††thanks: The first two authors contributed equally to this work.††thanks: The
first two authors contributed equally to this work.

# Unifying and benchmarking state-of-the-art quantum error mitigation
techniques

Daniel Bultrini  Theoretical Division, Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  Theoretische Chemie, Physikalisch-Chemisches Institut,
Universität Heidelberg, INF 229, D-69120 Heidelberg, Germany  Max Hunter
Gordon  Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid,
Madrid, Spain  Piotr Czarnik  Theoretical Division, Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  Andrew Arrasmith  Theoretical Division,
Los Alamos National Laboratory, Los Alamos, NM 87545, USA  Quantum Science
Center, Oak Ridge, TN 37931, USA  Patrick J. Coles  Theoretical Division, Los
Alamos National Laboratory, Los Alamos, NM 87545, USA  Quantum Science Center,
Oak Ridge, TN 37931, USA  Lukasz Cincio  Theoretical Division, Los Alamos
National Laboratory, Los Alamos, NM 87545, USA  Quantum Science Center, Oak
Ridge, TN 37931, USA

###### Abstract

Error mitigation is an essential component of achieving practical quantum
advantage in the near term, and a number of different approaches have been
proposed. In this work, we recognize that many state-of-the-art error
mitigation methods share a common feature: they are data-driven, employing
classical data obtained from runs of different quantum circuits. For example,
Zero-noise extrapolation (ZNE) uses variable noise data and Clifford-data
regression (CDR) uses data from near-Clifford circuits. We show that Virtual
Distillation (VD) can be viewed in a similar manner by considering classical
data produced from different numbers of state preparations. Observing this
fact allows us to unify these three methods under a general data-driven error
mitigation framework that we call UNIfied Technique for Error mitigation with
Data (UNITED). In certain situations, we find that our UNITED method can
outperform the individual methods (i.e., the whole is better than the
individual parts). Specifically, we employ a realistic noise model obtained
from a trapped ion quantum computer to benchmark UNITED, as well as state-of-
the-art methods, for problems with various numbers of qubits, circuit depths
and total numbers of shots. We find that different techniques are optimal for
different shot budgets. Namely, ZNE is the best performer for small shot
budgets ($10^{5}$), while Clifford-based approaches are optimal for larger
shot budgets ($10^{6}-10^{8}$), and for our largest considered shot budget
($10^{10}$), UNITED gives the most accurate correction. Hence, our work
represents a benchmarking of current error mitigation methods, and provides a
guide for the regimes when certain methods are most useful.

##  I Introduction

As new generations of quantum computers become larger and less noisy, the
practical application of quantum algorithms is just around the corner. It is
likely that any quantum algorithm that can achieve an advantage over its
classical counterpart will have to contend with substantial hardware error.
While quantum error correction may eventually allow one to eliminate hardware
noise, doing so will require many more qubits and significantly less noise.
Although full error correction will not be practical in the near term, it is
possible to partially mitigate errors with far lower resource requirements. A
number of such error mitigation techniques have been proposed Endo  _et al._
(2021); Cerezo  _et al._ (2020), each with their own benefits and trade-offs.

One of the most popular approaches to error mitigation is zero noise
extrapolation (ZNE) Temme  _et al._ (2017). ZNE involves collecting data at
different noise levels and fitting an extrapolation to the noiseless case. A
variety of methods have been introduced to amplify the noise as well as to
perform the extrapolation Temme  _et al._ (2017); Endo  _et al._ (2018). ZNE
has been shown to work well for small problem sizes Kandala  _et al._ (2019),
but has faced challenges for large problem sizes Czarnik  _et al._ (2020). Due
to the uncertainty of the extrapolation ZNE does not have performance
guarantees for regimes of high noise. Explicitly, for circuits involving many
gates, the lowest error points available are often too noisy for such fits to
be helpful.

