This allows for the errors $\\{E_{i}\\}$ afflicting the codespace to
be identified unambiguously and hence corrected for. Such codes are referred
to as _perfectly_ correctable codes. This property of a quantum code to be
perfectly correctable under the action of the errors $\\{E_{i}\\}$ is captured
by the Knill-Laflamme conditions, stated as knill97 ,

$PE_{i}^{\dagger}E_{j}P=\alpha_{ij}P,\;\forall i,j.$ (2)

Here, $P$ is the projection onto the codespace $\mathcal{C}$ and $\alpha_{ij}$
are complex elements of a Hermitian matrix $\alpha$. The algebraic conditions
in Eq. (2) are necessary and sufficient for the existence of a recovery (CPTP)
map $\mathcal{R}$ which can correct perfectly for the errors $\\{E_{i}\\}$. We
note here that information-theoretic conditions for perfect QEC have also been
formulated in terms of the coherent information infoqec_96 . In contrast to
the active approach to QEC involving a recovery operation, there exist passive
QEC strategies lidar , where the information to be protected is stored in
subspaces or subsytems of a larger Hilbert space that remain unaffected by the
noise, called Decoherence-free subspaces (DFS) and Noiseless subsystems (NS)
kribs , respectively. Such passive QEC schemes are outside the scope of this
review, here we will focus exclusively on active quantum error correction.

Since the QEC conditions in Eq. (2) are linear, a code that corrects perfectly
for the set $\\{E_{i}\\}$ will also correct perfectly for any linear
combination of the errors in the set. Thus, to correct for an arbitrary
single-qubit error, it suffices to find codes that can correct perfectly for a
unitary error basis such as the Pauli basis spanned by the single-qubit Pauli
operators $\\{I,X,Y,Z\\}$. This in turn leads to an elegant reformulation of
the theory of perfect QEC using the group theoretic algebra of Pauli operators
called the stabilizer formalism lidar ; nielsen . We refer to other review
articles in this special issue for further details on the stabilizer formalism
and its role in quantum code construction.

Based on the set of errors correctable by it, a quantum code is characterized
by three parameters. An $[[n,k,d]]$ quantum code is one the encodes $k$ qubits
into $n$ physical qubits and corrects for errors on upto $t$ qubits, where,
$d=2t+1$ is often referred to as the distance of the code. Stabilizer codes
such as the well known $9$-qubit Shor code Shor95 , the $7$-qubit Steane code
Steane97  and the $5$-qubit code laflamme_96  are all general-purpose quantum
codes, since they can correct for arbitrary single-qubit errors. However,
their error-detection and error-correction mechanisms work by discretizing
arbitrary errors in terms of the Pauli errors and distinguishing single-qubit
Pauli errors perfectly. These perfect QEC codes are in essence tailored to
protect against single-qubit Pauli noise, which in turn imposes rigid
constraints on the structure of such codes. In what follows, we will look at
how it is possible to construct newer classes of quantum codes both by
deviating from the demands of perfect QEC and by moving beyond the class of
Pauli noise channels.

###  2.2 Approximate Quantum Error Correction

In order to correct perfectly for arbitrary single-qubit errors by decomposing
them in terms of Pauli errors, a single qubit must be encoded into at least
five qubits knill97 . In violation of this quantum Hamming bound, the
pioneering work of Leung _et al._ leung  provided an example of $4$-qubit QEC
code that can correct for single-qubit amplitude-damping noise. It was further
shown that this $4$-qubit code does not satisfy the Knill-Laflamme conditions
in Eq. 2 exactly, but satisfies a _perturbed_ form of the QEC conditions. The
$4$-qubit code is thus the first example of an _approximate_ quantum code,
with a codespace that is spanned by,

$|0_{L}\rangle=\tfrac{1}{\sqrt{2}}(|0000\rangle+|1111\rangle),\quad|1_{L}\rangle=\tfrac{1}{\sqrt{2}}(|1100\rangle+|0011\rangle).$
(3)

Finally, it was shown that this approximate $4$-qubit code achieves a
comparable degree of protection as the standard $5$-qubit code, against
amplitude-damping noise leung . These observations generated a lot of interest
in looking for approximate codes and QEC protocols.

Figure 4: Approximate QEC: codespace gets mapped to overlapping subspaces due
to errors $\\{E_{i}\\}$ thesis .

In contrast to the idea of perfect QEC, approximate QEC is achieved by
relaxing the demands on the code. The codespace need not be mapped to distinct
subspaces under the action of the errors, but can instead be mapped to
overlapping subspaces as shown in Fig. 4. Furthermore, while the standard
recovery procedure for a perfect code simply involves applying the appropriate
Pauli operator once the error is detected and identified, an approximate code
often requires a non-trivial recovery, one that is also adapted to the noise,
so as to reverse the effects of the noise in the best possible way. Thus, we
may also refer to approximate codes and approximate QEC protocols as _noise-
adapted_ schemes, since the code and/or the recovery are tailored to a
particular noise model.

