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# Purification and correction of quantum channels by commutation-derived
quantum filters

Sowmitra Das<EMAIL_ADDRESS>Blackett Laboratory, Imperial
College London, London SW7 2AZ, United Kingdom  School of Data and Sciences,
Brac University, Dhaka 1212, Bangladesh  Jinzhao Sun
<EMAIL_ADDRESS>Blackett Laboratory, Imperial College London,
London SW7 2AZ, United Kingdom  Michael Hanks  Blackett Laboratory, Imperial
College London, London SW7 2AZ, United Kingdom  Bálint Koczor  Mathematical
Institute, University of Oxford, Oxford OX2 6GG, United Kingdom  M. S. Kim
Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom

###### Abstract

Reducing the effect of errors is essential for reliable quantum computation.
Quantum error mitigation (QEM) and quantum error correction (QEC) are two
frameworks that have been proposed to address this task, each with its
respective challenges: sampling costs and inability to recover the state for
QEM, and qubit overheads for QEC. In this work, we combine ideas from these
two frameworks and introduce an information-theoretic machinery called a
quantum filter that can purify or correct quantum channels. We provide an
explicit construction of a filter that can correct arbitrary types of noise in
an $n$-qubit Clifford circuit using $2n$ ancillary qubits based on a
commutation-derived error detection circuit. We also show that this filtering
scheme can partially purify noise in non-Clifford gates, such as T and CCZ
gates. In contrast to QEC, this scheme works in an error-reduction sense
because it does not require prior encoding of the input state into a QEC code
and requires only a single instance of the target channel. Under the
assumptions of clean ancillary qubits, this scheme overcomes the exponential
sampling overhead in QEM because it can deterministically correct the error
channel without discarding any result. The filter construction also provides a
concrete protocol to implement a large class of quantum superchannels. We
further propose an ancilla-efficient Pauli filter which can remove nearly all
the low-weight Pauli components of the error channel in a Clifford circuit
using only 2 ancillary qubits similar to flag error correction codes. We prove
that for local depolarising noise, this filter can achieve a quadratic
reduction in the average infidelity of the channel. The Pauli filter can also
be used to convert an unbiased error channel into a completely biased error
channel and thus is compatible with biased-noise QEC codes which have high
code capacity. These examples demonstrate the utility of the quantum filter as
an efficient error-reduction technique in quantum information processing.

##  I Introduction

A fundamental obstacle on the road to large-scale quantum computation is
protecting quantum information from errors. The theoretical framework proposed
for this is quantum error correction (QEC). In QEC, the state of a single
logical qubit is encoded into the larger Hilbert space of multiple physical
qubits. The errors act on the larger space in such a way that appropriate
measurements can determine the nature of the error and subsequent correction
operations can revert the system back to its initial state lidar2013quantum ;
nielsen2010quantum . However, the implementation of a fully error-corrected
quantum computer imposes lofty requirements on the qubit number and the
average fidelity of each gate operation. Quantum error mitigation (QEM) has
been proposed as an intermediate alternative to QEC, which mitigates errors in
expectation values by post-processing the results of noisy circuit-runs
temme_error_2017 ; li_efficient_2017 ; endo_practical_2018 ; sun2021mitigating
; huggins_virtual_2021 ; koczor_exponential_2021 ; bonet2018low ;
mcardle2019error ; yoshioka_generalized_2022 ; mcclean_decoding_2020 ;
strikis_learning-based_2021 ; czarnik2021error ; cai_quantum_2022 ;
endo_hybrid_2021 ; yang2024quantum ; huo_dual-state_2022 ; qin_error_2023 . As
a consequence, QEM cannot recover the noiseless state, in contrast to QEC.
More importantly, investigations invoking the data-processing inequality have
shown that exponentially many samples (in circuit width and depth) are
required to mitigate errors under typical noise models takagi2022fundamental ;
takagi2023universal ; tsubouchi2023universal ; quek2024exponentially , thus
defeating the advantage of quantum computing in the first place. This
indicates that purely relying on post-processing with no additional qubits
would not be adequate to scale these methods up. We would need to use
controlled amounts of quantum resources and intermediate forms of error-
correction to efficiently achieve error suppression without an exponential
sampling overhead, which is the major objective of this paper.

A number of recent works have tried to combine the ideas of QEM and QEC
lostaglio2021error ; piveteau2021error ; suzuki2022quantum , mostly focusing
on gate purification using probabilistic error cancellation (PEC). However,
since these methods apply QEM to a sub-routine of the QEC pipeline and do not
change the underlying structure of the QEM protocol, they still suffer from
the exponential sampling overhead induced by QEM. Recently there has been a
surge of interest in designing error-suppression protocols using repeated
calls to the circuit that we wish to purify, which can go beyond the regime of
just estimating expected values. Examples of such protocols are superposed QEM
miguel2023superposed , black-box error-suppression lee2023error , and virtual
channel purification liu2024virtual , which achieve this by coherently
controlling parallel or sequential calls to the original circuit. The amount
of error reduction obtained by these methods depends on the number of calls to
the noisy circuit. However, many of these protocols work probabilistically
lee2023error ; liu2024virtual , and they require many calls (at least two
calls miguel2023superposed ; lee2023error ; liu2024virtual ) to the noisy
channel to achieve a desired accuracy. From a practical standpoint, all these
protocols require complicated non-Clifford gates such as controlled SWAP
gates, which are hard to implement on current hardware and require additional
non-Clifford resources in the fault-tolerant regime.

In this work, we overcome the challenges set up by the limitations of QEM and
the above intermediate forms between QEM and QEC, under the assumptions of
perfect ancillary qubits and Clifford operations on ancillae. We introduce an
information-theoretic machinery called a quantum filter. Formally, a quantum
filter provides a general way to implement a large class of quantum
superchannels. As an application of this formalism, we propose a method to
detect or correct errors using a single query to the circuit and simple
controlled operations (specifically, controlled Pauli operations). We first
introduce the idea of the commutation-derived filter, which can probe a
channel by exploiting the commutation structure of the channel components, and
filter out unwanted components. Then, we extend this construction to correct
the noise component instead of just discarding it, such that we can
deterministically recover the noiseless state. Although we formulate the
correction scheme for a stochastic Pauli channel, we show that given the
commutation structure, it can be used to purify arbitrary quantum channels. We
give an explicit construction to completely purify Clifford channels, which
shares similarity with space-time codes of Clifford circuits
delfosse2023spacetime , though our filter can work for arbitrary input states.
The construction of the quantum filter, which satisfies the commutation
relations, is inspired by the linear-combination-of-unitaries circuits
childs2012hamiltonian ; sun2024high . We also show that the filter can purify
non-Clifford gates, such as T-gates and CCZ-gate, towards $Z$-biased noise. We
validate the theoretical analysis of the correction filter with numerical
results, where ancillary qubit noise is considered as well.

For near-term quantum devices, it is desirable to introduce fewer resources
than those required to correct the entire channel. We provide the construction
of an ancilla-efficient Pauli filter, which operates in the error-detection
mode using only 2 ancillae and can remove all the weight-1 Pauli errors in a
noisy Clifford channel, sharing similarities with flag error correction codes
chao2018quantum ; chao2020flag ; chamberland2018flag , optimal distance-$2$
code self2024protecting  and the Iceberg code self2024protecting . We show
that this ancilla-efficient Pauli filter can additionally remove a large
portion of the other low-weight Pauli components of the error channel.
Assuming a local depolarising noise model, this results in at least a
quadratic reduction in the infidelity of the channel, which exceeds the result
in lee2023error  requiring only a single query to the original circuit, albeit
for the restricted class of circuit and noise. Even in scenarios where we
cannot correct the entire channel, we show that we can bias the errors towards
a certain axis (either $X$, $Y$ or $Z$) using a Pauli filter. Therefore, this
protocol is compatible with biased-noise QEC codes which usually have high
code capacity tuckett2018ultrahigh ; bonilla2021xzzx ; tuckett2019tailoring ;
xu2023tailored . The results established in this work demonstrate the
versatility of the quantum filter as an error-reduction method - one can use
it either in error detection or correction mode, tune the amount of
purification based on the number of ancillary qubits available, select which
gates to purify and also which noise components to remove. Thus, flexibility
in configuring a quantum filter’s capabilities is shown depending on the
amount of resources at hand.

##  II Quantum Channel Filtration

In this section, we introduce the concept of the quantum filter, restricting
our examples to simple storage/communication channels. The main idea of
quantum channel filtration is that we can probe an erroneous quantum channel
by using an error-free operation that obeys a certain commutation relation
with respect to the ideal channel. The probing operation, if chosen properly,
separates the error in the original channel into sub-channels. Depending on
the measurement outcome, the computation is then either aborted or continued.
This is reminiscent of filtering communication channels by their frequency
components in classical communication theory, which motivates us to name this
scheme a quantum filter (this is different from a similarly named scheme
introduced in lee2023error ). We show the basic principles of this
construction in subsection II.1.

As a single probe may not be sufficient to filter out all the errors, we show
how to successively concatenate multiple probes to achieve higher purification
of the input channel in subsection II.2. Finally, we show in subsection II.3
that instead of discarding the erroneous channels, it is possible to correct
the errors based on the outcomes of the probes. Given the present formalism
depends only on the quantum channel to be purified—and has no dependence on
the input state—we refer to this framework as quantum channel filtration and
quantum channel correction.

###  II.1 The commutation-derived quantum filter

First, we develop the machinery of commutation structures that we are going to
exploit to detect and filter out errors in a quantum channel.

##### Commutation-derived error detection circuit

Suppose our aim is to ideally implement a unitary $U$ but in a way that we can
detect if certain errors have occurred during execution (but without prying
open $U$, or measuring the output state). To do this, suppose we have access
to a unitary $S$ which commutes with $U$, i.e., $[S,U]=0$. We then use the
following circuit to probe $U$ using $S$ as

$\ket{0}$$\ket{\psi}$$S$$U$$S^{\dagger}$$\ket{{\tt out}}$

It is straightforward to see that the output state $\ket{\tt out}$ of the
circuit (before the measurement) is:

$\ket{\tt
out}=\frac{1}{2}\ket{0}\otimes(U+S^{\dagger}US)\ket{\psi}+\frac{1}{2}\ket{1}\otimes(U-S^{\dagger}US)\ket{\psi}$
(1)

If $U$ commutes with $S$, the output state will be

$\ket{\tt out}=\ket{0}\otimes U\ket{\psi}.$ (2)

Hence, we always get the outcome $\ket{0}$ when measuring the ancilla. If we
incidentally get an outcome of $\ket{1}$, we can be certain that an error has
occurred in $U$ and we can discard the circuit-run to avoid the error.

##### From Detection to Filtration

Upon taking a closer look at Eq. 1, we can see that, based on the measurement
outcome of the ancilla, the qubits in the state $\ket{\psi}$ collapse to the
following states as

$\displaystyle\ket{0}:(U+{S}^{\dagger}US)\ket{\psi}$
$\displaystyle\ket{1}:(U-{S}^{\dagger}US)\ket{\psi}$

Now, if we assume that the unitary $U$ has the property that, it either
commutes or anti-commutes with another unitary $P$,

$PU=\pm UP$ (3)

then, if we plug in $P$ in place of $S$, the output of the circuit is

$\ket{\tt out}=\begin{cases}\ket{0}\otimes U\ket{\psi}&\text{if
}U\;\text{commutes with}\;P\\\ \ket{1}\otimes U\ket{\psi}&\text{if
}U\;\text{anti-commutes with}\;P\end{cases}$ (4)

Therefore, the measurement result on the ancilla reveals the commutation
structure of $U$ with respect to $P$, which can be used to detect errors in
the channel.

##### Filtering Quantum Channels

Now, we generalize the above error detection circuit further to a quantum
filter. Consider the stochastic channel,
$\mathcal{E}=\sum_{i}p_{i}\mathcal{U}_{i}$ where $p_{i}$ are probabilities and
$\mathcal{U}_{i}$ are unitary channels that act by conjugation as
$\mathcal{U}_{i}(\rho)=U_{i}\rho U_{i}^{\dagger}$. Applying our error
detection circuit in to this channel yields

$\ket{0}$$P$$\mathcal{E}$$P^{\dagger}$

Suppose that there exists a $P$, such that it either commutes or anti-commutes
with the unitaries $U_{i}$ as

$PU_{i}=\pm U_{i}P\qquad{\rm for\;all\;}i.$ (5)

This condition divides the components of $\mathcal{E}$ into 2 disjoint
subsets: those which commute, and, those which anti-commute with $P$. Let us
denote these components as $\\{U_{j}^{+}\\}$ and $\\{U_{k}^{-}\\}$ with
corresponding probabilities $\\{p_{j}^{+}\\}$ and $\\{p_{k}^{-}\\}$. Then,
conditioned on the measurement result, the channel components $U_{j}^{+}$ and
$U_{k}^{-}$ are transformed to

$\displaystyle\ket{0}$ $\displaystyle:U_{j}^{+}\mapsto U_{j}^{+},\qquad
U_{k}^{-}\mapsto 0$ $\displaystyle\ket{1}$ $\displaystyle:U_{j}^{+}\mapsto
0,\quad\qquad U_{k}^{-}\mapsto U_{k}^{-}$

Thus, the entire channel $\mathcal{E}$ is transformed to -

$\displaystyle\ket{0}:\mathcal{E}\mapsto\mathcal{E^{+}}=\frac{1}{p^{+}}\sum_{j}p_{j}^{+}\cdot\mathcal{U}_{j}^{+}$
(6)
$\displaystyle\ket{1}:\mathcal{E}\mapsto\mathcal{E^{-}}=\frac{1}{p^{-}}\sum_{k}p_{k}^{-}\cdot\mathcal{U}_{k}^{-}$
(7)

with $p^{+}=\sum_{j}p_{j}^{+}$, and $p^{-}=\sum_{k}p_{k}^{-}$. The probability
of the two outcomes is $p(\ket{0})=p^{+}$ and $p(\ket{1})=p^{-}$.