Alternatively, classically simulable circuits can be used to inform the
experimenter about noise in a quantum device Czarnik  _et al._ (2020); Strikis
_et al._ (2020); Montanaro and Stanisic (2021); Vovrosh  _et al._ (2021);
Urbanek  _et al._ (2021). The Clifford data regression (CDR) method Czarnik
_et al._ (2020) makes use of the Gottesman-Knill theorem Gottesman (1998)
which guarantees that quantum circuits containing only Clifford gates can be
classically simulated. CDR constructs a training set using data obtained from
classically simulable near-Clifford circuits similar to the generic quantum
circuit of interest. This set of data is then used to perform a regressive fit
on an ansatz mapping noisy to exact expectation values. The fitted ansatz is
used to estimate the value of the noiseless result. Furthermore, the
combination of CDR and ZNE, which can outperform either method separately, has
been considered and is called variable-noise CDR (vnCDR) Lowe  _et al._
(2020). vnCDR uses training data involving near-Clifford training circuits
evaluated at multiple noise levels and can be thought of as using Clifford
data to inform the extrapolation to the zero noise limit.

More recently, error mitigation via virtual distillation (VD) has been
introduced Koczor (2020); Huggins  _et al._ (2020). VD mitigates noise by
simultaneously preparing multiple copies of a noisy state, entangling them,
and making measurements to virtually prepare a state which is more pure than
the original noisy states. Further improvements have been made and analyses
have been performed Czarnik  _et al._ (2021); Koczor (2021); Huo and Li
(2021); Cai (2021a). While VD could achieve exponential error suppression in
some regime Koczor (2020); Huggins  _et al._ (2020); Czarnik  _et al._ (2021),
it has been noted that VD requires a number of shots that scales exponentially
in the number of qubits used for sufficiently large noise rates Czarnik  _et
al._ (2021).

Given the success of merging ZNE and CDR into vnCDR, in this paper we consider
whether or not it is beneficial to unify these other error mitigation methods
with VD. By considering the number of copies used in VD to be a noise-control
parameter, we use Clifford data to guide an extrapolation to the noiseless
case (corresponding to the many copy limit), much like vnCDR. While also
adding ZNE-like noise amplification we arrive at a framework unifying CDR,
ZNE, and VD, which we call UNIfied Technique for Error mitigation with Data
(UNITED).

In order to get a realistic idea of how ZNE, vnCDR, VD, and UNITED compare we
carry out a numerical study benchmarking the performance of each technique
when mitigating a local observable produced by a random quantum circuit. We
explore their performance on a simulator based on the current error rates of
an ion-trap quantum computer. We compare performance across various system
sizes and circuit depths, involving systems of up to $10$ qubits and depths of
up to $128$ Molmer-Sørensøn entangling gates layers. Furthermore, we
systematically compare the necessary shot cost for each method in its
mitigation task, providing a fair comparison to guide experimenters in this
field.

Overall, we show an advantage of vnCDR over VD and ZNE for all considered
regimes apart from the regime of shallow circuits with $10^{5}$ shots for
which ZNE performs best. For the largest considered shot budget ($10^{10}$)
our new unified method UNITED gives an advantage over all other methods
considered. ZNE and VD can give good corrections, but their corrections do not
significantly improve with a reasonable increase in the total shot budget. The
unified approaches of vnCDR and UNITED display significant improvement over
both ZNE and VD, but a higher total shot cost is necessary to unlock that
improvement. This suggests that the unified methods (vnCDR and UNITED) are the
more powerful techniques, and the advantage they offer becomes clear when
investing sufficient resources. Hence, we believe such unified methods are
crucial to explore.

##  II State-of-the-art error mitigation methods

###  II.1 Zero Noise Extrapolation

ZNE Temme  _et al._ (2017) uses noisy expectation values of an observable of
interest $X$ obtained from the quantum circuit of interest $U$ run at
increased noise to extrapolate to the noise free limit. Originally, Richardson
extrapolation Richardson and Gaunt (1927) was proposed to estimate the noise-
free observable Temme  _et al._ (2017). The method was applied successfully to
mitigate variational ground state quantum simulations with IBM quantum
computers Kandala  _et al._ (2019); Dumitrescu  _et al._ (2018).