Formally, a channel $\mathcal{E}$ is approximately correctible on codespace
$\mathcal{C}$ if and only if there exists a CPTP map $\mathcal{R}$ such that
$\mathbb{F}[\mathcal{C}\mathchar
24635\relax\;\mathcal{R}\circ\mathcal{E}]\approx 1$, where $\mathbb{F}$ is any
average or worst-case fidelity measure, quantifying how well the combination
of $\mathcal{E}$ followed by $\mathcal{R}$ protects the states in the
codespace. Before we proceed with our survey of approximate or noise-adapted
QEC, it may be useful to recall two standard fidelity measures that are used
to quantify the performance of any QEC protocol. The first is the
_entanglement fidelity_ $F_{e}(\rho,\mathcal{E})$, which describes how well
the noise channel $\mathcal{E}$ preserves entanglement. Specifically, let
$|\Psi_{AB}\rangle$ be a purification of the state $\rho_{A}$ onto the the
joint system $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. Then,
$F_{e}(\rho,\mathcal{E})$ defined as nielsen ,

$F_{e}(\rho,\mathcal{E})=\langle\Psi_{AB}|(\mathcal{E}_{A}\otimes\mathcal{I}_{B})(|\Psi_{AB}\rangle\langle\Psi_{AB}|)|\Psi_{AB}\rangle,$
(4)

where the noise $\mathcal{E}_{A}$ acts only on the subsystem
$\mathcal{H}_{A}$. In contrast to this, the _fidelity_
$F(\rho,\mathcal{E}(\rho))$ quantifies how well the channel preserves the
state $\rho$ in terms of the Bures’ metric, and is defined as nielsen ,

$F(\rho,\mathcal{E})=\mathrm{Tr}\sqrt{\sqrt{\rho}\mathcal{E}(\rho)\sqrt{\rho}}$
(5)

The performance of a QEC protocol described by the pair of encoding and
recovery maps $(\mathcal{W},\mathcal{R})$ for a noise $\mathcal{E}$ is then
quantified by the _worst-case fidelity_ , defined as,

$F_{\min}(\mathcal{W},\mathcal{R}\mathchar
24635\relax\;\mathcal{E})\equiv\min_{|\psi\rangle\in\mathcal{H}_{0}}F(|\psi\rangle,\mathcal{W}^{-1}\circ\mathcal{R}\circ\mathcal{E}\circ\mathcal{W}).$
(6)

##  3 Noise-adapted Quantum Error Correction

Much of the existing work on QEC focuses on general-purpose QEC codes that
assume no knowledge about the noise affecting the system. This allows one to
construct codes capable of correcting any arbitrary error by discretizing the
error in terms of Pauli operators, as explained earlier. However, such Pauli-
based QEC codes are resource intensive, whereas noise-adapted codes like the
$4$-qubit code demonstrated that it may be possible to achieve a comparable
degree of protection using fewer qubits. The problem of noise-adapted QEC is
to identify a pair $(\mathcal{W},\mathcal{R})$ of encoding and recovery maps
respectively, that achieve optimal protection against the noise map under
consideration.

For a given noise map $\mathcal{E}$ (assuming some dimension $d$ of the
physical system), the best QEC protocol is thus the solution to an
optimization over encoding isometries $\mathcal{W}$ and recovery maps
$\mathcal{R}$, for a chosen measure of fidelity $\mathbb{F}$.

$\operatorname*{argmax}_{\mathcal{W}}\operatorname*{argmax}_{\mathcal{R}}\mathbb{F}(\mathcal{W},\mathcal{R}\mathchar
24635\relax\;\mathcal{E}).$ (7)

This is in general a hard problem, since it involves a double optimization or
a triple optimization depending on whether the measure $\mathbb{F}$ is an
average fidelity measure or a worst-case fidelity measure such as the one
defined in Eq. (6). The problem of noise-adapted QEC becomes more tractable
when these two tasks are decoupled: we may instead ask what is the best
possible encoding assuming a fixed recovery map, or, we may try to solve for
the best recovery map for a given encoding. In what follows, we will first
survey the known results on noise-adapted recovery maps (Sec. 3.1) and then
proceed to discuss different approaches to search for and construct good
noise-adapted quantum codes (Sec. 3.2).

###  3.1 Noise-adapted recovery maps

One of the earliest analytical results on noise-adapted QEC was the
demonstration of the existence of a universal, near-optimal recovery map for
any noise channel, with optimality defined in terms of the average
entanglement fidelity barnum . This noise-adapted recovery map is based on an
original construction due to Petz petz2003 , and is often referred to as the
_Petz map_ today in the literature. Subsequently, this idea was used to obtain
conditions for approximate QEC beny  as a generalization of the Knill-Laflamme
conditions, which in turn provided a way to construct a near-optimal recovery
map in terms of the worst-case entanglement fidelity. In related work, it was
shown that a modified version of the Petz map is in fact a noise-adapted
recovery map for any combination of noise and codespace, achieving near-
optimal worst-case fidelity hui_prabha , leading to simple algebraic
conditions for approximate QEC as a perturbation of the Knill-Laflamme
conditions. We note here that information-theoretic conditions for approximate
QEC have also been formulated SW2002 , as a perturbation of the perfect
conditions.