As such, post-selecting on the outcome $\ket{0}$ ($\ket{1}$) of the ancilla
qubit removes all components of the channel $\mathcal{E}$ that anti-commute
(commute) with $P$. Given this circuit selects components of a channel based
on their commutation structure with respect to a unitary $P$, we will refer to
it as a commutation filter. More generally, we call any circuit which can
filter out certain components of a noise channel a quantum filter.
Furthermore, as the present quantum filter maps one quantum channel
($\mathcal{E}$) to another ($\mathcal{E^{+}}$ or $\mathcal{E^{-}}$), it is
formally a quantum superchannel chiribella2008transforming . A description of
quantum filters using the language superchannels is discussed in subsection
III.1.

##### Generalization

The commutation filter can be generalized to arbitrary channels $\mathcal{E}$
and more general operators $P$. We first provide the following proposition.

######  Proposition 1.

For an arbitrary operator $A$ and unitary operator $F$ such that, $F^{2}$
commutes with $A$, $[F^{2},A]=0$ we can write $A$ as the sum of two
components, $A=A_{+}+A_{-}$ such that, $FA_{\pm}=\pm A_{\pm}F$ i.e, one of
them commutes with $F$ and the other anti-commutes with $F$.

###### Proof.

We provide a proof by construction. Consider the operators,

$A_{\pm}:=(A\pm FAF^{\dagger})/2$ (8)

It is easy to check that $A_{+}+A_{-}=A$. Plugging these expressions into
$FA_{\pm}$, and using $F^{2}A=AF^{2}$, one can verify the equalities
$FA_{\pm}=\pm A_{\pm}F$, which concludes our proof. ∎

Let us assume an arbitrary channel $\mathcal{E}$ in terms of its Kraus
operators $E_{i}$ as $\mathcal{E}(\rho)=\sum_{i}E_{i}\rho E_{i}^{\dagger}$,
such that an operator $F$ exists whose square $F^{2}$ commutes with all Kraus
operators $E_{i}$ as

$[F^{2},E_{i}]=0\qquad\forall\,i.$ (9)

Due to the above proposition, it then follows that all Kraus operators $E_{i}$
indeed decompose as a sum of commuting and anti-commuting components as

$E_{i}=E_{i+}+E_{i-}.$ (10)

Plugging the channel $\mathcal{E}$ into a commutation filter via $P=F$, the
filter selects either the commuting or anti-commuting components depending on
the outcome of the ancilla measurement as

$\displaystyle\ket{0}:\mathcal{E}\mapsto\mathcal{E}^{+}=\frac{1}{e^{+}}\sum_{i}E_{i+}(\cdot)E_{i+}^{\dagger}$
(11)
$\displaystyle\ket{1}:\mathcal{E}\mapsto\mathcal{E}^{-}=\frac{1}{e^{-}}\sum_{i}E_{i-}(\cdot)E_{i-}^{\dagger}$
(12)

where $e^{\pm}$ are normalization factors,

$e^{\pm}=\sum_{i}\Tr[E_{i\pm}(\cdot)E_{i\pm}^{\dagger}].$ (13)

A subset of all admissible operators $F$ are involution operators whose square
is by definition the identity operator as $F^{2}=I$. Well-known examples of
such operators are Pauli strings and SWAP operators, which has been discussed
in liu2024virtual  in a similar context. For involution operators, Eq. 9 holds
trivially.

###  II.2 Channel purification by successive filtration

We have shown that given an appropriately chosen probing operator $P$, the
commutation filter can remove certain erroneous components of an input
channel. Usually, a single filtration step will not be able to remove all the
errors. Here, we show how to concatenate successive filtration steps to
potentially remove all the error components in the channel.

We return to our first example whereby we aim to implement a unitary channel
$\mathcal{U}_{0}(\rho)=U_{0}\rho U_{0}^{\dagger}$ and assume a noise model
such that the actual channel is
$\mathcal{U_{E}}(\rho)=\sum_{i=0}^{m}p_{i}\cdot\mathcal{U}_{i}$. Here the
index value $i>0$ indicates unitary $U_{i}$ as the noise-component. Now, if we
can find an operator $F$ which commutes with $U_{0}$ and anti-commutes with
all other $U_{i}$’s, we can filter out $U_{0}$ by plugging $\mathcal{E}$ into
a commutation filter with $P=F$. The probability of successful filtration will
be $p_{s}=p_{0}$.

In general, a single $F$ which commutes with $U_{0}$ and anti-commutes with
all other $U_{i}$ might not exist, or might be difficult to construct.
Instead, we may construct a set of multiple operators $\\{F_{i}\\}$ such that
only $U_{0}$ commutes with all of them, and all the other $U_{i}$’s anti-
commute with at least one $F_{i}$. Then, we can purify the channel
$\mathcal{U_{E}}(\rho)$ by successively filtering it using
$F_{1},F_{2},\ldots$. If all the rounds are successful, the component $U_{0}$
will survive all the filters, and all other components will be removed by at
least one filter. The total probability that all the rounds will be successful
is still $p_{0}$.

##### Noiseless Channel from a Noisy Pauli Channel

Consider the (single-qubit) stochastic Pauli-channel which would ideally
($p_{I}=1$) implement the identity operation as

$\mathcal{E}(\rho)=p_{I}\cdot\rho+p_{X}\cdot X\rho X+p_{Y}\cdot Y\rho
Y+p_{Z}\cdot Z\rho Z,$ (14)

where the channel components $I,X,Y$ and $Z$ are Pauli operators and
$p_{I}+p_{X}+p_{Y}+p_{Z}=1$. Consider the commutation filters
$\bm{\mathcal{F}}_{Z},\bm{\mathcal{F}}_{X}$ using the Pauli operators
$P\in\\{Z,X\\}$ implemented one after the other as in Fig. 1. If we filter
$\mathcal{E}$ first using $Z$, and then using $X$ (post-selecting on the
measurement 0 each time), the components that survive both steps are

$\\{I,X,Y,Z\\}\xrightarrow[\ket{0}]{\quad\bm{\mathcal{F}}_{Z}\quad}\\{I,Z\\}\xrightarrow[\ket{0}]{\quad\bm{\mathcal{F}}_{X}\quad}\\{I\\}.$
(15)

The probability of success of the first round is $p_{s1}=p_{I}{+}p_{Z}$ while
the probability of success of the second round, given that the first round was
successful, is $p_{s2|s1}={p_{I}}/(p_{I}+p_{Z})$. Indeed, the probability that
the entire filtration protocol is successful is given by

$p_{s}=p_{s1}\cdot p_{s1|s2}=p_{I}.$ (16)

$\ket{0}$$X$$\ket{0}$$Z$$\mathcal{E}$$Z$$\bra{0}$$X$$\bra{0}$ Figure 1:
Successive filtration scheme for purifying a single-qubit error channel.

As can be seen, we can probabilistically purify a single-qubit noisy Pauli
channel using a $Z$-Filter and an $X$-Filter in sequence. This can be easily
generalized to an arbitrary number of qubits. To purify an $n$-qubit Pauli
channel using Successive Filtration, we need $2n$ ancillary qubits, $n$
$Z$-Filters and $n$ $X$-Filters.

###  II.3 Quantum channel correction

In the previous section, we have shown that a channel can be purified by
successively discarding erroneous components at each filtration step. The part
of the channel that survives all the steps has all the corresponding errors
removed. Whether a certain step will be successful depends on the
probabilistic outcome of the probe. Here we show that, by slight modification
of the above scheme, instead of probabilistically purifying a quantum channel,
we can deterministically correct the channel. This stems from the observation
that if a round is unsuccessful, we can still use the information from the
measurement outcomes to correct for the corresponding error components.

Suppose that in example II.2, the first round with the $Z$-Filter is
unsuccessful, i.e, the outcome of the first ancilla is $\ket{1}$. Then, the
components of the channel which anti-commute with $Z$ survive resulting in the
mapping

$\\{I,X,Y,Z\\}\xrightarrow[\ket{1}]{\quad\bm{\mathcal{F}}_{Z}\quad}\\{X,Y\\}$
(17)

In this case, the channel $\mathcal{E}$ above is transformed to

$\mathcal{E}_{1}=\frac{1}{p_{X}+p_{Y}}\Big{(}p_{X}X(\cdot)X+p_{Y}Y(\cdot)Y\Big{)},$
(18)

and thus applying the Pauli operator $X$ converts all components of this
channel to ones that commute with $Z$ as

$X\cdot\\{X,Y\\}\rightarrow\\{I,Z\\},$ (19)

thereby deterministically converting all channel components to $\\{I,Z\\}$.
Now, if we repeat this procedure for the $X$-Filter, and apply the $Z$ gate
conditioned on the ancilla of this filter returning 1, we can
deterministically convert both components to $I$. As a result, at the end of
both rounds, the channel will be corrected to the identity channel. The
circuit that implements this scheme of quantum channel correction is shown in
Fig. 2.

$\ket{0}$$\ket{0}$$X$$Z$$\mathcal{E}$$Z$$X$$X$$Z$$(\mathcal{I})$ Figure 2: An
example of a Quantum Filter for channel correction.

Although we have outlined the scheme for Pauli errors on a single-qubit, this
is sufficient to correct any arbitrary errors on a qubit by a standard result
in the theory of quantum error correction shor1995scheme . We provide proof of
this result for the concatenated Pauli correction filter of Fig. 2 in Appendix
A. A consequence of this result is that, although many QEM schemes are limited
to correcting stochastic errors liu2024virtual ; lee2023error  or known types
of noise (which is required by PEC temme_error_2017 ), our quantum channel
correction scheme works for both stochastic Pauli errors and also for more
general error models that may include coherent errors. As our result applies
to arbitrary single-qubit error channel correction, it can be easily seen that
any $n$-qubit error channel can be corrected by running this scheme in
parallel on each qubit using $2n$ ancillary qubits.

##  III Construction of a quantum filter

In this section, we extend our description beyond simple communication
channels, providing an explicit construction to correct for noise in arbitrary
Clifford circuits. This requires that we first develop a more general
description of the filter circuit. We formulate this in the language of
quantum superchannels.

Briefly, a quantum superchannel $\bm{\mathcal{F}}$ (written in bold
calligraphic) is a map that takes quantum channels to quantum channels. A
superchannel $\bm{\mathcal{F}}$ is described by a set of super-Kraus elements
$\\{\bm{F}_{j}\\}_{j}$, such that, if $\bm{\mathcal{F}}$ acts on a quantum
channel $\mathcal{E}$ with Kraus operators $\\{E_{k}\\}_{k}$, the Kraus
operators of the transformed channel $\bm{\mathcal{F}}[\mathcal{E}]$ are

$\bm{\mathcal{F}}[\mathcal{E}]:\\{E_{k}\\}_{k}\longrightarrow\\{\bm{F}_{j}[E_{k}]\\}_{j,k}$
(20)

Each of the $\bm{F}_{j}$’s are linear maps on the space of operators $E_{k}$.
If $\mathcal{E}$ acts on a state $\rho$ as

$\mathcal{E}(\rho)=\sum_{k}E_{k}\rho E_{k}^{\dagger},$ (21)

the transformed channel $\bm{\mathcal{F}}[\mathcal{E}]$ acts on $\rho$
according to

$\bm{\mathcal{F}}[\mathcal{E}](\rho)=\sum_{j}\sum_{k}\bm{F}_{j}[E_{k}]\rho\bm{F}_{j}[E_{k}]^{\dagger}.$
(22)

If the set $\\{\bm{F}_{j}\\}$ contains a single element, we call
$\bm{\mathcal{F}}$ a unitary superchannel. Readers interested in this
formalism may consult chiribella2008transforming ; chiribella2008quantum ;
daly2022axiomatic  for further details.

###  III.1 Quantum filters as superchannels

First, we focus on the circuit of the basic quantum filter shown below as

Figure 3: Quantum filter outlining the filter supermap

As shown before in Eq. 1, the output of this circuit is

$\ket{\tt
out}=\frac{1}{2}\ket{0}\otimes(U+{P}^{\dagger}UP)\ket{\psi}+\frac{1}{2}\ket{1}\otimes(U-{P}^{\dagger}UP)\ket{\psi}$

Consider the part of the circuit outlined in blue, which we denote by
$\bm{F}_{P}$. This can be thought of as a circuit element that takes the
unitary $U$ as input, and produces an isometry that takes the system qubits to
the combined system-ancilla state. This map is described as

$\displaystyle\bm{F}_{P}[U]:$ $\displaystyle
U\mapsto\frac{1}{2}\ket{0}\otimes(U+{P}^{\dagger}UP)+\frac{1}{2}\ket{1}\otimes(U-{P}^{\dagger}UP)$
(23)

In this form, instead of thinking about how the input state is transformed, we
focus on how the filter transforms the unitary $U$ (or the channel
$\mathcal{U}=U(\cdot)U^{\dagger}$). We call the associated map
$\bm{\mathcal{F}}_{P}$, that transforms the channel $\mathcal{U}$ to
$\bm{\mathcal{F}}_{p}[\mathcal{U}]$, the Filter Superchannel
chiribella2008transforming .

$\bm{\mathcal{F}}_{p}[\mathcal{U}](\rho)=\bm{F}_{p}[U]\rho\bm{F}_{p}[U]^{\dagger}$
(24)

Since $\bm{\mathcal{F}}_{P}$ is defined by a single map $\bm{F}_{P}$, it is a
unitary supermap daly2022axiomatic . In this language, $\bm{F}_{P}$ is a
super-Kraus element of the superchannel $\bm{\mathcal{F}}_{P}$.