More formally, ZNE usually assumes that the noise can be characterized by a
single parameter $\lambda$ which is the noise strength. We consider a task of
estimating the expectation value $\mu=\bra{0}U^{\dagger}XU\ket{0}$ from
measurements on a noisy quantum device. Here $\ket{0}$ is the initial state.
To perform the extrapolation we measure the observable for $n+1$ noise
strengths $\lambda_{j}=c_{j}\lambda_{0}$, where $1=c_{0}<c_{1}<\dots<c_{n}$.
Here $\lambda_{0}$ is the smallest noise strength which can be obtained for
the quantum device. We call $c_{j}$ coefficients noise levels. For each noise
level we obtain the expectation value $\hat{\mu}_{j}$. The Richardson
extrapolation estimates the noise free value $\mu$ as

$\hat{\mu}^{\prime}=\sum_{j=0}^{n}\gamma_{j}\hat{\mu}_{j},$ (1)

where

$\sum_{j=0}^{n}\gamma_{j}=1,\quad\sum_{j=0}^{n}\gamma_{j}c_{j}^{k}=0\,\ {\rm
for}\,k=1,\ldots,n.$ (2)

Richardson extrapolation approximates $\mu$ with an error scaling as
$\mathcal{O}(\lambda_{0}^{n+1})$ Temme  _et al._ (2017). Nevertheless, the
Richardson extrapolation in the presence of shot noise leads to exponential
growth of $\hat{\mu}^{\prime}$ uncertainty with increasing $n$ Giurgica-Tiron
_et al._ (2020). Richardson extrapolation is equivalent to fitting a
polynomial of order $n$ on the various noisy expectations, treating the noise
levels $c_{j}$ as independent variables He  _et al._ (2020). To address
problems with reliability of Richardson extrapolation for large $n$ lower
order polynomial fits and exponential fits have been proposed Dumitrescu  _et
al._ (2018); Giurgica-Tiron  _et al._ (2020); Cai (2020). Furthermore,
approaches for noise characterized by multiple parameters have been postulated
Otten and Gray (2019).

The original proposal stretches pulses used to implement gates to increase the
hardware noise Temme  _et al._ (2017). Nevertheless, it is often more
convenient to increase noise performing so called identity insertions
Dumitrescu  _et al._ (2018); He  _et al._ (2020); Giurgica-Tiron  _et al._
(2020). These methods insert groups of gates that evaluate to the identity
into the circuit of interest. This does not change a state encoded by the
circuit $U\ket{0}$ but increases the number of gates in the circuit, scaling
the noise.

ZNE provides a very simple framework to mitigate errors introduced by noise on
some observable of interest. However, there are several technical challenges
which limit its power. Firstly, ZNE assumes that low enough noise levels are
experimentally accessible to enable successful extrapolation to the zero noise
limit. Furthermore, the best choice of the ansatz used to perform the
extrapolation is not known a priori Giurgica-Tiron  _et al._ (2020). Also,
controlled increase of the noise strength is hardware dependent and may be
challenging to implement in practice Kandala  _et al._ (2019).

###  II.2 Clifford Data Regression

We first introduce the CDR framework which provides background for the more
general vnCDR and UNITED methods. CDR Czarnik  _et al._ (2020) uses
classically simulable training circuits consisting mainly of Clifford gates to
learn effects of the noise on an observable of interest. For the training
circuits the expectation value $X$ is evaluated classically and on noisy
quantum hardware. This data is used to train a function mapping the noisy to
the exact expectation values, which is then used to mitigate the expectation
value $X$ for some circuit of interest. The method was successfully applied to
quantum simulations of condensed matter systems with IBM quantum computers
Czarnik  _et al._ (2020); Sopena  _et al._ (2021).