Given the important role played by the Petz map, it might be useful to briefly
review its definition and properties here. The map was originally introduced
in an information-theoretic setting, in the context of saturating the
monotonicity of the quantum relative entropy Petz . For a density operator
$\rho$ and noise channel $\mathcal{E}$ with Kraus operators
$\\{E_{i},i=1,2,\ldots,N\\}$, the Petz map is defined via its Kraus operator
decomposition as follows.

$\mathcal{R}_{\rho}\sim\\{R_{i}\equiv\sqrt{\rho}E_{i}^{\dagger}\mathcal{E}(\rho)^{-1/2}\\}.$
(8)

Note that this map recovers the state $\rho$ perfectly after the action of the
noise $\mathcal{E}$. This _state-specific_ Petz map was generalized for an
ensemble of states in barnum , to obtain a near-optimal recovery map in terms
of the average entanglement fidelity.

Subsequently, a _code-specific_ Petz map $\mathcal{R}_{P}$ was defined for the
noise channel $\mathcal{E}$ with Kraus operators $\\{E_{i},i=1,2,\ldots,N\\}$
and a codespace $\mathcal{C}$, as,

$\mathcal{R}_{P}\equiv\\{PE_{i}^{\dagger}\mathcal{E}(P)^{-1/2}\\},\;i=1,2,\ldots,N,$
(9)

where, $P$ is the projector onto the codespace and
$\mathcal{E}(P)=\sum_{i}E_{i}PE_{i}^{\dagger}$. The action of the code-
specific Petz map is thus defined on the support of $\mathcal{E}(P)$, as,

$\mathcal{R}_{P}(\cdot)\equiv\sum_{i=1}^{N}PE_{i}^{\dagger}\mathcal{E}(P)^{-1/2}(\cdot)\mathcal{E}(P)^{-1/2}E_{i}P.$
(10)

Note that $\mathcal{R}_{P}$ can be thought of as being composed of three CP
maps as follows.

Normalizer map :
$\displaystyle\quad\mathcal{E}(\mathcal{P}_{\mathcal{C}})^{-1/2}(.)\mathcal{E}(\mathcal{P}_{\mathcal{C}})^{-1/2}$
Adjoint map : $\displaystyle\quad\mathcal{E}^{\dagger}(.)$ Projector map :
$\displaystyle\quad\mathcal{P}_{\mathcal{C}}\,(.)\mathcal{P}_{\mathcal{C}}$
(11)

This map was shown to achieve close to optimal worst-case fidelity for the
codespace $\mathcal{C}$ affected by the noise channel $\mathcal{E}$ hui_prabha
. Furthermore, in case the codespace $\mathcal{C}$ is perfectly correctable
for noise $\mathcal{E}$, it can be shown that the map $\mathcal{R}_{P}$ is
indeed the standard recovery map of perfect QEC! We note here that the code-
specific Petz map construction was further generalised as a near-optimal
recovery for subsystem codes mandayam2012 . On a related note, approximate QEC
has also been illustrated in a different setting, where the noise channel is
assumed to act jointly on system and bath hui_2 .

In recent times, the Petz map which is a close analogue of the classical
Bayesian reversal map petz_bayesian  has been used to recover information in
the case of non-Markovian dynamics petz_nonmarkovian . A physical protocol
which implements the Petz map in the form of Hamiltonians and jump operators
has been given in Ref. petz_physicalprocess  . The Petz map construction for
the case of continuous variable case, namely the bosonic Gaussian channel has
also been studied in Ref. wilde . Finally, a quantum algorithm to implement
Petz map based on the quantum singular value decomposition was proposed
recently gilyen2022 .

We conclude this section with a summary of numerical studies of optimal noise-
adapted recovery maps. Solving for the optimal recovery map in terms of the
worst-case fidelity is computationally hard given that Eq. (7) requires a
triple optimization in this case. However, upon relaxing certain constraints,
it was shown that this triple optimization is tractable via a semidefinite
program (SDP), although the recovery map thus obtained is typically suboptimal
yamamoto . The entanglement fidelity measure defined in Eq. (4), on the other
hand, is amenable to convex optimization techniques kosut . In terms of the
worst-case entanglement fidelity, the problem of finding the optimal noise-
adapted code for a fixed recovery and the problem of finding the optimal
noise-adapted recovery for a fixed code can both be recast as semidefinite
programs, which are simply dual to each other fletcher_rec ; fletcher_thesis .

###  3.2 Noise-adapted quantum codes

Starting with the $4$-qubit code leung  described in Eq. (3) above, several
analytical constructions of approximate codes have been obtained in the
literature. These include the class of _cat-codes_ catcode1 ; catcode2  and
_binomial codes_ binomial2016  which are tailored towards protecting
information stored in bosonic modes against dominant noise processes such as
photon-loss and amplitude-damping. There are a few analytical constructions of
noise-adapted codes for amplitude-damping noise shor_lang2007 ; shor2011  and
generalized amplitude-damping cafaro2014 , which make use of the structure of
the Kraus operators of the respective channels to get the desired error
mitigation properties.