####  III.1.1 Symmetric (Unitary) Pauli Filters

We illustrate the above formulation by analyzing the (single-qubit) Pauli $Z$
and Pauli $X$ filters, i.e, $P~{}\in~{}\\{Z,X\\}$. We denote these filters
simply by $\bm{\mathcal{F}}_{Z},\bm{\mathcal{F}}_{X}$ respectively. For
convenience, we define $(I,Z,X,Y)=(P_{00},P_{01},P_{10},P_{11})$. Using Eq.
23, we can show that,

$\displaystyle\bm{F}_{Z}[P_{u}]=\ket{u_{0}}\otimes P_{u}$ (25)
$\displaystyle\bm{F}_{X}[P_{u}]=\ket{u_{1}}\otimes P_{u}$ (26)

where $u=u_{0}u_{1}\in\\{00,01,10,11\\}$. Successive filtration by these two
filters (Fig. 1) is equivalent to concatenating $\bm{\mathcal{F}}_{Z}$ and
$\bm{\mathcal{F}}_{X}$. The resulting super-Kraus element $\bm{F}_{Z;X}$ is
defined (on the Paulis) as

$\bm{F}_{Z;X}[P_{u}]=\bm{F}_{Z}[\bm{F}_{X}[P_{u}]]=\ket{u_{0}}\otimes\ket{u_{1}}\otimes
P_{u}=\ket{u}\otimes P_{u}$

Now, if the Pauli Transfer Matrix $\chi_{u,v}$ of a (single-qubit) quantum
channel $\mathcal{E}$ is given, such that

$\mathcal{E}(\rho)=\sum_{u,v}\chi_{u,v}P_{u}\rho P_{v},$

by applying the filter $\bm{\mathcal{F}}_{Z;X}$ on this channel, we have,

$\displaystyle\bm{\mathcal{F}}_{Z;X}[\mathcal{E}](\rho)$
$\displaystyle=\sum_{u,v}\chi_{u,v}\bm{F}_{Z;X}[P_{u}]\rho\bm{F}_{Z;X}[P_{v}]^{\dagger}$
$\displaystyle=\sum_{u,v}\chi_{u,v}\outerproduct{u}{v}\otimes P_{u}\rho P_{v}$
(27)

Hence, for any stochastic Pauli channel of the form

$\mathcal{E}=p_{I}\mathcal{I}+p_{Z}\mathcal{Z}+p_{X}\mathcal{X}+p_{Y}\mathcal{Y}$
(28)

where $\mathcal{X}(\rho)=X\rho X$, etc. and for which the matrix $\chi_{u,v}$
is diagonal, we have,

$\displaystyle\bm{\mathcal{F}}_{Z;X}[\mathcal{E}]=\,$ $\displaystyle
p_{I}\outerproduct{00}{00}\otimes\mathcal{I}+p_{Z}\outerproduct{01}{01}\otimes\mathcal{Z}$
$\displaystyle+$ $\displaystyle
p_{Y}\outerproduct{11}{11}\otimes\mathcal{Y}+p_{X}\outerproduct{10}{10}\otimes\mathcal{X}$
(29)

Thus, the ancilla now carries information about the channel components, which,
as we have seen, can be used to filter them out or perform further post-
processing. Interestingly, Eq. 27 suggests that the coefficients $\chi_{u,v}$
might be found out by performing state-tomography on the ancillary qubits.
This would be equivalent to performing process tomography of the channel
$\mathcal{E}$. The quantum filter formalism can be directly applied to process
tomography, and we leave it to future work.

####  III.1.2 Non-Unitary Filters

In the above treatment, we considered only unitary operators as part of the
filter circuitry. However, we may extend this treatment further to include
non-unitary operations such as measurements and conditional operations as
well. Suppose, we choose to discard the filtering qubits from the circuit in
Fig. 3, as shown in Fig. 4.

Figure 4: Non-unitary Quantum Filter

The resulting superchannel is $\widetilde{\bm{\mathcal{F}}}$. The action of
$\widetilde{\bm{\mathcal{F}}}$ may be found by tracing out the ancilla in Eq.
24,

$\displaystyle\widetilde{\bm{\mathcal{F}}}[\mathcal{U}](\rho)$
$\displaystyle=\bra{0}\bm{F}[U]\rho\bm{F}[U]^{\dagger}\ket{0}+\bra{1}\bm{F}[U]\rho\bm{F}[U]^{\dagger}\ket{1}$
$\displaystyle=\bm{F_{1}}[U]\rho\bm{F_{1}}[U]^{\dagger}+\bm{F_{2}}[U]\rho\bm{F_{2}}[U]^{\dagger}$
(30)

where in the first line, the inner product is over the ancillary qubits, and
in the second line, we define -

$\displaystyle\bm{F_{1}}[U]$
$\displaystyle=\frac{1}{2}IUI+\frac{1}{2}P^{\dagger}UP$ (31)
$\displaystyle\bm{F_{2}}[U]$
$\displaystyle=\frac{1}{2}IUI-\frac{1}{2}P^{\dagger}UP$ (32)

The maps $\bm{F_{1}},\bm{F_{2}}$ are super-Kraus elements of the filter
$\widetilde{\bm{\mathcal{F}}}$.

Thus, we have translated the Quantum Filter formalism in terms of the theory
of quantum superchannels, which gives a natural language for our scheme. In
Appendix A.1, we give an analysis of the channel-correction filter using this
formalism, which may be used to provide an alternate proof of the channel
correction property mentioned at the end of Section II.3.

###  III.2 Beyond symmetry: general construction of filters

Now, we extend the construction of a symmetric commutation filter to more
general quantum filters. First, we observe that our original Error-Detector
(conditioned on the output of the ancilla being 0) maps $U$ to

$U\mapsto\frac{1}{2}U+\frac{1}{2}{P}^{\dagger}UP$ (33)

This can be seen as a convex combination of $IUI$ and ${P}^{\dagger}UP$ (with
equal probabilities). The equal coefficients come from the use of the Hadamard
gate, and, $I$, $P$ and ${P}^{\dagger}$ come from the Controlled-$P$ and
Controlled-${P}^{\dagger}$ gates. In general, we might think of mapping $U$ to
a general convex combination of the form

$U\mapsto\sum_{i}p_{i}\,Q_{i}UP_{i}$ (34)

for unitaries $P_{i},Q_{i}$ and probabilities $p_{i}$. This transformation can
be achieved by using the circuit in Fig. 5, whose construction is inspired
from the LCU circuits proposed by Childs and Wiebe childs2012hamiltonian .

Figure 5: A schematic of a quantum filter specified by the prepare unitary V
and the controlled unitaries $\textsc{Select}_{P}$ and $\textsc{Select}_{Q}$.

This quantum filter $\bm{\mathcal{F}}$ is specified by the preparation unitary
$V$ and the controlled unitaries $\textsc{Select}_{P}$ and
$\textsc{Select}_{Q}$. The ancilla can be in general a multi-qubit register,
which we call the _Filtering Qubits_. The probabilities $p_{i}$ are encoded in
the unitary $V$, which is defined as

$V\ket{0}=\sum_{i}\sqrt{p_{i}}\ket{i}$ (35)

and, the $P_{i}$’s and $Q_{i}$’s are encoded in the Controlled Unitaries
$\textsc{Select}_{P}$ and $\textsc{Select}_{Q}$, which are defined as

$\displaystyle\textsc{Select}_{P}=\sum_{i}\outerproduct{i}{i}\otimes P_{i}$
(36) $\displaystyle\textsc{Select}_{Q}=\sum_{i}\outerproduct{i}{i}\otimes
Q_{i}$ (37)

where $\ket{i}$’s are the (computational) basis states of the filtering
qubits. Using $V\ket{j}=\sum_{i}v_{ij}\ket{i}$, one can easily check that, the
output of the circuit (before the measurement) is

$\displaystyle\ket{\rm out}$
$\displaystyle=\left(\sum_{i}\sum_{j}v_{i0}v_{ij}^{*}\ket{j}\otimes
Q_{i}UP_{i}\right)\ket{\psi}$ (38)
$\displaystyle=\ket{0}\otimes\left(\sum_{i}p_{i}\,Q_{i}UP_{i}\right)\ket{\psi}+{\rm
other\;terms}.$ (39)

So, if the measurement outcome on the ancilla is 0, we can achieve the
transformation of Eq. 34. In general, for any Quantum Channel $\mathcal{E}$
with Kraus Elements $E_{k}$, this filter (probabilistically) transforms
$\mathcal{E}$ into:

$\bm{\mathcal{F}}[\mathcal{E}]:E_{k}\xrightarrow{\;\ket{0}\;}\sum_{i}p_{i}\,Q_{i}E_{k}P_{i}$
(40)

In the special case, where all the $P_{i}$’s and $Q_{i}$’s are Paulis, we call
$\bm{\mathcal{F}}$ a _Pauli filter_. If
$\textsc{Select}_{P}=\textsc{Select}_{Q}$, we call this a _symmetric_ filter.
Although this implementation of a superchannel is probabilistic, the
probability of success may be arbitrarily improved by using amplitude
amplification methods grover1996fast ; brassard2002quantum .

If we post-select on the outcome $\ket{j}$ on the ancilla, the complete super-
Kraus representation of the general quantum filter $\bm{\mathcal{F}}$ in Fig.
5 is given by

$\bm{\mathcal{F}}[\mathcal{E}]:\bm{F_{j}}[E_{k}]\xrightarrow{\;\ket{j}\;}\sum_{i}\alpha_{ij}\,Q_{i}E_{k}P_{i}$
(41)

where the coefficients are $\alpha_{ij}=v_{i0}v_{ij}^{*}$. Hence, this method
gives us a way to implement a number of super-channels in one go.

###  III.3 Application: Clifford circuit purification using Pauli Filters

In this section, we focus on a specific application of the (general) quantum
filter constructed in the previous section in purifying Clifford Circuits. The
filtering scheme we provide uses only Hadamards and controlled-Pauli
operations, thus enabling us to purify (noisy) Clifford circuits using purely
Clifford gates.

Consider the following scenario: we are running a circuit compiled in the
Clifford+T gate-set. The circuit is arranged in layers of Clifford gates and T
gates (Fig. 6). We consider a local depolarising noise $\mathcal{D}_{p}$ of
strength $p$ on each qubit:

$\mathcal{D}_{p}(\rho)=(1-p)\cdot\rho+p\cdot\frac{1}{3}(X\rho X+Y\rho Y+Z\rho
Z)$ (42)

dots[inner sep=0pt]$\vdots$
$\bm{C}_{1}$$\mathcal{D}_{p}$$\bm{T}_{1}$$\bm{C}_{2}$$\mathcal{D}_{p}$$\bm{T}_{2}$$\cdots$$\bm{C}_{m}$$\mathcal{D}_{p}$$\bm{T}_{m}$

Figure 6: Schematic of a Circuit Compiled in the Clifford+$T$ gate-set.
$\bm{C}_{i}$ and $\bm{T}_{i}$ are circuit layers composed purely of Clifford
and T gates respectively. $\mathcal{D}_{p}$ is a local depolarising channel
with error-rate $p$.

The circuit in Fig. 6 has $m$ layers. Each layer $L_{i}$ is composed of a
Clifford unitary $\bm{C}_{i}$ and, a unitary $\bm{T}_{i}$ composed only of T
Gates. Each Clifford layer is followed by a layer of local depolarising noise.
We assume that the number of T Gates in each $\bm{T}_{i}$ is small (compared
to the number of gates in the $\bm{C}_{i}$’s), and thus the noise in them is
negligible.

If the number of qubits in the circuit is $n$, the combined channel after each
Clifford layer is :

$\displaystyle\mathcal{D}_{p}^{\otimes n}\circ\mathcal{C}_{i}(\rho)=$
$\displaystyle(1-p)^{n}\cdot\bm{C}_{i}\rho\bm{C}_{i}^{\dagger}+\sum_{w=1}^{n}(p/3)^{w}(1-p)^{n-w}\times$
$\displaystyle\sum_{k}N_{k}[w]\bm{C}_{i}\rho\bm{C}_{i}^{\dagger}N_{k}[w]^{\dagger}$
(43)

where $N_{k}[w]$ are Pauli operators with weight $w$ 111A Pauli of weight $w$
acts non-trivially on $w$ qubits. For example, the Pauli
$I_{1}X_{2}Y_{3}I_{4}Z_{5}$ has weight 3 and acts non-trivially on the qubits
2, 3, and 5., and

$\mathcal{C}_{i}(\rho)=\bm{C}_{i}\rho\bm{C}_{i}^{\dagger}$ (44)

is the (pure) Clifford channel. So, the components of the combined channel
consists of the noiseless Clifford term $\bm{C}_{i}$ and noise-terms
$N_{k}[w]\bm{C}_{i}$. In the following, we focus on a single such layer of
Clifford circuits under Pauli noise.

Consider one such component of the channel plugged into a Pauli filter

$\ket{0}$$P$$\bm{C}$$N$$Q$

If $P$ is a Pauli operator, this circuit is equivalent to

$\ket{0}$$\bm{C}$$\vec{P}$$N$$Q$

where we have defined

$\overrightarrow{P}:=\bm{C}P\bm{C}^{\dagger}$ (45)

as the forward propagation (in time) of the Pauli $P$ through the the Clifford
$\bm{C}$. Since the Clifford group is the normalizer of the Pauli group,
$\overrightarrow{P}$ is also a Pauli operator.

Now, if we select $P=\bm{C}^{\dagger}Q\bm{C}=\overleftarrow{Q}$ in the
original filter circuit (which is the backward propagation of $Q$ through
$\bm{C}$), then $\overrightarrow{P}=Q$ and the resulting circuit becomes
equivalent to

$\ket{0}$$\bm{C}$$Q$$N$$Q$

As a result, the noise-channel $N$ sees a symmetric Pauli filter
$\bm{\mathcal{F}}_{Q}$. In subsection II.3, we have already shown that an
arbitrary noise channel can be deterministically purified with $2n$ ancillae
by using a symmetric Pauli filter of the following configuration

$\displaystyle V$ $\displaystyle=H^{\otimes 2n}$ (46)
$\displaystyle\textsc{Select}_{Q}$
$\displaystyle=\sum_{i}\outerproduct{i}{i}\otimes Q_{i}$ (47)

where the sum on $i$ runs over all the $n$-qubit Pauli strings. Thus, the
noisy Clifford $\bm{C}$ will also be completely purified if we choose,

$\textsc{Select}_{P}=\textsc{Select}_{\overleftarrow{Q}}=\sum_{i}\outerproduct{i}{i}\otimes\bm{C}^{\dagger}Q_{i}\bm{C}$
(48)

If we are given access to $k$ ancillary qubits, we can completely clean $k/2$
system qubits using this filter configuration. Hence, the output fidelity
scales with the number of ancilla as

$F_{o}=(1-p)^{n-k/2}=F_{i}\cdot\left(1/\sqrt{1-p}\right)^{k}$ (49)

which is an exponential increase with respect to $k$.