The training circuits $\mathcal{S}=\\{V_{i}\\}_{i=1}^{N_{t}}$ are chosen to be
similar to $U$. The simplest strategy to construct the training set is to
replace most of non-Clifford gates in the circuit of interest by a Clifford
gate which is similar Czarnik  _et al._ (2020). Alternatively, the usage of
Markov Chain Monte Carlo was proposed to construct training sets where the
training circuits have a low cost function value in the case of variational
quantum algorithms Czarnik  _et al._ (2020). To ensure classical simulability
of the training circuits, they contain at most $N$ non-Clifford gates, with
the cost of their classical simulation growing exponentially with $N$. The
training set is constructed as $\mathcal{T}_{\rm CDR}=\\{(x_{i},y_{i})\\}$,
where $x_{i}$ and $y_{i}$ are the noisy and exact expectation values of $X$
for the near-Clifford training circuits. This data is used to fit a linear
function,

$\displaystyle f_{\rm CDR}(x)=a_{1}x+a_{2},$ (3)

which approximates a relation between the noisy and exact expectation values.
Values of parameters $a_{1},a_{2}$ are found by least-squares linear
regression on $\mathcal{T}_{\rm CDR}$. The trained ansatz is then applied to
estimate the noise-free result for the circuit of interest $\hat{\mu}_{\rm
CDR}=f_{\rm CDR}(\hat{\mu}_{0})$.

The primary challenge in CDR is the construction of the near-Clifford training
circuits as the optimal procedure to construct the training set is not known
in general. Furthermore, the method requires evaluation of the observable of
interest for the training circuits which may result in a large shot cost of
the mitigation for large training sets.

###  II.3 Variable noise Clifford Data Regression

vnCDR conceptually unifies ZNE with CDR by using a training set involving
near-Clifford circuits evaluated at several noise levels to mitigate $X$ for
$U$ which in general can not be simulated classically Lowe  _et al._ (2020).
This data is used to learn the parameters that lead to the best extrapolation
to the zero noise limit. The method was successfully implemented for quantum
dynamics simulations with IBM quantum computers Sopena  _et al._ (2021).

The training set is defined as $\mathcal{T}_{\rm
vnCDR}=\\{(\bm{x}_{i},y_{i})\\}_{i=1}^{N_{t}}$, where
$\bm{x}_{i}=(x_{i,0},\dots,x_{i,n})$ is the vector of noisy estimates of the
expectation value $X$ , with length $n+1$. $x_{i,j}$ is evaluated for the
circuit $V_{i}$ with noise scaled by a factor $c_{j}$. For $n=0$ the vnCDR
training set becomes equivalent to the CDR training set. This data is then
used to fit a linear function $f_{\rm vnCDR}$,

$f_{\rm vnCDR}(\bm{x})=\sum_{j=0}^{n}a_{j}x_{j}\,.$ (4)

where $f_{\rm vnCDR}$ approximates relation between the noisy and the exact
expectation values and is fitted using least-squares regression on the data
from vectors given by training circuits $V_{i}$ for all $i$, who’s index has
been dropped in the equation above. This leads to the best fit parameters
$a^{*}_{j}$.

The fitted parameters $a^{*}_{j}$ are then used to obtain vnCDR estimate of
$X$ for $U$, $\hat{\mu}_{\rm vnCDR}=f_{\rm vnCDR}(\bm{\hat{\mu}})$, where
$\bm{\mu}=(\hat{\mu}_{0},\dots,\hat{\mu}_{n})$ is comprised of the $n+1$ noisy
expectations for the circuit of interest.

vnCDR shares several common features with CDR and ZNE. The functional form of
the ansatz is a linear combination of noisy estimates which resembles
Richardson extrapolation. Therefore, vnCDR can be understood as performing an
extrapolation guided by near-Clifford circuits with a similar structure to the
circuit of interest. Nevertheless, the fitted ansatz do not require explicit
knowledge of the noise levels unlike ZNE ansatze, potentially easing ZNE
requirements for precise control of the noise strength. Both CDR and vnCDR use
training on data produced by near-Clifford quantum circuits. Consequently,
vnCDR can be also treated as adding additional relevant data features to the
CDR training set.

As in the case of CDR the construction of an informative training set is the
primary technical challenge. Furthermore, adding additional noise levels to
the vnCDR training set increases shot cost of the mitigation. Therefore, the
best choice of noise levels is not known a priori.