Constructing noise-adapted codes for an arbitrary quantum channels is in
general a hard problem. Recently, a systematic way of constructing optimal
noise-adapted approximate codes for general noise models using the Cartan
decomposition was provided in Ref. ak_cartan . Furthermore, in contrast to
earlier numerical approaches, the worst-case fidelity was used as the figure
of merit to obtain the optimal codes. We briefly summarize this approach here
before surveying other numerical approaches to finding noise-adapted quantum
codes.

As a first step towards simplifying the search for good quantum codes, the
recovery map is simply fixed to be the code-specific recovery map
$\mathcal{R}_{P}$ defined in Eq. 10. Specifically, for a qubit noise channel
$\mathcal{E}$ and two-dimensional code $\mathcal{C}$, using the Petz map leads
to a simple optimization problem for the worst-case fidelity, namely ak_cartan
,

$F_{\rm
min}(\mathcal{W})=1-\frac{1}{2}{\left[1-t_{\min}(\mathcal{W})\right]}.$ (12)

where $t_{\rm min}$ refers to the smallest eigenvalue of a $3\times 3$ matrix.
Therefore, the encoding unitary $\mathcal{W}$ which fixes the optimal
codespace is the one that maximizes the fidelity, or, minimizes the fidelity
loss $\eta_{\mathcal{W}}=1-F_{\rm min}(\mathcal{W})$ over all encoding
unitaries.

In order to solve Eq. 12 each element of $SU(2^{n})$ is then parameterized
using the so-called _Cartan decomposition_ khaneja ; earp . The Cartan
decomposition provides a way to represent any element of $SU(2^{n})$, upto
local unitaries, as a product of operators from $SU(2^{n-1})\otimes SU(2)$ and
unitary operators _nonlocal_ on the entire $n$-qubit space. Such a
decomposition can be applied recursively, to further decompose each
$SU(2^{m})$ operator as a product of local and nonlocal unitary operators.

Figure 5:  Performance of $4$-qubit noise-adapted codes for amplitude-damping
channel, obtained using the numerical search procedure based on the Cartan
decomposition.

The Cartan form is naturally useful in the context of searching for good
codespaces, especially in the regime where the noise assumed to be independent
and local. The Cartan decomposition allows one to look for potential
codespaces by directly searching over the non-local unitaries in the
decomposition, while keeping the local unitaries fixed. For instance, in the
case of a $4$-qubit encoding, the total number of parameters required to
describe any element in $SU(2^{4})$ is $363$ using the Cartan decomposition.
However, since we only need to search over the intermediate non-local
unitaries to find good codespaces, one needs to search over only $82$
parameters. Therefore, this Cartan-form-based search leads to a quick and
efficient way of obtaining optimal noise-adapted codes for arbitrary noise
channels.

Fig 5 compares the performance of various quantum codes for the amplitude-
damping channel, by plotting the worst-case fidelity as a function of the
noise strength $\gamma$. We see that the best $4$-qubit codes obtained via the
numerical search outperform the perfect $5$-qubit code (solid line) as well as
the noise-adapted $4$-qubit code (dot-dashed line). It further emphasizes the
point made earlier that it suffices to search only over the non-local parts of
the unitary in the Cartan decomposition (structured search, cross-line), since
the best code thus obtained performs comparably to the best code obtained by
searching over the entire parameter space in the Cartan decomposition
(unstructured search, dashed line).

###  3.3 Learning-based approaches to construct adaptive QEC codes

We conclude this section on noise-adapted quantum codes with a survey of some
recent works that propose to use learning-based approaches in searching for
good QEC protocols. On the one hand, Ref. qvector  demonstrates a variational
quantum algorithm to obtain quantum codes specific to the device hardware, in
a manner that does not require prior knowledge about the dominant noise model
in the system. A simple enough cost function, namely, the average fidelity has
been optimized in this algorithm. Alternately, Ref. cao2022quantum  develops a
variational quantum algorithm to obtain arbitrary quantum codes, not
necessarily the channel-adapted ones, but also those with specific code
parameters such as distance. In this case, the cost function is defined by
generalizing the Knill-Laflamme condition in Eq. (2) and the efficacy of their
algorithm is demonstrated by identifying various known codes.

There have also been reinforcement based approaches in finding quantum codes.
For example, Ref. neural_networks  show how neural network based learning can
be used to construct quantum codes for various quantum channels. On a
different note, Ref. qeccodes  demonstrates a reinforcement-based learning
approach to adapting QEC codes by benchmarking them against the desired
logical error rate. Finally, Ref. reinforcement  demonstrates how to train a
network-based agent to learn QEC strategies from scratch and hence protect a
set of qubits from noise.

##  4 Applications of noise-adapted QEC

It is well known that ideas from quantum error correction have today permeated
diverse fields in physics, from the toric code toric_code  which has given
rise to several interesting topological models in condensed matter
baskaran2007  to the more recent connections between stabilizer codes, tensor
networks ferris2014  and holography happy2015 . In this section, we will
describe some interesting connections between the theory of approximate or
noise-adapted QEC and areas like many body physics and the AdS/CFT
correspondence.