###  III.4 Implementation of ${\textsc{Select}_{P}}$ and
${\textsc{Select}_{Q}}$

A problem of practical matters needs to be addressed regarding implementing
the unitaries $\textsc{Select}_{P}$ and $\textsc{Select}_{Q}$ in Eq. 47, Eq.
48. If the $2n$ filtering qubits are labelled as
$f_{1,1},f_{1,2},f_{2,1},f_{2,2},\ldots f_{n,1},f_{n,2}$, and the system
qubits as $s_{1},s_{2},\ldots s_{n}$, then, one choice of
$\textsc{Select}_{Q}$ is

$\textsc{Select}_{Q}=\bigotimes_{i=1}^{n}\textsf{C}_{f_{i,1}}Z_{s_{i}}\cdot\textsf{C}_{f_{i,2}}X_{s_{i}}$
(50)

where $\textsf{C}_{f_{i,1}}Z_{s_{i}}$ is a Controlled-$Z$ gate with control-
qubit $f_{i,1}$ and target-qubit $s_{i}$, etc. This requires $n$ $CZ$ and $n$
$CX$ gates.

Then, if we are trying to correct a Clifford $\bm{C}$, the corresponding
$\textsc{Select}_{P}$ will be

$\displaystyle\textsc{Select}_{P}$
$\displaystyle=\bm{C}^{\dagger}(\textsc{Select}_{Q})\bm{C}$
$\displaystyle=\bigotimes_{i=1}^{n}\bm{C}^{\dagger}(\textsf{C}_{f_{i,1}}Z_{s_{i}})\bm{C}\cdot\bm{C}^{\dagger}(\textsf{C}_{f_{i,2}}X_{s_{i}})\bm{C}$
$\displaystyle=\bigotimes_{i=1}^{n}\textsf{C}_{f_{i,1}}\overleftarrow{Z}_{s_{i}}\cdot\textsf{C}_{f_{i,2}}\overleftarrow{X}_{s_{i}}$

where

$\overleftarrow{Z}_{s_{i}}=\bm{C}^{\dagger}Z_{s_{i}}\bm{C}\qquad\overleftarrow{X}_{s_{i}}=\bm{C}^{\dagger}X_{s_{i}}\bm{C}$
(51)

are the backward propagation of the single-qubit Pauli-$X$ and Pauli-$Z$’s
through the Clifford $\bm{C}$.

In the worst case, each $\overleftarrow{Z}_{s_{i}},\overleftarrow{X}_{s_{i}}$
can be full-weight Paulis even though $Z_{s_{i}},X_{s_{i}}$ are single-weight
Paulis. So, each $\textsf{C}_{f_{i,1}}\overleftarrow{Z}_{s_{i}}$ may require
$n$ Controlled-$X$ gates to implement. As a result, to implement the entire
$\textsc{Select}_{P}$ unitary, we may need $2n^{2}$ $CX$-gates.

An interesting thing to note is that both the unitaries $\textsc{Select}_{P}$
and $\textsc{Select}_{Q}$ are composed entirely of Clifford gates. Combined
with the fact that the PREPARE unitaries of the ancilla for a Pauli filter are
$H^{\otimes 2n}$, we can see that the entire Pauli filter is a Clifford
circuit. So, we have shown that we can correct Clifford circuits using purely
Clifford circuits.

###  III.5 Partial purification of non-Clifford gates

In the above section, we have outlined how we can purify Clifford circuits
using Clifford gates, particularly controlled-Paulis. This scheme can be
adapted to the purification of non-Clifford gates such as T Gates, and more
generally, $R_{z}(\theta)=e^{-i\theta Z}$ gates which have applications in
quantum dynamics simulation by Trotterisation or eigenstate preparation
georgescu2014quantum . However, we will need non-Clifford gates to completely
purify circuits composed of T Gates. In the regime of fault tolerant quantum
computing, this is an expensive resource (where Clifford gates can be
considered almost free). Here, we outline a method to partially purify T Gates
using less expensive Clifford gates.

Consider the following circuit

$\ket{0}$$Z$$\mathcal{D}$$Z$$X$

where we model a noisy T-gate as an ideal T-gate followed by a local
depolarising error model $\mathcal{D}$

$\mathcal{D}=(1-p)\mathcal{I}+p_{X}\mathcal{X}+p_{Y}\mathcal{Y}+p_{Z}\mathcal{Z}$
(52)

with $p_{X}+p_{Y}+p_{Z}=p$, and $\mathcal{X}(\rho)=X\rho X$, etc. This error
model is motivated by probabilistic compilation techniques
Kliuchnikov2023shorterquantum ; koczor2024sparse ; PhysRevLett.132.130602 .
Since T commutes with $Z$, this circuit is equivalent to

$\ket{0}$$Z$$\mathcal{D}$$Z$$X$$\equiv$$\tilde{\mathcal{D}}$

Hence, the noise-channel $\mathcal{D}$ sees the symmetric Pauli filter
$\bm{\mathcal{F}}_{Z}$. As shown before, $\bm{\mathcal{F}}_{Z}$ converts
$\mathcal{D}$ into the channel $\tilde{\mathcal{D}}$, where

$\tilde{\mathcal{D}}=\bm{\mathcal{F}}_{Z}[\mathcal{D}]=(1-p+p_{X})\mathcal{I}+(p_{Y}+p_{Z})\mathcal{Z}$
(53)

Thus, the fidelity of the channel increases by $p_{X}$. This approach is
particularly relevant for architectures where the noise is biased
aliferis2008fault ; webster2015reducing , i.e, where one of the components
among $p_{X},p_{Y},p_{Z}$ is much greater than the other two. So, if the noise
on the system qubits is biased in such a way that $p_{X}\gg p_{Z},p_{Y}$,
then,

$\tilde{\mathcal{D}}\sim\mathcal{I}$

and we have an (almost) perfect T-gate. If the dominant error is the $Y$
error, i.e, $p_{Y}\gg p_{X},p_{Z}$, then we could replace the feedback
operation of $CX$ after the measurement with $CY$, as below,

$\ket{0}$$Z$$\mathcal{D}$$Z$$Y$

This will result in the channel,

$\tilde{D}=(1-p+p_{Y})\mathcal{I}+(p_{X}+p_{Z})\mathcal{Z}\sim\mathcal{I}$
(54)

However, we cannot use this technique if the qubits are biased towards the $Z$
error, since $Z$ commutes T.

$\ket{0}$$\ket{0}$

$\vec{\reflectbox{\\!$Z^{\otimes n}$}}$

$\vec{\reflectbox{\\!$X^{\otimes n}$}}$

$\bm{C}$$\mathcal{D}_{p}^{\otimes n}$$X^{\otimes n}$$Z^{\otimes
n}$$\ket{0}\otimes\ket{0}\;({\tt Postselect})$$(\tilde{\mathcal{C}})$ Figure
7: An ancilla-efficient Pauli filter for purifying a Clifford Circuit $\bm{C}$
(affected by an $n$-qubit local depolarising channel).
$\reflectbox{$\vec{\reflectbox{\\!$P$}}$}=\bm{C}^{\dagger}P\bm{C}$ is the
backward propagation (in time) of the Pauli $P$ through the Clifford $\bm{C}$.

A depolarising channel $\mathcal{D}_{p}$ with $p_{X}=p_{Y}=p_{Z}=\frac{1}{3}p$
is converted into the infinitely biased channel as

$\tilde{\mathcal{D}}(\rho)=(1-2p/3)\cdot\rho+2p/3\cdot Z\rho Z,$ (55)

which is therefore amenable to techniques developed for biased noise models
tuckett2018ultrahigh ; bonilla2021xzzx ; xu2023tailored .

Beyond biasing the error channel, the fidelity is increased by $p/3$ as the
incoherent $X$ and $Y$ components of the noise-channel are removed. If we
purify $n$ T Gates in parallel using this scheme (with $n$ ancillae), the
residual noise-channel will be $\tilde{\mathcal{D}}^{\otimes n}$ with a
fidelity of $(1-2p/3)^{n}$, which is a $1/3$-rd reduction in the decay rate
compared to the raw channel. This is still exponentially small in $n$, a
signature of the partial nature of the purification. This scheme can be
adapted to partially purify the 3-qubit CCZ gate, which is shown in Appendix
B. Subject to depolarising noise model on each qubit described above, this
increases the fidelity of a CCZ gate from $(1-p)^{3}$ to $(1-2p/3)^{3}$.

The approach taken for non-Clifford gates in this subsection shares a similar
spirit to quantum error correction enhanced quantum metrology - it is nearly
impossible to protect against errors along the same axis as the signal (which
may produce arbitrarily small rotation angles), but if the error and the
signal occur in orthogonal directions, then error correction can mitigate the
effects of noise in a similar vein in quantum metrology shettell2021practical
; zhou2018achieving ; demkowicz2017adaptive .

###  III.6  Conjecture of efficient filtration

Using $2n$ additional qubits to cancel errors for an an $n$-qubit channel
might seem expensive in some cases. If we want to deterministically remove all
the noise components of a general $n$-qubit Pauli Channel, unfortunately, $2n$
filtering qubits are necessary. This is because, in order to remove noise from
such a channel we need $4^{n}$ distinct syndromes to distinguish the $4^{n}$
Pauli Channel components (so that we can perform the correction for the
corresponding component). To do this, we need at least $\log_{2}(4^{n})=2n$
qubits.

However, for the local depolarising noise model considered above, all the
noise components do not have equal strength. The strength of a (Pauli) noise-
component with weight $w$, is

$p^{w}(1-p)^{n-w}$ (56)

Assuming that the depolarising strength $p$ is small, components with very
high weights will be less likely. Therefore, we may be content with removing
components with weight up to some threshold $k$. The number of $n$-qubit
Paulis with weight $\leq k$ is,

$\sum_{i=0}^{k}\binom{n}{i}3^{i}\sim O\left(3^{k}\binom{n}{k}\right)$ (57)

The number of ancillary qubits that are needed to correct channel components
up to order $k$ is thus lower bounded by

$\\#(\text{ancilla})\sim\Omega\left(\log_{2}\left(\frac{3en}{k}\right)^{k}\right)\sim\Omega(k\log_{2}(n/k))$
(58)

where the inequality $\binom{n}{k}<\left(\frac{\text{e}n}{k}\right)^{k}$ is
used. It remains to be seen whether this number of ancillary qubits is
sufficient for stochastic Pauli noise correction.

##  IV Ancilla-efficient Pauli filter

In near-term applications, instead of correcting a channel, probabilistically
filtering noise components using fewer resources might be desirable. In this
section, we give a filter configuration that can remove weight-1 Paulis in any
$n$-qubit Clifford Circuit using just 2 extra qubits. This is quite useful
because weight-1 Paulis are the dominant noise component (in the regime
$n<1/p$), and the ancilla overhead is very small.

Consider the following filter $\bm{\mathcal{F}}_{AE}$ specified by:

$\displaystyle V$ $\displaystyle=H^{\otimes 2}$ (59)
$\displaystyle\textsc{select}_{Q}$ $\displaystyle=\outerproduct{0}{0}\otimes
I^{\otimes n}+\outerproduct{1}{1}\otimes Z^{\otimes n}$
$\displaystyle+\outerproduct{2}{2}\otimes X^{\otimes
n}+\outerproduct{3}{3}\otimes Y^{\otimes n}$
$\displaystyle\textsc{select}_{P}$
$\displaystyle=\textsc{select}_{Q^{\star}}=\bm{C}\left(\textsc{Select}_{Q}\right)\bm{C}^{\dagger}$

This filter maps a noise component $N$ to

$N\xrightarrow{\bm{\mathcal{F}}_{AE}}I^{\otimes n}NI^{\otimes n}+Z^{\otimes
n}NZ^{\otimes n}+X^{\otimes n}NX^{\otimes n}+Y^{\otimes n}NY^{\otimes n}$ (60)

If $N$ is a weight-1 Pauli $S_{i}$, $S\in\\{X,Y,Z\\}$, which acts non-
trivially only on the $i$-th qubit, we have,

$S_{i}\xrightarrow{\bm{\mathcal{F}}_{AE}}I_{i}S_{i}I_{i}+Z_{i}S_{i}Z_{i}+X_{i}S_{i}X_{i}+Y_{i}S_{i}Y_{i}\quad\forall
i$ (61)

This sum is always 0, since each weight-1 $S_{i}$ commutes with 2 of these
terms and anti-commutes with the other two. So, all the weight-1 Pauli’s are
filtered out. The effect of this is an improvement in the fidelity of the
channel.

The (average) fidelity of a noisy channel $\tilde{\mathcal{V}}$ compared to
the ideal channel $\mathcal{V}$ is defined as

$F_{\rm
avg}(\tilde{\mathcal{V}};\mathcal{V})=\int\differential{\psi}\Tr[\tilde{\mathcal{V}}(\psi)\mathcal{V}(\psi)]$
(62)

where the integral is over all pure states $\psi=\outerproduct{\psi}{\psi}$ in
the state-space. For a local depolarising error channel of the form of Eq. 42,
for which the ideal channel is the identity, the average fidelity reduces to
$F_{\rm avg}(\mathcal{D}_{p};\mathcal{I})=1-\frac{2}{3}p$, where $p$ is the
probability of an error occurring horodecki1999general . The average
infidelity then becomes $\epsilon_{\rm avg}=1-F_{\rm
avg}=\frac{2}{3}p=\Theta(p)$. In the following analysis, we ignore the
unimportant scaling factor of $2/3$, and define the infidelity of a stochastic
Pauli error channel as

$\epsilon=p_{\rm error}:=1-p_{\mathcal{I}}$ (63)

where $p_{\mathcal{I}}$ is the probability of the identity component of the
channel. Using this metric, the removal of weight-1 Paulis by the ancilla-
efficient filter reduces the infidelity of the channel according to the
following theorem.