###  II.4 Virtual Distillation

Virtual distillation (VD) Koczor (2020); Huggins  _et al._ (2020) uses $M>1$
copies of a noisy state $\rho$ corresponding to a quantum circuit $U$ to
obtain expectation values of an observables X for a purer, “distilled” state.
VD mitigates its expectation value by computing

$\hat{\mu}_{\rm VD}=\frac{{\rm Tr}[\rho^{M}X]}{{\rm Tr}[\rho^{M}]}.$ (5)

We consider the eigenvalue decomposition of $\rho$

$\rho=\sum_{i=0}^{2^{Q}-1}\lambda_{i}|\psi_{i}\rangle\langle\psi_{i}|,$ (6)

with $\lambda_{i}$ decreasing with increasing $i$, and $Q$ being the number of
the qubits. Then,

$\hat{\mu}_{\rm
VD}=\frac{\langle\psi_{0}|X|\psi_{0}\rangle}{1+\sum_{i=0}^{2^{Q}-1}(\lambda_{i}/\lambda_{0})^{M}}+\frac{\sum_{i=1}^{2^{Q}-1}(\lambda_{i}/\lambda_{0})^{M}\langle\psi_{i}|X|\psi_{i}\rangle}{1+\sum_{i=0}^{2^{Q}-1}(\lambda_{i}/\lambda_{0})^{M}}.$
(7)

Therefore, with increasing $M$, VD exponentially suppresses contributions to
$\hat{\mu}_{\rm VD}$ from eigenvectors of $\rho$ other than the dominant one
$|\psi_{0}\rangle$. While increasing $M$ suppresses the errors it also
increases the shot cost of the estimation of $\hat{\mu}_{\rm VD}$
exponentially with a given accuracy for large error rates Czarnik  _et al._
(2021).

VD implicitly assumes that the dominant eigenvector $|\psi_{0}\rangle$ is
close to the noise-less state $|\psi_{\rm exact}\rangle$. Therefore, the
quality of the VD correction is limited by a noise floor

$\epsilon=\langle\psi_{0}|X|\psi_{0}\rangle-\langle\psi_{\rm
exact}|X|\psi_{\rm exact}\rangle.$ (8)

The noise floor can be bounded from above using a quantity called the coherent
mismatch

$c=1-|\langle\psi_{\rm exact}|\psi_{0}\rangle|^{2},$ (9)

i.e.,

$|\epsilon|\leq 2c||X||_{\infty},$ (10)

where $||X||_{\infty}$ is the largest eigenvalue of $X$ Koczor (2021). For
realistic noise, $\epsilon$ usually does not equal $0$ Koczor (2020).
Nevertheless, it was shown that for sufficiently small error rates the noise
floor does not prevent VD from significantly reducing the impact of noise
Koczor (2021).

${\rm Tr}[\rho^{M}X]$ can be computed using $M$ copies of the noisy state
$\rho$ and an ancillary qubit with a controlled permutation of the copies
which changes position of each copy (so-called controlled derangement).
Consequently, $QM+1$ qubits are required to mitigate a $Q$-qubit state. This
number can be reduced to $2Q+1$ by employing qubit resets and increasing the
depth of the circuit performing the controlled derangement Czarnik  _et al._
(2021). The controlled derangement can be implemented as a circuit of $QM$
controlled swaps. Therefore, the derangement introduces errors to the
estimates of ${\rm Tr}[\rho^{M}X]$ and ${\rm Tr}[\rho^{M}]$ which may require
correction with other error mitigation methods Koczor (2020).

Recently, VD-inspired methods were proposed which further reduce required
qubit counts Huo and Li (2021); Cai (2021a) and conceptually unify VD with
both ZNE and subspace expansion techniques enabling error mitigation beyond
the noise floor Xiong  _et al._ (2021); Yoshioka  _et al._ (2021); Cai
(2021b).