###  4.1 Quantum state transfer over spin chains

In the context of quantum many body systems, it was recently shown that
approximate quantum codes can occur naturally as ground spaces of certain
topologically-ordered, gapped Hamiltonian systems elizabeth ; elizabeth_2 .
Another interesting connection between quantum channels, quantum error
correction and many body systems arises in the context of state transfer over
spin chains bose2007 . Here, we would like to elaborate on this connection and
emphasize the role played by approximate QEC in achieving information transfer
over spin chains with a high degree if fidelity.

Quantum communications using interacting spins is one of the areas that has
generated a lot of interest over the recent years. Following the pioneering
work of Bose bose  who showed that such a state transfer could be viewed as
transmitting quantum information over a quantum channel, multiple protocols
demonstrating _perfect_ and _pretty-good_ state transfers using spin chains
have been proposed. Rather than use a single spin chain, it was subsequently
shown that conclusive and perfect state transfer could be achieved using a
pair of spin-$1/2$ chains with dual-rail encoding Burgarth . As opposed to a
perfect transfer, in the case of pretty good transfer one finds an optimal
scheme to transfer the information with high fidelity, using permanently
coupled spin chains osborne . Going beyond the $2$-qubit dual-rail code, it
has been shown that state transfer can be achieved over disordered spin chains
beyond the localization length, by using a concatenated form of the standard
$5$-qubit code allcock . Alternately, the role of QEC in enhancing fidelity of
state transfer under local noise affecting each individual spin in the chain
has also been studied kay_2016 .

Since the underlying quantum channel that arises in the context of state
transfer over spin chains is typically not of the Pauli form, a natural
question to ask is whether noise-adapted QEC may enable information transfer
using spin chains over longer distances. This question was addressed recently
ak_statetransfer , where it was shown that pretty good state transfer can be
achieved using adaptive QEC protocols on spin chains. We will now briefly
describe such a noise-adapted protocol for quantum state transfer over a $1$-d
Heisenberg chain.

Consider the problem of transmitting a single qubit worth information across a
spin-$1/2$ chain with the interaction Hamiltonian given by,

$\mathcal{H}=-\sum_{k}J_{k}\left(\sigma^{k}_{x}\sigma^{k+1}_{x}+\sigma^{k}_{y}\sigma^{k+1}_{y}\right)-\sum_{k}\tilde{J}_{k}\sigma_{z}^{k}\sigma^{k+1}_{z}+\sum_{k}B_{k}\sigma_{k}^{z},$
(13)

where, $\\{J_{k}\\}>0$ and $\\{\tilde{J}_{k}\\}>0$ are site-dependent exchange
couplings of a ferromagnetic spin chain, $\\{B_{k}\\}$ denote the magnetic
field strengths at each site, and,
$(\sigma^{k}_{x},\sigma^{k}_{y},\sigma^{k}_{z})$ are the Pauli operators at
the $k^{\rm th}$ site. The standard state transfer protocol works by
initialising a single spin at site $s$ (_sender_) to the quantum state to be
transferred and then _receiving_ it at site $r$. It is easily shown that if
the spins are interacting via the Hamiltonian in Eq. (13), this leads to a
quantum channel $\mathcal{E}$ from the input spin to the receiver’s end with
Kraus operators,

$E_{0}=\left(\begin{array}[]{cc}1&0\\\
0&f_{r,s}^{N}(t)\end{array}\right),\;E_{1}=\left(\begin{array}[]{cc}0&\sqrt{1-|f_{r,s}^{N}(t)|^{2}}\\\
0&0\end{array}\right).$ (14)

The Kraus operators in Eq.(14) lead to a channel that has the same structure
as the amplitude-damping channel described in Eq. (1), but is more general
since the parameter $f_{r,s}^{N}(t)$ characterizing the noise in the channel
is complex. The parameter $f_{r,s}^{N}(t)$ is the so-called _transition
amplitude_ , $r$ refers to the receiver’s site and $s$ refers to the senders
site and $N$ refers to the total number of spins on the spin chain.

Since the quantum channel $\mathcal{E}$ looks very similar to amplitude-
damping channel, the adaptive QEC protocol in ak_statetransfer  picks the
$4$-qubit code in Eq. (3) and the Petz recovery map in Eq. (10) adapted to
this code to achieve state transfer with pretty good fidelity. The plots in
Fig. 6 describes the performance of such an adaptive QEC protocol on a spin
chain of length $N$ subject to the $XXX$ interaction, with $\\{B_{k}=0\\}$ and
$\\{J_{i}=J\\}$ in Eq. (13).