######  Theorem 1 (Quadratic reduction in the infidelity with ancilla-
efficient Pauli filters).

Consider the ancilla-efficient Pauli filter $\bm{\mathcal{F}}_{AE}$ defined by
the unitaries in Eq. 59 which can purify an $n$-qubit Clifford Circuit
$\bm{C}$ using 2 ancillary qubits (Fig. 7). If the noise model on the system
qubits is such that, the Clifford $\bm{C}$ is followed by an $n$-qubit local
depolarising channel $\mathcal{D}_{p}^{\otimes n}$ of strength $p$, where

$\mathcal{D}_{p}(\rho)=(1-p)\cdot\rho+p\cdot\frac{1}{3}(X\rho X+Y\rho Y+Z\rho
Z)$ (64)

with an input infidelity $\epsilon_{\tt in}:=1-(1-p)^{n}$ defined in Eq. 63.
Then, the ancilla-efficient Pauli filter succeeds with a probability $p_{s}$
that is upper bounded by

$p_{s}\leq 1-p^{2}/\epsilon_{\tt in}$ (65)

Conditioned on its success, the infidelity of the purified channel is bounded
by

$\epsilon_{\tt out}\leq 2\epsilon_{\tt in}^{2}=O(\epsilon_{\tt in}^{2})$ (66)

Hence, there is at least a quadratic reduction in the infidelity. A proof of
this result can be found in Appendix C.3. In a recent paper by Lee et al.
lee2023error , they have shown a black-box error suppression method which can
be modified to achieve a quadratic reduction in the infidelity using
$\log(1/\epsilon)$ ancilla and $(1/\epsilon)$ queries to the circuit. Our
scheme can do this with only 2 ancillae and 1 query, although for the specific
noise model that we consider and only for the restricted case of Clifford
circuits. Nevertheless, this is an important class of circuits. If we restrict
the ancilla count to 2 qubits, Lee et al. ’s scheme achieves an output
infidelity of $\epsilon_{\tt out}=\epsilon_{\tt in}/4$. Thus, our ancilla-
efficient filter outperforms their scheme in the regime $2\epsilon_{\tt
in}^{2}<\epsilon_{\tt in}/4$, i.e, when $\epsilon_{\tt in}<1/8$. In Appendix
C.4, we show that this result also holds for an exponentially decaying global
Pauli noise model.

All of these bounds are conservative, and the ancilla-efficient filter
performs even better since it can remove many of the higher-weight Paulis as
well.

######  Theorem 2 (Filtration of low-weight Pauli errors).

Suppose the ancilla-efficient Pauli filter $\bm{\mathcal{F}}_{AE}$ defined in
Eq. 59 is used to purify an $n$-qubit stochastic Pauli channel (with $4^{n}$
Pauli components). Let $\\#(w)$ be the number of Pauli-components of weight
$w$ that $\bm{\mathcal{F}}_{AE}$ can remove (if successful). Then, $\\#(w)$ is
lower-bounded by

$\\#(w)\geq\binom{n}{w}3^{w}/w^{2}$ (67)

$\bm{\mathcal{F}}_{AE}$ can remove at least $4^{n}/n^{2}$ of all the $4^{n}$
Pauli components of the channel. Moreover, the average weight of all the
Paulis that can be removed is (asymptotically) $n/2$.

Proof of this result and a more detailed treatment of the ancilla-efficient
filter $\bm{\mathcal{F}}_{AE}$ can be found in Appendix C. We note that the
ancilla-efficient filter configuration given in Eq. 59, gives significant
error reduction only in the regime where most of the error is concentrated in
the low-weight Paulis. In cases where there are correlated errors over
multiple qubits (resulting in error-concentration in high-weight Paulis), the
reduction in error is small.

Figure 8: Numerical results for noisy implementation of the channel correction
filter (Fig. 2) to purify local depolarising noise on 4 qubits. (a) Average
Fidelity of the corrected channel for various combinations of $(p_{c},p_{t})$.
(b) Adjusted Fidelity, $\Delta
F=F(p_{c},p_{t})-F(\max(p_{c},p_{t}),\max(p_{c},p_{t}))$ showing the relative
gain in fidelity for the error-rates on control qubit vs. target qubit,
compared to the case where the error-rates on both the system and the ancilla
are the same. As can be seen, the filter performance is more sensitive to
noise on the ancilla (i.e, control qubits).

##  V Numerical Results

Figure 9: Performance of the ancilla-efficient Pauli filter (Fig. 7) for
filtering layered brickwork circuits of 12 qubits composed of CNOT gates only.
For the plots (a)-(d), we use a common noise model for the filtering qubits
and system qubits, and, uniform error-rates on the control and target qubits
of 2-qubit gates, i.e, we set $p_{c}({\rm filter})=p_{t}({\rm
filter})=p_{c}({\rm circuit})=p_{t}({\rm circuit})=p$. (a) Average Fidelity vs
Circuit Depth for the original (unfiltered) circuit. (b1) Average Fidelity of
the Filtered Circuit, (b2) Post-selection Rate, or, Success Probability of the
ancilla-efficient filtering protocol. (c1), (c2) show the dependence of the
Gain in Fidelity ($F_{\rm filtered}/F_{\rm original}$) vs the number of layers
and error-rate respectively. (d) Crossover Depth (depth at which fidelity of
the filtered circuit surpasses the fidelity of the original circuit) vs error-
rate. Plots (e1),(e2) show the performance of the filter, if the filtering
qubits are cleaner than the system qubits. They show dependence of the
fidelity on $p_{c}({\rm filter})$, where we set $p_{t}({\rm
filter})=p_{c}({\rm circuit})=p_{t}({\rm circuit})=0.01$. Finally plot (f)
shows the performance of a noiseless filter, i.e, $p_{c}({\rm
filter})=p_{t}({\rm filter})=0$ and $p_{c}({\rm circuit})=p_{t}({\rm
circuit})=p_{e}$.

In the above theoretical analysis, we analyse the performance of quantum
filters assuming that the filter operations are noiseless. In this section, we
present some numerical results for noisy implementations of the quantum
filters. We consider noisy 2-qubit gates and assume that the error rates of
single-qubit gates are negligible. For any noisy 2-qubit gate, we model the
noise by assuming that any application of the gate is followed by a local
depolarising noise channel on the control and target qubits

$\mathcal{D}_{p}(\rho)=(1-p)\rho+p\cdot\frac{1}{3}(X\rho X+Y\rho Y+Z\rho Z)$

If $\mathcal{U}_{i;j}$ is an ideal 2-qubit gate (with control on the $i$-th
qubit and target on the $j$-th qubit), we model the noisy version of this gate
$\tilde{\mathcal{U}}_{i;j}$ as

$\tilde{\mathcal{U}}_{i;j}=\left(\mathcal{D}_{p_{c}}^{(i)}\otimes\mathcal{D}_{p_{t}}^{(j)}\right)\circ\mathcal{U}_{i;j}$
(68)

where $p_{c}$ and $p_{t}$ are error rates on the control and target qubits,
respectively. For this noise model on the 2-qubit operations, if we want to
purify a local depolarising channel on a single qubit using the channel
correction filter in Fig. 2, the fidelity of the corrected channel is bounded
by

$\tilde{F}\geq(1-p_{c})(1-p_{t})(1-2p_{c}/3)^{2}(1-2p_{t}/3)^{2}(1-2p_{c}/3)$

A derivation of this bound is given in Appendix D. This shows that the output
fidelity is more sensitive to noise on the ancillae. The above lower bound
gives a pseudo-threshold on the error rate of the noise channel, above which
using the filter is beneficial.

In Fig. 8, we provide numerical results that support the above
characteristics. Fig. 8(a) shows the average fidelity of the output state for
purifying local depolarising noise on 4 qubits, for various combinations of
$p_{t}$ and $p_{c}$. Fig. 8(b) shows the relative gain in fidelity $\Delta
F=F(p_{c},p_{t})-F(p^{*},p^{*})$, where $p^{*}=\max(p_{c},p_{t})$, obtained by
keeping either the ancillae or the system qubits cleaner. As numerical
evidence suggests, for the same error budget, it is more advantageous to keep
the controls cleaner than the targets.

Next, we provide some performance results for the ancilla-efficient filter.
The circuits that we consider for filtering are layered brickwork circuits
composed of CNOT gates, which are simply used for benchmarking.

We consider 3 scenarios (i) the filtering qubits are subject to the same noise
model as the system qubits ($p_{c}=p_{t}>0$), (ii) the filtering qubits are
cleaner than the system qubits ($0<p_{c}<p_{t}$) and (iii) the filtering
qubits and operations are noiseless ($p_{c}=p_{t}=0$). We compare the
dependence of the fidelity of the filtered circuits on various parameters.

The results for the regime (i) are shown in the plots Fig. 9(a)-(d), where we
set $p_{c}({\rm filter})=p_{t}({\rm filter})=p_{c}({\rm circuit})=p_{t}({\rm
circuit})=p$. Fig. 9(a) shows the average fidelity of the output state vs
circuit depth for the original (unfiltered) circuit, (b1) the average fidelity
of the output state for the filtered circuit, and (b2) the post-selection rate
or the success probability of the ancilla-efficient filter. As can be seen,
the filtered output is consistently better than the original output, although
the post-selection rate decreases with the error rate as expected. Fig. 9(c1),
(c2) shows the dependence of the Gain in Fidelity ($F_{\rm filtered}/F_{\rm
original}$) vs the number of layers and error-rate respectively. The relative
gain in fidelity is significantly higher for more noisy circuits and higher
depths. Fig. 9(d) shows the critical depth or crossover depth at which the
fidelity of the filtered circuit surpasses the fidelity of the original
circuit. If the filter and system qubits are subject to the same noise-
strength, the crossover happens at deeper circuits for higher error rates.

Fig. 9(e1),(e2) show the performance of the filter in regime (ii), if the
filtering qubits are cleaner than the system qubits. They show dependence of
the fidelity on $p_{c}({\rm filter})$, where we set $p_{t}({\rm
filter})=p_{c}({\rm circuit})=p_{t}({\rm circuit})=0.01$. As before, for the
same depth, this results in a higher output fidelity with a corresponding
penalty in the post-selection rate.

Finally, Fig. 9(f) shows the performance of the ancilla-efficient filter in
regime (iii), where we set $p_{c}({\rm filter})=p_{t}({\rm filter})=0$ and
$p_{c}({\rm circuit})=p_{t}({\rm circuit})=p_{e}$. This version of the filter
has the best performance of all the cases which, as before, is expected.

From all the results, it is evident that the performance of the filter is more
sensitive to the error rate on the ancillae. In all these cases, we see that
if the depth of the circuit is above a certain level, the ancilla-efficient
filter provides a modest gain in fidelity. This holds true even if the
filtering qubits are as noisy as the system qubits.

##  VI Further remarks

Different interpretations of our scheme. The construction of commutation
filters can also be understood from other perspectives. One can think of the
idea as the use of a coherent variant of quantum phase estimation
kitaev1995quantum , with the syndrome measurements of quantum error detection
schemes. The coherence allows us to measure only changes in syndrome, avoiding
the complexity of encoding, at the cost of requiring high-quality ancillary
qubits and controlled gates. It also shows some similarity to the measurement
of out-of-time-order-correlators (OTOCs) for scrambling mi2021information ,
though the connection is not that straightforward. It is possible to interpret
our scheme as an application of indefinite causal order to error detection
chiribella2021indefinite , though with an additional Clifford pre-processing
step. For pure storage channels, we can also view our scheme as an example of
entanglement-assisted QEC brun2014catalytic , or as a coherent example of the
conversion between QEC codes and entanglement distillation schemes
aschauer2005quantum  (for which the only requirement on the Pauli frame is
that it matches between the two parties).

Quantum filters, as proposed in this work, can map a quantum channel to
another and thus provides a way to realise quantum channel transformations.
Although we mainly discuss this in the context of channel purification, error
reduction and correction, it can find applications in the context of
implementing quantum superchannels chiribella2008transforming . In particular,
the general quantum filter construction introduced in subsection III.2 can be
used to realise a large class of quantum superchannels via LCU-inspired
circuits childs2012hamiltonian . Other types of quantum superchannels may be
designed inspired by this idea, which could be an interesting direction for
future work.

The error detection circuit, which is part of the quantum filter, shares
similarities with flag fault-tolerant error correction (see
chamberland2018flag ; chao2020flag ; chao2018quantum . It also shares similar
spirits with spacetime checks in the recently proposed spacetime codes of
Clifford circuits delfosse2023spacetime , although our scheme works for
arbitrary input states. Compared to QEC codes, our method can detect or
correct errors in a circuit without explicitly encoding the input state. The
circuit described in Fig. 7 is also similar to the ‘Iceberg’ code used in
self2024protecting , and Rains’ optimal distance-$2$ code rains1999quantum
for even $n$. As proposed in self2024protecting , four ancillas are used in
total with two for encoding and two for syndrome measurement which can detect
any single-qubit error though at the cost of a discard rate.

Our quantum filtering scheme might have several attractive features compared
to QEC in terms of near-term applications. The scheme is input-state agnostic,
and the filter construction depends only on the channel that we are trying to
purify. As such, we do not need to encode the system qubits a priori, or
convert the target channel into large multi-qubit logical gates. We can plug
in the original circuit of our computation into the filter as is, treating it
almost like a black box.

When the computation runs on the system qubits, the filtering qubits remain
idle. They are acted upon only at the start and end of the computation. As a
result, we can think of keeping the filtering qubits isolated while the
computation is running, so that they are not affected by noise. In fact, it
might also be possible to use two physically different types of qubits for
these two registers, stable qubits with long coherence times for the ancilla,
and qubits with fast gate execution for the main system. In addition, since
only specific types of gates act on the filtering qubits (Hadamards and only
control of Controlled-Paulis in our Pauli Filtering scheme), and they don’t
take part in general-purpose computation, we can optimize these qubits only
for these specific gates.

From a quantum communication viewpoint, we might think of scenarios where we
have a small number of noiseless channels, and a large number of noisy
channels. Then we can use the noiseless channels as filters to clean the noisy
channels. This method can also be used as a quantum error mitigation technique
where we can use a small number of clean qubits to perform gate-purification
dur2003entanglement  on a larger number of noisy qubits. This paradigm of
using extra qubits which are cleaner than the system qubits has also been
explored under the moniker of error suppression lee2023error  and partial
error correction koukoulekidis2023framework .