##  III Unifying variable noise Clifford Data Regression methods with Virtual
Distillation

In this section we unify VD into the framework provided by mitigation
strategies based on training on Clifford circuit data. These learning based
error mitigation methods like CDR and vnCDR assume that effect of the noise
can be learned from near-Clifford circuits evaluated on both a noisy quantum
device and a classical simulator. Results obtained with vnCDR show that it is
often beneficial to supplement the CDR training data with data from the
training circuits run under increased noise. Similarly, one can consider
supplementing the vnCDR training set with results obtained by VD to improve
its performance. This can be understood as using Clifford data to first
extrapolate VD results to the limit of large $M$, and then to perform an
extrapolation to the zero noise limit. This conceptually unifies VD, ZNE and
CDR into one mitigation method which we call UNIfied Technique for Error
mitigation with Data (UNITED).

To motivate the form of the UNITED ansatz we use Richardson extrapolation to
perform the aforementioned double extrapolation to the zero noise limit. In
the following we neglect the errors introduced by imperfect derangement and
using a finite number of shots. The error of mitigating an observable $X$
using VD with $M$ copies, can be expressed as

$|\mu-\hat{\mu}_{M}|=\mathcal{O}(\xi^{M}+\epsilon),$ (11)

where $\mu$ is the exact value, $\hat{\mu}_{M}$ is the VD mitigated
expectation value obtained using $M$ copies, $|\xi|<1$ corresponds to the
error introduced by a part of the noisy state orthogonal to its dominant
eigenvector $|\psi_{0}\rangle$ and $\epsilon$ is the noise floor.

We use Richardson extrapolation in the number of copies up to copy $M_{\rm
max}$. Taking results obtained for each value of $M$ as corresponding to an
noise level $c_{M}$ we obtain a mitigated expectation value

$\hat{\mu}^{\prime}=\sum_{M=1}^{M_{\rm max}}\gamma_{M}\hat{\mu}_{M},$ (12)

where

$\sum_{M=1}^{M_{\rm max}}\gamma_{M}=1,\quad\sum_{M=1}^{M_{\rm
max}}\gamma_{M}c_{M}^{k}=0\,,$ (13)

for $k=1,\ldots,M_{\rm max}$. The mitigated estimate $\hat{\mu}^{\prime}$ has
an error of order $\mathcal{O}(\xi^{M_{\rm max}+1}+\epsilon)$. Therefore,
extrapolation on the copy data improves on VD suppression of the incoherent
error for $M=M_{\rm max}$ without requiring $M>M_{\rm max}$.

Similarly to vnCDR one can use near-Clifford circuits to fit the Richardson
extrapolation coefficients (12). Therefore, we use a training set constructed
analogously to the vnCDR training set replacing $c>1$ noisy expectation values
with VD mitigated multi-copy expectation values. Furthermore, we use the
linear ansatz used by vnCDR to learn large $M$ VD-mitigated expectation
values. We call this approach Clifford Guided Virtual Distillation (CGVD).
This motivation comes with the caveat that we do not have classical access to
the noise floor for near-Clifford circuits so we propose to use their exact
expectation values instead implicitly assuming that the noise floor is
negligible.

The CGVD training set for an observable $X$ and a circuit of interest $U$ is
defined as $\mathcal{T}_{\rm CGVD}=\\{(\bm{x}_{i},y_{i})\\}$, where
$\bm{x}_{i}=(x_{i,1},\dots,x_{i,M_{\rm max}})$ is the vector of the noisy
$x_{i,1}$ and VD-mitigated $(x_{i,2},\dots,x_{i,M_{\rm max}})$ estimates using
numbers of copies from $2$ to $M_{\rm max}$, evaluated from the near-Clifford
training circuit $V_{i}$. We note that the ansatz can be generalized to apply
to other choices of noisy and VD-mitigated training data. The training set is
fitted with a linear ansatz to approximate the relation between the vector of
noisy values, the VD-mitigated expectation values and the exact expectation
value

$f_{\rm CGVD}(\bm{x})=\sum_{m=1}^{M_{\rm max}}b_{m}x_{m}\,,$ (14)

where the parameters $b_{m}$ are fitted using least-squares regression on the
training data consisting of vectors $\bm{x}_{i}$ produced by training circuits
$V_{i}$ for all $i$, who’s index has been dropped in the above equation. This
leads to the best fit parameters $b^{*}_{j}$ that are then used to correct the
noisy and VD-mitigated estimates of $X$ expectation value for the circuit of
interest. In Appendix A we show that CGVD perfectly mitigates global
depolarizing noise. The perfect correction is possible as for this noise model
the noise floor is zero.