Figure 6: a) Spin transfer on an ideal XXX chain (left). b) Spin transfer on a
disordered XXX chain (right)

In these plots, the performance of the state transfer protocol from the first
to the $N^{\rm th}$ site is characterised using the worst-case fidelity as
function of the length $N$ of the spin chain. The first plot in Fig. 6 shows
the state transfer fidelities for an ideal $XXX$ chain, whereas the second
plot shows the fidelities obtained for disordered spin chain with disorder
strength $\delta$. In this case, the disorder-averaged worst-case fidelity
$\langle F_{\rm min}^{2}\rangle_{\delta}$ is the performance metric, and the
noise-adapted Petz recovery is constructed in terms of the disorder-averaged
transition amplitude $\langle f_{r,s}^{N}(t)\rangle_{\delta}$. It is shown
both analytically as well as numerically that for small disorder strengths the
disorder-averaged transition amplitude is the same as the no-disorder case
ak_statetransfer  and the variance in the transition amplitude is vanishingly
small. This makes a convincing case for using the noise-adapted Petz recovery
to achieve pretty good state transfer over ideal as well as disordered spin
chains.

###  4.2 The Petz map in Ads/CFT

Moving on from many body systems to holography, we now provide a brief summary
of the interesting emerging connections between perfect/approximate quantum
codes and the bulk-boundary correspondence in Ads/CFT. The AdS/CFT duality
proposes a correspondence between a theory of quantum gravity on a
$d+1$-dimensional (asymptotically) anti-deSitter (AdS) spacetime and a
conformal field theory (CFT) in one less spatial dimension defined on its
boundary. It was shown in the early works of Almheiri _et al_ almheiri2015
and Pastawski _et al_ happy2015 ; pastawski2017  that the framework of quantum
error correction provides an elegant characterization of this bulk-boundary
correspondence. The mathematical formalism of QEC is naturally suited to
answer the question of bulk reconstruction, namely, can one find a
representation of operators in the bulk gravity theory as operators acting on
a subregion of the boundary.

Viewing the AdS/CFT correspondence as a map from the bulk to the boundary, a
noisy quantum channel arises from tracing over a subregion of the boundary.
The problem of bulk reconstruction while having access only to a subregion of
the boundary, is then the same as the problem of recovering from this erasure
noise channel. Formally, it has been shown that such a reconstruction is
possible when there is an _exact_ equivalence of the bulk relative entropy and
boundary relative entropy dong2016 ; jlms2016 . In practice one can only
expect _approximate_ equality of the bulk and boundary relative entropies, and
this naturally leads to the question of whether approximate or noise-adapted
recovery maps like the Petz map might find use in the context of the bulk-
boundary reconstruction.

We will conclude with a quick survey of some recent results in this area and
refer to holography_qec  for a detailed review of this emerging field. The
first concrete proposal for an approximate recovery map that could solve the
problem of bulk reconstruction involved a variant of the Petz map, often
called the _twirled_ Petz map cotler2019 . Recall that the original state-
dependent Petz map was originally derived as the recovery map that
characterizes saturation of monotonicity of relative entropy under a noise
channel. The twirled Petz map is a generalization of the original Petz
construction which was shown to characterize approximate saturation of the
monotonicity of relative entropy under a noise channel junge2018 .
Subsequently, it was shown that approximate reconstruction of the bulk
operators can be achieved via an averaged version of the Petz map chen2020 .
In related work, a Petz-based reconstruction map was demonstrated for toy
models of holography based on random tensor networks jia2020petz . Ultimately,
a complete understanding of the role of approximate recovery maps in the
holographic setting requires a construction of such maps for general von
Neumann algebras. There has been some progress in this regard recently, with a
modified Petz map for arbitrary von Neumann algebras faulkner2022 , but the
quest is on to find a suitable physical construction of a universal,
approximate recovery map that can solve the bulk reconstruction puzzle in
holography.

##  5 Noise-adapted techniques for quantum fault tolerance

A practical quantum computer suffers from decoherence due to the inevitable
system-environment interactions. The theory of fault-tolerant quantum
computing preskill1998  provides a framework by which reliable and long
computations can be done even in the presence of noise, which could arise
either from gate imperfections or noisy quantum systems. A fault-tolerant
computation proceeds by first encoding the information into a quantum error
correcting code and then performing encoded gate operations on them in a
noise-resilient way. The errors generated are then to be removed periodically,
before they can cause irreparable damage.

Central to this theory is the _threshold theorem_ which provides a critical
threshold value, namely, the noise strength below which fault tolerance scheme
is successful and the encoded operations outperform the unencoded operations
aliferis2006 . This threshold number depends crucially on the QEC protocol
being used and the noise model assumption. For example, Ref. cross2009
provides a comparative study of the threshold values achieved by different
quantum codes for the case of depolarizing noise, wheres Ref. campbell2017
discusses the thresholds achievable by topological codes such as the surface
codes. The threshold numbers obtained thus far against various noise models by
specific quantum codes are tabulated in Table. 1.

As with standard QEC, much of the work on fault tolerance (FT) in the past has
been based on _perfect_ quantum codes that are essentially based on Pauli
noise models. However, in most cases, the dominant system noise is non-Pauli,
and in these cases the standard fault-tolerance scheme is too resourceful to
be implemented on current NISQ hardware. Recently, there have been efforts
towards developing noise-adapted fault tolerance schemes that are tailored to
the dominant noise processes in physical systems, and these will form our
focus in the following sections.