Partial Purification and Biased Noise In practice, physical qubits can have a
biased noise channel (i.e, one of the Pauli $Z$ or $X$ components of the
channel is exponentially suppressed compared to the other). Examples of such
biased-noise qubits are superconducting fluxonium qubits pop2014coherent ,
quantum-dot spin qubits watson2018programmable , etc. If the noise is biased,
the filter operations in our scheme can be simplified. For example, if the
noise is biased towards $Z$ errors (for which $X$ and $Y$ errors are
exponentially suppressed), we can remove all the $Z$ errors from $n$ qubits
deterministically using $n$ Pauli $X$-Filters. Note that this requires $n$
ancillae and thus has a factor of 2 reduction from the complete correction
filter. Recently, there has been progress in designing QEC codes for biased-
noise qubits, which are shown to have high code capacity tuckett2019tailoring
; tuckett2018ultrahigh ; bonilla2021xzzx ; xu2023tailored . The partial
purification scheme outlined in subsection III.5 can be used to convert an
unbiased noise channel to a channel biased towards either $Z$ or $X$ errors.
Therefore, it is compatible with the biased-noise QEC codes. Our scheme may
also find applications in the context of multi-core QC based on Bell pair
injections, in which case biased noise is common as found in experiments
jnane2022multicore .

In applications where one needs to find the expectation values of observables
(which is the case in many practical quantum algorithms, like VQE
tilly2022variational  and quantum simulation georgescu2014quantum ), the final
result is naturally robust to certain types of errors. More concretely, if an
observable $H$ commutes with all the Kraus elements of a noise channel
$\mathcal{E}$, then the expectation value $\expectationvalue{H}$ is invariant
to the effect of $\mathcal{E}$ on a state $\rho$, i.e,
$\Tr[\mathcal{E}(\rho)H]=\Tr[\left(\sum_{k}E_{K}^{\dagger}E_{k}\right)\rho
H]=\Tr[\rho H]$. In these cases, it is sufficient to remove the components of
$\mathcal{E}$ which anti-commute with $H$. For example, if we want to find the
expectation of an observable with only Pauli $Z$ operators, it will be robust
to $Z$ errors and mainly be affected by $X$ and $Y$ errors.

Practical Considerations Some factors need to be considered for practical
implementations of our scheme. The Select operations needed to implement the
Clifford Circuit purification scheme (outlined in subsection III.3) require
$2n^{2}+2n=O(n^{2})$ 2-qubit gates (in the worst case). Current state-of-the-
art methods for synthesizing $n$-qubit Clifford Operations also require
$O(n^{2})$ 2-qubit gates bravyi2021hadamard ; schneider2023sat  (albeit with a
higher pre-factor). As a result, although there is always some amount of error
reduction due to the filter, if the filtering circuit and the main Clifford
circuit are subject to the same strength of noise, the asymptotic rate of
error-reduction might not be very high. This just highlights the point that
the ancilla being cleaner than the main qubits is an essential assumption
required to obtain significant gains in fidelity.

For the ancilla-efficient filter, significant error reductions are obtained in
the regime where $n<1/p$ ($p$ being the single-qubit depolarising error-rate)
and where most of the error strength is concentrated in low-weight Paulis. For
long-range entangling gates which quickly spread errors over multiple qubits,
the reduction in error is likely to be small. Nevertheless, the ancilla-
efficient scheme still gives a low-cost method to reduce errors in
sufficiently deep Clifford circuits.

If the filter operations are noisy, any noise on the system part of the Select
operations can be considered as part of the target channel. Nevertheless, such
a treatment would not work for noise on the filtering qubits. This, as
mentioned before, is a limitation of our method for practical applications.
Although we have provided a basic treatment of the effect of these errors on
the accuracy of the filter in Appendix D, it is instructive to ponder how to
avoid or potentially correct such errors. This might lead to a theory
analogous to fault tolerance, which might be a direction for future work.

##  VII Discussions

In this work, we proposed an information-theoretic device called a quantum
filter that can be used to purify and correct noisy quantum channels. We
provide explicit constructions of Pauli filters which exploit the commutation
structures between Pauli and Clifford gates to achieve this. We also provide a
general construction that can realize quantum filters as superchannels. This
quantum filter can reduce the errors in a noisy quantum channel and can be
regarded as an intermediate form of QEM and QEC. Under the assumptions of
clean ancillary qubits and ancillary operations in a similar vein to
lee2023error ; liu2024virtual ; miguel2023superposed , the filtering scheme
that we propose can fully recover the ideal channel without the exponential
sampling overhead in QEM. Finally, we propose an ancilla-efficient Pauli
filter which can detect and remove nearly all the low-weight Pauli errors in a
noise channel using up to two ancillary qubits. In cases where the noise is
biased, the filtering operations can be simplified. The Pauli filter can be
applied to convert unbiased noise into fully biased noise. We perform
numerical simulations to investigate the performance of our scheme with
consideration of ancillary qubit noise, and find conditions which show that
our scheme can still show modest gains in fidelity. In Appendix D, we analysed
the effect of ancilla noise on the fidelity of the corrected channel.  The
Pauli filter protocol can be applied as an alternative to QEM kim2023evidence
; guo2024experimental ; kim_scalable_2023 ; o2023purification  in the near-
term or early fault-tolerant quantum computation, with the advantage of
recovering the quantum state, not just the expectation values.

In the specific schemes outlined in the work, we have restricted ourselves to
using Clifford gates in the filtering circuits (although the general
construction of a quantum filter does not have that restriction). We have
focused on this case because it allows us to exploit the underlying
commutation structure of Pauli operators to the greatest extent. In addition,
from a practical standpoint, Clifford gates are easier to implement in a
fault-tolerant manner in error-corrected quantum computation. However, this
restriction does imply that we can perform only a partial purification of non-
Clifford gates.  It is worth noting that we can convert the noise associated
with non-Clifford gates into fully biased one. It is an interesting direction
to extend the filter formalism beyond commutation structures and find ways to
completely purify non-Clifford resources as well. We might also explore the
possibilities of joint quantum channel correction and state correction
schemes, where we encode both states and channels to reduce errors further.

###### Acknowledgements.

J.S. and B.K. would like to thank Zhenyu Cai and Zhenhuan Liu for the early
discussions related to this project. S.D. thanks David Jennings and Samson
Wang for helpful discussions. We thank the UK EPSRC for their financial
support through EP/T0010621/1, EP/Y004752/1, EP/W032643/1, and EP/W003463/1.
We also thank the Samsung GRC Grant and the KIST Open Innovation Grant. J.S.
would like to thank the Schmidt AI in Science Fellowship supported by Schmidt
Futures. B.K. thanks the University of Oxford for a Glasstone Research
Fellowship, Lady Margaret Hall, Oxford for a Research Fellowship and thanks
the UK EPSRC for financial support through EP/Y004655/1 (SEEQA).

##  Appendix A Proof of the channel correction property

Here, we provide a simple proof of the channel correction property mentioned
at the end of subsection II.3, that is, a channel correction scheme that can
correct Pauli errors on a single qubit is sufficient for correcting an
arbitrary error. The proof is a re-packaging of the standard result from QEC,
which states that if a QEC code can correct $Z$-errors and $X$-errors, it can
correct arbitrary errors. Nevertheless, it is illustrative to go through this
proof in the framework that we have developed.

$\ket{0}$$\ket{0}$$\ket{\psi}$$X$$Z$$P_{u}$$Z$$X$$\ket{\tt out}$ Figure 10: An
example of quantum filter for channel correction

Consider the circuit in Fig. 10, where a Pauli error happens in the middle of
the filter. For convenience, we define
$(P_{00},P_{01},P_{10},P_{11})=(I,Z,X,Y)$. We have already shown in subsection
II.3 that, the output state $\ket{\tt out}$ of this circuit is -

$\ket{\psi}\to\ket{\tt out}=\ket{u}\otimes P_{u}\ket{\psi}$ (69)

where $u\in(\mathbb{Z}_{2})^{2}$. Now, assume that an error $\mathcal{E}$
happens in the filter which is described by Kraus elements $\\{E_{k}\\}$, i.e,
$\mathcal{E}(\rho)=\sum_{k}E_{k}\rho E_{k}$. Since the Pauli operators form a
basis on the space of operators, we can write the $E_{k}$’s as

$E_{k}=\sum_{u\in(\mathbb{Z}_{2})^{2}}\alpha_{k,u}P_{u}$ (70)

where the sum is over all the (single-qubit) Paulis. Hence, if $P_{u}$ is
replaced by $E_{k}$ in Fig. 10, by linearity, the output-state will be

$\ket{\psi}\xrightarrow{E_{k}}\ket{{\tt
out}_{k}}=\sum_{u\in(\mathbb{Z}_{2})^{2}}\alpha_{k,u}\ket{u}\otimes
P_{u}\ket{\psi}$ (71)

Hence, the output state for the channel $\mathcal{E}$ plugged into the filter
is

$\displaystyle\rho_{\tt out}$ $\displaystyle=\sum_{k}\outerproduct{{\tt
out}_{k}}{{\tt out}_{k}}$ (72)
$\displaystyle=\sum_{k}\left(\sum_{u}\alpha_{k,u}\ket{u}\otimes
P_{u}\ket{\psi}\right)\left(\sum_{v}\alpha_{k,v}^{*}\bra{v}\otimes
P_{v}\bra{\psi}\right)$ (73)
$\displaystyle=\sum_{k}\sum_{u,v}\alpha_{k,u}\alpha_{k,v}^{*}\outerproduct{u}{v}\otimes
P_{u}(\outerproduct{\psi}{\psi})P_{v}$ (74)

Now, in the correction phase, the ancillae are measured in the computational
basis (which is equivalent to a syndrome measurement) and a corresponding
Pauli is applied to the main qubit to correct for the error. As has been
shown, if the measurement result on the two ancillae is
$\ket{u_{0}}\otimes\ket{u_{1}}=\ket{u}$, then the corresponding correction
Pauli is $P_{u}$. If the measurement projectors are
$\Pi_{u}=\outerproduct{u}{u}$, the correction channel is given by

$\displaystyle\mathcal{C}(\rho)=\sum_{u}(\Pi_{u}\otimes
P_{u})\rho(\Pi_{u}\otimes P_{u})^{\dagger}$ (75)

Therefore, the post-correction state is

$\displaystyle\mathcal{C}(\rho_{\tt out})$
$\displaystyle=\sum_{k}\sum_{u,v,w}\alpha_{k,u}\alpha_{k,v}^{*}(\Pi_{w}\outerproduct{u}{v}\Pi_{w})\otimes
P_{w}(P_{u}\outerproduct{\psi}{\psi}P_{v})P_{w}$ (76)
$\displaystyle=\left(\sum_{k}\sum_{u}\absolutevalue{\alpha_{k,u}}^{2}\outerproduct{u}{u}\right)\otimes\outerproduct{\psi}{\psi}$
(77)

Hence, the system qubit is returned to its original state and the proof is
complete. The proof is easily extended for mixed input states. And for
multiple qubits, one can just use a filter for each qubit separately, and
repeat the procedure given above.

###  A.1 Proof of the channel correction in terms of superchannels

Below, we provide an alternate proof of the channel correction property using
the quantum superchannel formalism. Consider the single-qubit channel
correction filter shown in Fig. 11, highlighted as a superchannel. Instead of
considering the correction cycle separately as in Appendix A, we consider the
correction operations as part of the filter.

Figure 11: Single-qubit quantum channel correction filter

We use the same notation developed in subsection III.1. The super-Kraus
elements of the $Z$-correction and $X$-correction filters are

$\displaystyle\bar{\bm{\mathcal{F}}}_{Z}:\;$
$\displaystyle\bm{F}_{Z0}[E]=I\cdot(IEI+ZEZ)/2,\;$
$\displaystyle\bm{F}_{Z1}[E]=X\cdot(IEI-ZEZ)/2$ (78)
$\displaystyle\bm{\mathcal{\bar{F}}}_{X}:\;$
$\displaystyle\bm{F}_{X0}[E]=I\cdot(IEI+XEX)/2,\;$
$\displaystyle\bm{F}_{X1}[E]=Z\cdot(IEI-XEX)/2$ (79)

The combined correction-filter is
$\bm{\mathcal{F}}_{X}\circ\bm{\mathcal{F}}_{Z}$ whose super-Kraus elements are

$\bm{\mathcal{F}}_{X}\circ\bm{\mathcal{F}}_{Z}:\bm{F}_{u}[E]=\bm{F}_{Xu_{0}}[\bm{F}_{Zu_{1}}[E]]$
(80)

One can check that,

$\bm{F}_{u}[P_{v}]=\delta_{\bm{u},\bm{v}}I$ (81)

for ${u},{v}\in(\mathbb{Z}_{2})^{2}$. Hence, $\bm{F}_{u}[E]\propto I$ for all
the $u$ and arbitrary $E$. Thus, $\bm{\mathcal{F}}[\mathcal{E}]=\mathcal{I}$
for arbitrary channels $\mathcal{E}$. Although we have shown this result for a
single-qubit, it is straightforward to extend it to the case of multiple
qubits. This proves that the channel correction filter can indeed correct
arbitrary channels.

##  Appendix B Partial Purification of CCZ Gates

Figure 12: Pauli Filter for partially purifying a CCZ gate. (Left) Noisy CCZ
gate, modelled as an ideal CCZ gate followed by partial depolarising noise on
all the qubits, plugged into a Pauli $Z$ filter on 3 qubits. (Right) Since CCZ
commutes with $Z$’s, we can pull the ideal gate out of the filter. The
residual depolarising noise sees the $Z$-filter, which filters out the
$X$-component of the noise.

Since the CCZ gate commutes with $Z\otimes Z\otimes Z$, we can adapt the
scheme of subsection III.5 to partially purify them. Consider the circuit in
Fig. 12. A noisy CCZ gate is modeled as an ideal CCZ gate followed by a local
depolarising channel $\mathcal{D}^{\otimes 3}$, with $\mathcal{D}$ defined as
in Eq. 52
($\mathcal{D}=(1-p)\mathcal{I}+p_{X}\mathcal{X}+p_{Y}\mathcal{Y}+p_{Z}\mathcal{Z}$).
Since the single-qubit Paulis $Z_{1},Z_{2},Z_{3}$ commute with the CCZ gate,
we can pull the CCZ gate out of the filter. Then, following the analysis of
subsection III.5, the channel $\mathcal{D}^{\otimes 3}$ is transformed to the
channel $\tilde{\mathcal{D}}^{\otimes 3}$, with $\tilde{\mathcal{D}}$ defined
in Eq. 53
($\tilde{\mathcal{D}}=(1-p_{Y}-p_{Z})\cdot\mathcal{I}+(p_{Y}+p_{z})\cdot\mathcal{Z}$).
For the case $p_{X}=p_{Y}=p_{Z}=p/3$, this results in an increase in fidelity
of the CCZ gate from $(1-p)^{3}$ to $(1-2p/3)^{3}$.