As previously mentioned, the CGVD method can be further generalized to
mitigate effects of the noise floor by unifying it with vnCDR. Evaluating
observables mitigated by VD for different levels of state preparation noise
will scale the noise floor. In general, the stronger the noise, the larger the
effect of coherent mismatch. Taking $n+1$ state preparation noise strengths
$\lambda_{j}=c_{j}\lambda_{0}$, where $1=c_{0}<c_{1}<\dots<c_{n}$, we can
express the error of an observable mitigated using Richardson extrapolation on
the copy data (12) at noise level $c_{j}\lambda$ as

$|\mu-\hat{\mu}^{\prime}_{j}|=\mathcal{O}(\xi_{j}^{M_{\rm
max}+1}+\epsilon_{j}).$ (15)

Assuming the noise scaling causes the coherent mismatch to be scaled in a
controlled manner, we can undertake a further Richardson extrapolation on the
noise in the state preparation

$\displaystyle\hat{\mu}^{\prime\prime}$
$\displaystyle=\sum_{j=0}^{n}\gamma_{j}\hat{\mu}^{\prime}_{j}\ ,$ (16)

where

$\sum_{j=0}^{n}\gamma_{j}=0,\quad\sum_{j=0}^{n}\gamma_{j}c_{j}^{k}=0\,\ {\rm
for}\,k=1,\ldots,n.$ (17)

This can be expanded as

$\displaystyle\hat{\mu}^{\prime\prime}$
$\displaystyle=\sum_{j=0}^{n}\sum_{M=1}^{M_{\rm
max}}\gamma_{j}\gamma_{M}\hat{\mu}_{M,j}$ (18)
$\displaystyle=\sum_{j=0}^{n}\sum_{M=1}^{M_{\rm
max}}\gamma^{\prime}_{j,M}\hat{\mu}_{M,j}\ ,$ (19)

where $\hat{\mu}_{M,j}$ is the observable mitigated by VD with $M$ copies at
state preparation noise strength $c_{j}\lambda$. Assuming $\xi_{j}^{M_{\rm
max}+1}\ll\epsilon_{j}$, this further extrapolation reduces the error of the
noise free estimate to $\mathcal{O}(\epsilon^{n+1})$. This appears to be a
promising approach to the zero noise limit as VD-mitigated estimates of
expectation values are much less affected by noise than single copy noisy
estimates.

We propose to use near-Clifford circuits to perform a guided extrapolation to
the noise-free limit aided by the values of observables mitigated by VD at
various levels of state preparation noise and different numbers of copies. We
supplement the CGVD training set with the noisy and VD-mitigated expectation
values for $c_{j}>1$ and fit the training set using a linear ansatz motivated
by the form of (19).

More explicitly, we use the most general training set $\mathcal{T}_{\rm
UNITED}=\\{(\bm{\tilde{x}}_{i},y_{i})\\}$, where
$\bm{\tilde{x}}_{i}=(\tilde{x}_{i,0,1},\dots,\tilde{x}_{i,n,M})$ is the vector
of noisy and VD mitigated estimates with copies $1$ to $M_{\rm max}$, and
state preparation levels $c_{0}$ to $c_{n}$, evaluated from the circuit
$V_{i}$. The ansatz we train is,

$f_{\rm UNITED}(\bm{x})=\sum_{j=0}^{n}\sum_{M=1}^{M_{\rm
max}}d_{j,m}x_{j,m}\,,$ (20)

using least-squares regression on the vectors denoted $\bm{\tilde{x}}_{i}$
produced by training circuits $V_{i}$, who’s index has been dropped in the
above equation. This outputs the fitted parameters $d^{*}_{j,m}$ that define
the best possible mapping from the noisy to exact values in the training set
on average.