Fault-tolerant Scheme | Noise model | Threshold
---|---|---
General | Simple stochastic noise | $\sim 10^{-6}$
Polynomial codes (CSS codes) | general noise model | $10^{-6}$
Concatenated $7$-qubit code | Adversarial stochastic noise | $2.7\times 10^{-5}$
Error detecting concatenated codes ($C_{4}/C_{6}$) | depolarizing noise | $1\%$
Concatenated $7$-qubit code | Hamiltonian noise | $10^{-8}$
Concatenated repetition code | biased dephasing noise |  $0.5\%$, $0.24\%$
Surface codes | depolarizing |  $0.75\%$ , $0.5\%-1\%$
$[[4,2,2]]$ concatenated toric code | depolarizing | $0.41\%$
Table 1:  Noise thresholds for various quantum fault tolerance schemes thesis
.

###  5.1 Quasi-exact fault tolerance

One interesting approach that goes beyond the standard prescription of quantum
fault tolerance (FT) is the theory of _quasi-exact fault tolerance_ wang ;
quasi . This scheme demonstrates fault tolerance using a class of approximate
quantum codes called _quasi codes_. These are approximate quantum codes
defined in terms of scaling parameters, which when tuned can lead to exact
codes in some limit. The advantage of this framework of quasi-exact does is
that it allows one to interpolate between approximate and perfect codes and
correspondingly define parameters like (quasi) code distance which cannot be
defined for other classes of approximate codes. Examples of quasi codes
include valence-bond solid codes and symmetry-protected topological states
that occur naturally in quantum many body systems.

Quasi-exact FT further invokes a weaker version of universality referred to as
_quasi universality_. This is a weakening of the usual universality, which
induces a coarse-graining structure on the unitary group by identifying a
suitable cut-off. Specifically, a set of logical unitary operators, $U$ $\in$
$SU(d)$ is now realized as a coarse-grained unitary $U^{\prime}$ $\in$
$SU(d)_{\eta}$, where $\eta$ is the accuracy, which is the distance between
$U^{\prime}$ and $U$, and $SU(d)_{\eta}$ $\in$ $SU(d)$. One interesting aspect
of quasi-exact FT is that it opens up the possibility of circumventing the no-
go theorem for regular FT, namely, that transversality and universality cannot
coexist for exact codes eastin . The theory of quasi-exact codes and quasi-
exact fault tolerance does allow for transversality with _quasi-universality_
, but, the computations allowed by quasi-exact FT are only finite length. The
notion of quasi-exact FT is strictly weaker than the standard fault tolerance
since the uncorrectable errors accumulate with the number of operations
performed.

###  5.2 Achieving fault tolerance using noise-adapted quantum codes

Much of the work on quantum fault tolerance assumes that the noise is of the
Pauli type and aims to build encoded units that are tolerant against Pauli-
type faults. Even in cases where the fault tolerance scheme is tailored to a
specific noise model, the dominant noise has been typically of the Pauli type.
For instance, Ref. biased_noise  presents a fault-tolerant scheme specific to
dephasing noise using the $3$-qubit phase flip code. In contrast to such
approaches, a recent work demonstrates the possibility of biased-noise fault-
tolerance based on a noise model as well as QEC code that is non-Pauli ak_ft .
Specifically, this work assumes that the dominant noise affecting the physical
qubits is amplitude-damping channel. Expanding the amplitude damping channel
in Eq. (1) in the limit of small values of the noise parameter $p$,

$\mathcal{E}_{\mathrm{AD}}(\,\cdot\,)=\tfrac{1}{4}{\left(1+\sqrt{1-p}\right)}^{2}\mathcal{I}(\,\cdot\,)+p\mathcal{F}(\,\cdot\,)+{\left[\tfrac{1}{4}p^{2}+O(p^{3})\right]}Z(\,\cdot\,)Z,$
(15)

where $\mathcal{I}(\cdot)\equiv(\cdot)$ is the identity channel, and
$\mathcal{F}(\cdot)$ is the TP (but not CP) channel defined by,

$\mathcal{F}(\,\cdot\,)\equiv\tfrac{1}{4}{\left[(\,\cdot\,)Z+Z(\,\cdot\,)\right]}+E(\,\cdot\,)E^{\dagger}\equiv\tfrac{1}{2}\mathcal{F}_{z}(\,\cdot\,)+\mathcal{F}_{a}(\,\cdot\,).$
(16)

In Eq. (16), $\mathcal{F}_{Z}$ refers to an off-diagonal error leading to a
non-state operator and $\mathcal{F}_{a}$ refers to a damping error.