##  Appendix C Analysis of the ancilla-efficient Pauli filter

Here, we find out some properties of the ancilla-efficient Pauli filter
$\bm{\mathcal{F}}_{g}$ introduced in Section IV, particularly how many high-
weight Paulis it can remove, its success probability for a local depolarising
noise on $n$-qubits, and fidelity of the output channel if it succeeds.

###  C.1 Number of Paulis removed by the ancilla-efficient Pauli filter

Suppose, a weight $w$ Pauli contains $(x,y,z)$ number of $X$’s, $Y$’s and
$Z$’s, i.e,

$w=x+y+z$ (82)

We will denote such a Pauli as $P_{x,y,z}$. By Eq. 60$,\bm{\mathcal{F}}_{AE}$
can remove $P_{x,y,z}$, if and only if

$I^{\otimes n}(P_{x,y,z})I^{\otimes n}+Z^{\otimes n}(P_{x,y,z})Z^{\otimes
n}+X^{\otimes n}(P_{x,y,z})X^{\otimes n}+Y^{\otimes n}(P_{x,y,z})Y^{\otimes
n}=0$ (83)

Using the anti-commutation property of the Paulis, this reduces to

$\displaystyle 1+(-1)^{x+y}+(-1)^{y+z}+(-1)^{(z+x)}=0$ (84)
$\displaystyle\Rightarrow$
$\displaystyle(-1)^{w}+(-1)^{x}+(-1)^{y}+(-1)^{z}=0$ (85)

Eq. 85 will be satisfied if 2 of the numbers $w,x,y,z$ are odd and the other 2
are even.

First, let us consider the case where $w$ is odd. Then, among $x,y,z$ one must
be odd, and the other two must be even. Let, $S_{w}[o,e,e]$ be the set of all
such tuples $(x,y,z)$. Then, the number of such Paulis of weight-$w$ that
satisfy Eq. 85 is

$\\#(w)=\binom{n}{w}\cdot\sum_{(x,y,z)\in S_{w}[o,e,e]}\binom{w}{x,y,z}$ (86)

Similarly, we may define a set $S_{w}[o,o,o]$ of tuples $(x,y,z)$ such that,
they sum to $w$ and all three of them are odd. If $w$ is odd, any tuple
$(x,y,z)$ of non-negative integers which sum to $w$ will belong to one of
these two sets. Therefore, we have,

$\sum_{(x,y,z)\in S_{w}[o,e,e]}\binom{w}{x,y,z}+\sum_{(x,y,z)\in
S_{w}[o,o,o]}\binom{w}{x,y,z}=\sum_{x,y,z}\binom{w}{x,y,z}=3^{w}$ (87)

Finding an exact expression for $\\#(w)$ might be difficult in general.
Therefore, we will try to bound this quantity instead.

Consider a tuple $(x,y,z)\in S_{w}[o,e,e]$. Let us call the odd element of
this tuple $o$, and the even elements $e_{1},e_{2}$. We define a map $f$ which
takes $o$ to $(o-2)$ and $e_{1},e_{2}$ to $e_{1}+1,e_{2}+1$. Notice that, all
3 of these numbers are odd. Therefore, $f$ maps a tuple of $S_{w}[o,e,e]$ to a
tuple of $S_{w}[o,o,o]$ (with the possibility that $(o-2)$ might be negative).
Then, if we define $\binom{w}{x,y,z}$ to be 0 if any one of $x,y,z$ is
negative, we can write

$\displaystyle\sum_{(x,y,z)\in S_{w}[o,o,o]}\binom{w}{x,y,z}$
$\displaystyle=\sum_{(x,y,z)\in S_{w}[o,e,e]}\binom{w}{f[(x,y,z)]}$ (88)
$\displaystyle=\sum_{(x,y,z)\in
S_{w}[o,e,e]}\left[\frac{o(o-1)}{(e_{1}+1)(e_{2}+1)}\right]_{(x,y,z)}\binom{w}{x,y,z}$
(89) $\displaystyle\leq\left(\max_{(x,y,z)\in
S_{w}[o,e,e]}\frac{o(o-1)}{(e_{1}+1)(e_{2}+1)}\right)\sum_{(x,y,z)\in
S_{w}[o,e,e]}\binom{w}{x,y,z}$ (90) $\displaystyle=w(w-1)\sum_{(x,y,z)\in
S_{w}[o,e,e]}\binom{w}{x,y,z}$ (91)

where in the second line, $(o,e_{1},e_{2})_{(x,y,z)}$ are the odd and even
elements of the tuple $(x,y,z)$ respectively.

Using this, we have,

$3^{w}=\sum_{(x,y,z)\in S_{w}[o,e,e]}\binom{w}{x,y,z}+\sum_{(x,y,z)\in
S_{w}[o,o,o]}\binom{w}{x,y,z}\leq(w^{2}-w+1)\sum_{(x,y,z)\in
S_{w}[o,e,e]}\binom{w}{x,y,z}$ (92)

Plugging this in Eq. 87 gives us the bound,

$\\#(w)\geq\binom{n}{w}\frac{3^{w}}{1+w(w-1)}$ (93)

This argument can be translated verbatim for the case where $w$ is even.

Therefore, the total number of components that are removed by
$\bm{\mathcal{F}}_{AE}$ satisfies

$\sum_{w=1}^{n}\\#(w)\geq\sum_{w=1}^{n}\binom{n}{w}\frac{3^{w}}{1+w(w-1)}$
(94)

As a conservative estimate, we may bound the terms $1/(1+w(w-1))$ by their
maximum value $1/(1+n(n-1))$. This gives us the bound,

$\displaystyle\sum_{w=1}^{n}\\#(w)\geq\frac{1}{1+n(n-1)}\sum_{w=1}^{n}\binom{n}{w}3^{w}=\frac{4^{n}}{n^{2}-n+1}\sim\Omega(4^{n}/n^{2})$

##### Average weight of removed Paulis

The average weight of the Paulis that are removed is bounded by

$\displaystyle\bar{w}=\frac{\sum_{w=1}^{n}w\cdot\\#(w)}{\sum_{w=1}^{n}\\#(w)}\geq\frac{\sum_{w=1}^{n}\frac{1}{w}\binom{n}{w}}{\sum_{w=1}^{n}\frac{1}{w^{2}}\binom{n}{w}}$

Asymptotically, the dominant term in these sums is $\binom{n}{n/2}$. Hence, in
the large-$n$ limit, we have,

$\displaystyle\bar{w}\sim\frac{\frac{1}{n/2}\binom{n}{n/2}}{\frac{1}{(n/2)^{2}}\binom{n}{n/2}}=n/2$

###  C.2 Fidelity of Purified Channel

Throughout this treatment, we will use the term fidelity to mean the
entanglement fidelity of a channel, or the co-efficient of the identity
element of its Pauli transfer matrix.

First, we find the probability that our protocol succeeds. The number of
Paulis that are not removed by $\bm{\mathcal{F}}_{AE}$ is

$\\#^{\prime}(w):=\binom{n}{w}3^{w}-\\#(w)\leq\binom{n}{w}3^{w}\left(1-\frac{1}{1+w(w-1)}\right)$
(95)

Since $w\leq n$, some simple algebra gives us the bound,

$\\#^{\prime}(w)\leq n\cdot\frac{n^{2}-1}{n^{3}+1}\binom{n}{w}3^{w}$ (96)

The probability that a particular Pauli of weight $w$ occurs is

$(p/3)^{w}q^{n-w}$

where $q=1-p$. Therefore, the probability of success is

$\displaystyle p_{s}$ $\displaystyle\leq
q^{n}+n\cdot\frac{n^{2}-1}{n^{3}+1}\sum_{w=1}^{n}\binom{n}{w}3^{w}(p/3)^{w}q^{n-w}$
(97) $\displaystyle=q^{n}+n\cdot\frac{n^{2}-1}{n^{3}+1}(1-q^{n})$ (98)
$\displaystyle=1-(1-q^{n})\frac{1+n}{1+n^{3}}$ (99)

Therefore, the fidelity of the purified output channel is

$\displaystyle
F_{o}=q^{n}/p_{s}\geq\frac{q^{n}}{1-(1-q^{n})\frac{1+n}{1+n^{3}}}\geq
q^{n}\left[1+(1-q^{n})\frac{1+n}{1+n^{3}}\right]\geq
q^{n}\left[1+(1-q^{n})\frac{1}{n^{2}}\right]$

In the limit that $n$ is large (and $q^{n}$, the input fidelity, is
vanishingly small), the rate of increase is approximately,

$F_{o}/F_{i}\simeq 1+\frac{1}{n^{2}}$ (100)

###  C.3 Quadratic Reduction in Infidelity

Let, the infidelity of the input channel be $\epsilon$, i.e,

$F_{i}=q^{n}:=1-\epsilon$ (101)

Then, the number of channel qubits in terms of the infidelity is

$n=\frac{\ln{(1-\epsilon)}}{\ln{q}}=\frac{\ln(1-\epsilon)}{\ln(1-p)}$ (102)

Using the relation,

$\epsilon\leq-\ln(1-\epsilon)<\frac{\epsilon}{1-\epsilon}\qquad\epsilon<1$
(103)

we have,

$\frac{\epsilon/p}{1-p}\leq n\leq\frac{\epsilon/p}{1-\epsilon}$ (104)

Now, the output fidelity of the channel is bounded by

$\displaystyle F_{o}\geq\frac{q^{n}}{1-npq^{n-1}}$ (105)

where we have considered removing only the weight-1 Paulis. Substituting
$q^{n}=1-\epsilon$ and $n\geq\epsilon/pq$, we get ,

$\displaystyle F_{o}$
$\displaystyle\geq\frac{1-\epsilon}{1-\epsilon(1-\epsilon)/q^{2}}$ (106)
$\displaystyle\geq\frac{1-\epsilon}{1-\epsilon(1-\epsilon)}$
$\displaystyle[\because q^{2}<1]$ (107)
$\displaystyle\geq(1-\epsilon)(1+\epsilon(1-\epsilon))$
$\displaystyle[\because\epsilon(1-\epsilon)<1]$ (108) $\displaystyle\geq
1-2\epsilon^{2}$ (109)

Hence, up to leading order, the output infidelity is $O(\epsilon^{2})$, which
is a quadratic reduction in the infidelity.

In fact, Eq. 105 is a conservative estimate of the output fidelity. A better
asymptotic (i.e, large $n$) estimate can be found by using Eq. 99. In terms of
the infidelity, we have,

$\displaystyle p_{s}\leq 1-(1-q^{n})\frac{1}{n^{2}}\leq
1-\epsilon\cdot\frac{p^{2}}{\epsilon^{2}}\leq 1-p^{2}/\epsilon$

Using this bound, the output fidelity becomes,

$\displaystyle F_{o}$ $\displaystyle\geq\frac{1-\epsilon}{1-p^{2}/\epsilon}$
(110)

which gives a better estimate in the regime,

$p^{2}/\epsilon>\epsilon(1-\epsilon)$ (111)

But,

$\epsilon^{2}(1-\epsilon)\leq(1-\epsilon)=q^{n}=e^{n\ln{q}}\leq e^{-np}$ (112)

Hence, a sufficient condition for Eq. 110 to give a better bound on the output
fidelity is if

$p^{2}>e^{-np}\Rightarrow n>2(1/p)\ln(1/p)$ (113)

###  C.4 A Global Pauli Noise Model

In this section, we consider the performance of the ancilla-efficient filter
for an exceptionally decaying global Pauli noise-model on $n$-qubits. Suppose,
the probability that a weight-$w$ Pauli error occurs is

$p(w)\propto\epsilon^{w}$ (114)

for some $\epsilon<1$. The constant of proportionality is
$F=(1-\epsilon)/(1-\epsilon^{n})$ which is found from normalization. This is
also equal to the fidelity of the channel, which is the probability of no
errors occuring ($p(0)$). Hence, the infidelity of the channel is,

$1-F=1-\frac{1-\epsilon}{1-\epsilon^{n}}=\epsilon-O(\epsilon^{n})$ (115)

which, up to leading order is $\epsilon$. The probability measure on the
weight-1 Paulis is $F\epsilon$. Hence, if we consider removing these using the
ancilla-efficient filter, the output fidelity of the channel is

$\displaystyle F_{\rm out}$
$\displaystyle\geq\frac{F}{1-F\epsilon}=\frac{1-\epsilon}{1-\epsilon^{n}-(1-\epsilon)\epsilon}\geq\frac{1-\epsilon}{1-(1-\epsilon)\epsilon}\geq
1-2\epsilon^{2}$ (116)

Hence, the output infidelity is again $O(\epsilon^{2})$. We see that, the
quadratic reduction in infidelity also holds up in the case of this model of
decaying global noise.

##  Appendix D Error Analysis

So far, we have considered the scenario that the filter operations are
noiseless. However, it would be instructive to have an estimate of the
performance of the filter if the filter operations introduce errors
themselves. In order to do this, we consider the single-qubit Pauli correction
filter, shown in Fig. 13, with measurements moved at the end of the circuit
for the ease of the analysis that is about to follow.

qubit q1; qubit q2; qubit r; init $\ket{0}$ q1; init $\ket{0}$ q2; h q1, q2;
box $X$ r—q1; box $Z$ r—q2; [shape=yquant-rectangle, rounded corners=.3em,
fill=red!20]box $\mathcal{E}$ r;

box $Z$ r—q2 ; box $X$ r—q1; h q1, q2;

measure q1; measure q2;

box $X$ r—q2; box $Z$ r—q1;

discard q1; discard q2; [$\equiv$] discard q2; init $\ket{0}$ q1;

h q1; box $X$ r—q1;

init $\ket{0}$ q2; h q2; box $Z$ r—q2; [shape=yquant-rectangle, rounded
corners=.3em, fill=red!20]box $\mathcal{E}$ r;

box $Z$ r—q2 ; h q2; box $X$ r—q2; measure q2; discard q2;

box $X$ r—q1; h q1; box $Z$ r—q1; measure q1;

discard q1;

Figure 13: single-qubit Pauli correction filter (with measurements pushed to
the end).