Following the basic principles of quantum fault tolerance developed in
aliferis2006 , Ref. ak_ft  constructs encoded units tolerant against faults of
upto $O(p)$ in Eq. (15) arising from amplitude-damping noise. Starting with
faulty physical units, and using the $4$-qubit code given in Eq. (3), it
describes a fault-tolerant universal encoded gate set comprising of
$\\{S,T,H,\textsc{cphase},\textsc{ccz}\\}$ and a fault-tolerant error
correction unit. Furthermore, going against the conventional wisdom of quantum
fault tolerance, this work shows that there could be transversal gates that
are not fault tolerant, such as the cnot gate in the case of amplitude-damping
noise. The transversal cphase gate turns out to be fault-tolerant to single
faults arising from ampltiude-damping noise. Thus, cphase gate is a _noise-
structure preserving_ gate, which is then used to implement the set of other
single-qubit encoded gates via teleportation. We note here that a similar idea
of identifying noise-bias-preserving gates has been recently discussed in the
context of systems that are subject to biased Pauli noise puri2020bias ;
xu2022 .

Finally, we briefly discuss the nosie-adapted recovery circuit used to
construct the fault-tolerant error correction (ec) unit in ak_ft . Since the
$4$-qubit code is approximate and adapted to a non-Pauli kind of noise, in
order to perform QEC one has to measure more than just the stabilizer set
defining the codespace. Specifically one needs to measure the set of two qubit
nearest neighbour $Z$ operators— $\\{ZZII,IIZZ\\}$ and single qubit $Z$
operator — $\\{ZIII\\}$ in order to find exactly where the damping has
occurred. Finally one measures $XXXX$ to restore the superposition and to kill
the off-diagonal errors. The basic units required to build the error
correction unit are shown in Fig. 7. These units are then strung up fault-
tolerantly ak_ft  with ancilla qubits and flag qubits to build a fault-
tolerant ec-unit. These fault-tolerant circuit constructions lead to an
estimation of a pseudo-threshold value for amplitude-damping noise. This was
estimated as $5.13\times 10^{-5}$ for the memory unit and $2.26\times 10^{-5}$
for the encoded cz-unit ak_ft .

Figure 7: Basic units of error correction using approximate $4$-qubit code.

##  6 Summary and Outlook

The current era of NISQ devices presents both a challenge and an opportunity
for theorists and experimentalists alike, from the perspective of error
correction and fault tolerance. On the one hand, in order to fully tap into
the potential of the available quantum technologies, one has to ascend the
ladder of _scalability_ and _control_ over the qubits simultaneously
jurcevic2021 . On the other hand, these NISQ devices also provide an
opportunity to rework our theoretical frameworks and perform experiments that
leverage NISQ devices to test theoretical ideas. These include, for instance,
experiments on the quantum cloud that validate the theory of quantum error
correction pokharel2018demonstration ; ghosh2018 , while also providing some
insight into the nature of the dominant noise process in the system repibm
and studies that demonstrate applications to quantum chemistry (see
nisq_review  for a recent review) and nuclear structure dumitrescu .

In this review, we have surveyed QEC approaches that deviate from the standard
QEC and fault tolerance formalism, and instead aim to correct for the dominant
noise affecting the quantum devices under consideration. Such noise-adaptive
QEC strategies are known to be less resourceful leung ; fletcher_AD ;
hui_prabha , but they also require non-trivial circuit constructions
gilyen2022  and syndrome extractions ak_ft  in contrast to the simpler Pauli
measurements for standard stabilizer codes. Thus, an immediate question to
address is whether such noise-specific techniques can be scaled up to be able
to suppress more errors and perform long computations reliably. One possible
approach towards such a scale up could be to concatenate a noise-adapted code
tailored to the dominant noise in the system with another QEC code of the same
or different type. Alternately, one could use channel-adapted techniques on
lattice-based codes like the Bacon-Shor code piedrafita2017reliable  and
surface codes which have the ability to tolerate more errors depending on
their lattice size.

The NISQ era has also spawned the development of newer ideas such as _quantum
error mitigation_ cao2021nisq  which focuses on the limited goal of reducing
the effective noise levels in near-term quantum devices, rather than
implementing a complete QEC protocol. This in turn opens up the possibility of
merging such hardware-specific error mitigation strategies from standard QEC
and fault tolerance qem2022 . Going forward, efficient and powerful
optimization techniques could lead us to better adaptive QEC strategies with
optimal encodings and recovery. For instance, optimization involving machine
learning reinforcement  and other learning-based approaches including a
quantum variational strategy cao2022quantum  are already being explored.

In the coming years, we expect to realize qubits with longer coherence times
and gates with higher accuracy in the lab. In the meantime, learnings from
noise-adapted QEC and fault tolerance will continue to give us insights on how
well we can make use of NISQ devices, and develop strategies that will enable
us to make the transition from noisy quantum devices to robust and scalable
quantum computing architectures.

##  7 Acknowledgements

This work is supported in part by a Seed Grant from the Indian Institute of
Technology Madras, as part of the Centre for Quantum Information,
Communication and Computing. P.M. acknowledges financial support by the
Department of Science and Technology, Govt. of India, under Grant No.
DST/ICPS/QuST/Theme-3/2019/Q59.

##  8 Declarations

Funding and/or Conflicts of interests/Competing interests: On behalf of both
authors, the corresponding author states that there is no conflict of
interest.

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