We assume noisy 2-qubit $CZ$ and $CX$ gates, with local depolarising noise on
the control and target qubits after each application of the gate.

$(\tilde{CZ}/\tilde{CX})=(\mathcal{D}_{p_{c}}\otimes\mathcal{D}_{p_{t}})\circ(CZ/CX)$
(117)

qubit q1; qubit q2; [this subcircuit box style=draw, dashed, rounded corners,
fill=red!20, inner ysep=5pt, inner xsep=0pt, register/default name=]
subcircuit qubit q[2]; box $Z$ q[1]—q[0]; (q1, q2); [$\equiv$]
operator/separation=2mm box $Z$ q2—q1 ; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box $p_{c}$ q1; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box $p_{t}$ q2; [ ] [this subcircuit box style=draw, dashed,
rounded corners, fill=red!20, inner ysep=5pt, inner xsep=0pt, register/default
name=] subcircuit qubit q[2]; box $X$ q[1]—q[0]; (q1, q2); [$\equiv$]
operator/separation=2mm box $X$ q2—q1 ; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box $p_{c}$ q1; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box $p_{t}$ q2;

Figure 14: Noisy $CZ$ and $CX$ gates. Each noisy 2-qubit gate is modelled as
an ideal gate, followed by local depolarising error on the control and target
qubits (with probability $p_{c}$ and $p_{t}$ respectively).

To find the effect of these errors, we use the (Pauli) error propagation rules
for the $CZ$ and $CX$ gates outlined in Fig. 15.

Figure 15: Pauli error propagation rules for the $CX$ (left) and $CZ$ (right)
gates.

Consider the filtering unitary $F_{Z}$, outlined in Fig. 16, where we have
marked locations of possible (Pauli) faults. Then, we denote by
$F_{Z}[U|P_{A}]$ by the unitary that results if we replace the location $A$ by
a Pauli $P_{A}$ (which can be either $I,X,Y,Z$). We denote $F_{Z}[I_{A}]$
simply by $F_{Z}$, which is the unitary without any faults. Using the
propagation rules, we can propagate this Pauli error $P_{A}$ forward in time
through all the gates that come after it. This results in an expression for
what errors remain on the final state after the circuit is executed with a
fault at location $A$. For example, one can check that,

$\bm{F_{Z}}[U|X_{A}]=Z_{1}Z_{2}\cdot\bm{F_{Z}}[U]$ (118)

Therefore, as the error $X_{A}$ propagates through the circuit, it results in
$Z$ errors on both the qubits (on top of the final state output from the
filter). In this way, we might find out how $Z$ and $X$ faults at the sites
$A,C,D$ propagate to the end of the filter. We can skip the calculations for
the site $C$, since this may be considered as part of the error channel
$\mathcal{E}$ that we are trying to correct. The result of this is shown in
Table 1.

qubit q; qubit r; init $\ket{0}$ q;

[this subcircuit box style=draw, dashed, rounded corners, fill=cyan!20, inner
ysep=6pt, inner xsep=0pt, ”$\bm{F}_{Z}$” below, register/default name=]
subcircuit qubit q; qubit r; h q; box $Z$ r—q; [shape=yquant-circle, radius =
2.5mm, fill=red!20]box A q; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box B r; [fill=white]box $U$ r; box $Z$ r—q; [shape=yquant-circle,
radius = 2.5mm, fill=red!20]box C q; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box D r; h q; box $X$ r—q; (q, r); measure q; discard q; [ ] init
$\ket{0}$ q;

[this subcircuit box style=draw, dashed, rounded corners, fill=cyan!20, inner
ysep=6pt, inner xsep=0pt, ”$\bm{F}_{X}$” below, register/default name=]
subcircuit qubit q; qubit r; h q; box $X$ r—q; [shape=yquant-circle, radius =
2.5mm, fill=red!20]box A q; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box B r; [fill=white]box $U$ r; box $X$ r—q; [shape=yquant-circle,
radius = 2.5mm, fill=red!20]box C q; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box D r; h q; box $Z$ r—q; (q, r); measure q; discard q;

Figure 16: Filter unitary for the Pauli-$Z$ (left) and Pauli-$X$ (right) Correction Filters with possible (Pauli) faults at locations A, B, C, D. Error | $X_{A}$ | $Z_{A}$ | $X_{C}$ | $Z_{C}$ | $X_{D}$ | $Z_{D}$
---|---|---|---|---|---|---
$F_{Z}$ | $q_{1}$ | $Z_{1}$ | $X_{1}$ | $Z_{1}$ | $X_{1}$ | $I_{1}$ | $Z_{1}$
$q_{2}$ | $Z_{2}$ | $X_{2}$ | $I_{2}$ | $X_{2}$ | $X_{2}$ | $Z_{2}$
$F_{X}$ | $q_{1}$ | $Z_{1}$ | $X_{1}$ | $Z_{1}$ | $X_{1}$ | $Z_{1}$ | $I_{1}$
$q_{2}$ | $X_{2}$ | $Z_{2}$ | $I_{2}$ | $Z_{2}$ | $X_{2}$ | $Z_{2}$
Table 1: Propagation of Pauli faults through the filter circuit.

Then, if we assume depolarising noise at all the fault sites and the error
channel, the faulty filter is equivalent to (Fig. 17)

$\tilde{\bm{\mathcal{F}}}[\mathcal{D}_{p_{e}}]=\mathcal{D}_{p_{0}}\circ\mathcal{D}^{(X)}_{p_{X}}\circ\mathcal{D}^{(Z)}_{p_{Z}}$
(119)

where $\mathcal{D},\mathcal{D}^{(X)},\mathcal{D}^{(Z)}$ are partially
depolarising, bit-flip and phase-flip channels respectively. The error
probabilities of these channels are

$\displaystyle p_{0}$ $\displaystyle=1-(1-p_{c})(1-p_{t})$
$\displaystyle\approx p_{c}+p_{t}$ (120) $\displaystyle p_{X}$
$\displaystyle=1-(1-2p_{c}/3)^{2}(1-2p_{t}/3)^{2}$ $\displaystyle\approx
2(p_{c}+p_{t})/3$ (121) $\displaystyle p_{Z}$ $\displaystyle=2p_{c}/3$ (122)

where the approximations hold for $p_{c},p_{t}\ll 1$. Then, the fidelity of
the output from the faulty filter is

$\displaystyle\tilde{F}$
$\displaystyle=(1-p_{0})(1-p_{X})(1-p_{Z})+p_{0}(p_{X}+p_{Z}+p_{Z}p_{X})/3$
(123)
$\displaystyle\geq(1-p_{c})(1-p_{t})\cdot(1-2p_{c})^{2}(1-2p_{t})^{2}\cdot(1-2p_{c}/3):=F_{\rm
critical}$ (124)

Eq. 124 shows that the lower bound on the output fidelity is more sensitive to
$p_{c}$ (it has an extra factor $(1-2p_{c}/3)$ of dependence on $p_{c}$, other
than which the terms involving $p_{c}$ and $p_{t}$ are symmetrical). As a
result, this provides analytical evidence for the intuition that, the fidelity
of the output channel is more sensitive to the error rate on the ancillae. In
addition, Eq. 124 also provides an upper bound $F_{\rm critical}$ on the
fidelity of the error channel $\mathcal{E}$, below which, using the filter
provides an advantage over using the unfiltered circuit.

qubit q2; qubit r; init $\ket{0}$ q2; h q2; [this subcircuit box style=draw,
dashed, rounded corners, fill=red!20, inner ysep=3pt, inner xsep=0pt,
register/default name=] subcircuit qubit q[2]; box $Z$ q[1]—q[0]; (q2, r);
[shape=yquant-rectangle, rounded corners=.3em, fill=red!20]box $\mathcal{D}$
r; [this subcircuit box style=draw, dashed, rounded corners, fill=red!20,
inner ysep=3pt, inner xsep=0pt, register/default name=] subcircuit qubit q[2];
box $Z$ q[1]—q[0]; (q2, r); h q2; measure q2; box $X$ r—q2; discard q2;
[$\equiv$] init $\ket{0}$ q2; h q2;

box $Z$ r—q2; [shape=yquant-circle, radius = 2.5mm, fill=red!20]box $p_{c}$
q2; [shape=yquant-circle, radius = 2.5mm, fill=red!20]box $p_{t}$ r;

[shape=yquant-rectangle, rounded corners=.3em, fill=red!20]box $\mathcal{D}$
r;

box $Z$ r—q2 ; [shape=yquant-circle, radius = 2.5mm, fill=red!20]box $p_{c}$
q2; [shape=yquant-circle, radius = 2.5mm, fill=red!20]box $p_{t}$ r;

h q2;

box $X$ r—q2;

measure q2; discard q2; [$\equiv$] discard q2; hspace 1mm r; [shape=yquant-
circle, radius = 2.5mm, fill=blue!20, ”$Z$” below]box $\bar{p}_{t}$ r;
[shape=yquant-rectangle, rounded corners=.3em, fill=blue!20, ”$Z$” below]box
$\mathcal{D}$ r; [shape=yquant-circle, radius = 2.5mm, fill=red!20]box
${p}_{t}$ r; [shape=yquant-circle, radius = 2.5mm, fill=red!20]box ${p}_{c}$
r; [shape=yquant-circle, radius = 2.5mm, fill=green!20, ”$X$” below]box
$\bar{p}_{c}$ r; hspace 1mm r;
qubit q1; qubit q2; qubit r; init $\ket{0}$ q1; init $\ket{0}$ q2; h q1, q2;
[this subcircuit box style=draw, dashed, rounded corners, fill=red!20, inner
ysep=3pt, inner xsep=0pt, register/default name=] subcircuit qubit q[2]; box
$X$ q[1]—q[0]; (q1, r);

[this subcircuit box style=draw, dashed, rounded corners, fill=red!20, inner
ysep=3pt, inner xsep=0pt, register/default name=] subcircuit qubit q[2]; box
$Z$ q[1]—q[0]; (q2, r); [shape=yquant-rectangle, rounded corners=.3em,
fill=red!20]box $\mathcal{E}$ r;

[this subcircuit box style=draw, dashed, rounded corners, fill=red!20, inner
ysep=3pt, inner xsep=0pt, register/default name=] subcircuit qubit q[2]; box
$Z$ q[1]—q[0]; (q2, r);

[this subcircuit box style=draw, dashed, rounded corners, fill=red!20, inner
ysep=3pt, inner xsep=0pt, register/default name=] subcircuit qubit q[2]; box
$X$ q[1]—q[0]; (q1, r);

h q1, q2;

measure q1; measure q2;

box $X$ r—q2; box $Z$ r—q1;

discard q1; discard q2; [$\equiv$] discard q2; init $\ket{0}$ q1;

h q1;

[this subcircuit box style=draw, dashed, rounded corners, fill=red!20, inner
ysep=3pt, inner xsep=0pt, register/default name=] subcircuit qubit q[2]; box
$X$ q[1]—q[0]; (q1, r);

[shape=yquant-circle, radius = 2.5mm, fill=blue!20, ”$Z$” below]box
$\bar{p}_{t}$ r; [shape=yquant-rectangle, rounded corners=.3em, fill=blue!20,
”$Z$” below]box $\mathcal{D}$ r; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box ${p}_{t}$ r; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box ${p}_{c}$ r; [shape=yquant-circle, radius = 2.5mm,
fill=green!20, ”$X$” below]box $\bar{p}_{c}$ r;

[this subcircuit box style=draw, dashed, rounded corners, fill=red!20, inner
ysep=3pt, inner xsep=0pt, register/default name=] subcircuit qubit q[2]; box
$X$ q[1]—q[0]; (q1, r);

h q1; measure q1; box $Z$ r—q1; discard q1;
qubit q1; qubit r; [$\equiv$] init $\ket{0}$ q1; h q1; box $X$ r—q1;
[shape=yquant-circle, radius = 2.5mm, fill=red!20]box $p_{c}$ q1;
[shape=yquant-circle, radius = 2.5mm, fill=red!20]box $p_{t}$ r;

[shape=yquant-circle, radius = 2.5mm, fill=blue!20, ”$Z$” below]box
$\bar{p}_{t}$ r; [shape=yquant-rectangle, rounded corners=.3em, fill=blue!20,
”$Z$” below]box $\mathcal{D}$ r; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box ${p}_{t}$ r; [shape=yquant-circle, radius = 2.5mm,
fill=red!20]box ${p}_{c}$ r; [shape=yquant-circle, radius = 2.5mm,
fill=green!20, ”$X$” below]box $\bar{p}_{c}$ r;

box $X$ r—q1; [shape=yquant-circle, radius = 2.5mm, fill=red!20]box $p_{c}$
q1; [shape=yquant-circle, radius = 2.5mm, fill=red!20]box $p_{t}$ r;

h q1; box $Z$ r—q1; measure q1; discard q1; [$\equiv$] discard q1;

[shape=yquant-circle, radius = 2.5mm, fill=green!20, ”$X$” below]box
$\bar{p}_{t}$ r; [shape=yquant-circle, radius = 2.5mm, fill=green!20, ”$X$”
below]box $\bar{p}_{t}$ r; [shape=yquant-circle, radius = 2.5mm,
fill=green!20, ”$X$” below]box $\bar{p}_{c}$ r; [shape=yquant-circle, radius =
2.5mm, fill=green!20, ”$X$” below]box $\bar{p}_{c}$ r; [shape=yquant-circle,
radius = 2.5mm, fill=red!20]box $p_{c}$ r; [shape=yquant-circle, radius =
2.5mm, fill=red!20]box $p_{t}$ r; [shape=yquant-circle, radius = 2.5mm,
fill=blue!20, ”$Z$” below]box $p_{c}$ r; hspace 1mm r;

Figure 17: Error propagation through the channel correction filter. Red, blue
and green circles represent partially depolarising, phase-flip and bit-flip
channels respectively, with corresponding error probabilities written inside
the circles. $\bar{p}_{c}=2p_{c}/3,\bar{p}_{t}=2p_{t}/3$.

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