The trained ansatz, with parameters $d^{*}_{j,m}$, is then used to correct
data obtained by similarly applying VD for different numbers of copies and
changing state preparation noise levels for the circuit of interest. In this
way we use near-Clifford circuits mitigated with VD to guide VD applied to the
circuit of interest while also using noise data to extrapolate from the VD
mitigated observables to the zero-noise limit. We note that under this
framework, vnCDR is a special case of UNITED. This special case is recovered
by taking $M_{\rm max}=1$.

It is important to note that although Richardson extrapolation motivates the
form of the fitted ansatz, it does not guarantee good performance of the new
method for several reasons. Firstly, all methods to scale physical noise are
only approximate so the noise levels will be imperfectly controlled, leading
to imperfect extrapolated zero noise estimates. Nevertheless, successful
applications of ZNE and vnCDR show that imperfect noise scaling control does
not preclude successful error mitigation. Furthermore, training a model based
on near-Clifford circuit data is not equivalent to performing a Richardson
extrapolation, so performance of UNITED depends on the ability to construct
informative training sets. On the other hand, analogous to the case of vnCDR,
we expect that fitting ansatz coefficients (20) instead of performing an
extrapolation is beneficial as it does not require knowledge of noise levels,
reducing impact of imperfect noise control. Additionally, adding copy data to
the training set further increases the shot cost of the mitigation. Therefore,
the best choice of noise levels and the optimal number of copies is not known
a priori. Despite this we found that the method gives mitigated estimates
which remain stable with increasing $n$ and $M_{\rm max}$ and that it either
outperforms or matches vnCDR performance for sufficiently large shot numbers.

##  IV Numerical Results

In this section we present the performance of ZNE, vnCDR, VD and UNITED
applied to noisy outputs obtained from simulations of an trapped-ion quantum
computer with a realistic noise model. We consider the task of mitigating the
output of random quantum circuits to benchmark the method for various system
sizes and circuit depths. We study several total shot budgets per mitigated
expectation value, highlighting strengths and weaknesses of each method.

Figure 1: Random quantum circuit structure used for our benchmark simulations.
Here we show a layer of an exemplary random quantum circuit for $Q=4$. The
layer is built from two layers of alternating nearest-neighbor Molmer-Sørensøn
gates $XX(\delta)=e^{-i\delta\sigma_{X}^{j}\sigma_{X}^{j+1}}$ gates, where
$\sigma_{X}^{j}$ is a Pauli operator acting on qubit $j$. The $XX$ gates are
decorated with general single-qubit unitaries
$v(\alpha,\beta,\gamma)=R_{Z}(\alpha)R_{Y}(\beta)R_{Z}(\gamma)$. Here
$R_{Z}(\alpha)=e^{-i\alpha/2\sigma_{Z}}$,
$R_{Y}(\beta)=e^{-i\beta/2\sigma_{Y}}$, and $\sigma_{Z},\sigma_{Y}$ are Pauli
operators. We choose angles $\alpha,\beta,\gamma,\delta$ randomly for each
random quantum circuit instance of the gates. We remark that $R_{Y},R_{Z},XX$
are native gates of a trapped-ion quantum computer.  Figure 2: Comparing error
mitigation methods for $L=Q$. Mean and maximal absolute errors of the
expectation value of $\sigma_{Z}$ measured at the first qubit of $30$
instances of random quantum circuits for noisy estimates and estimates
obtained with error mitigation methods. In (a), the results for total number
of shots per mitigated (noisy) expectation value $N_{\rm tot}=10^{5}$. In (b),
$N_{\rm tot}=10^{6}$. In (c), $N_{\rm tot}=10^{8}$. In (d), $N_{\rm
tot}=10^{10}$. For $N_{\rm tot}=10^{5}$, we find that ZNE gives the best
results. For $N_{\rm tot}=10^{6}$, $10^{8}$ vnCDR gives the best results while
for $N_{\rm tot}=10^{10}$ UNITED gives the best results.

###  IV.1 Random quantum circuits

We investigate random quantum circuits for various numbers of qubits
$Q=4,6,8,10$ and several depths. We consider random quantum circuits built of
the native gates of the trapped-ion quantum computer, namely of $L$ layers of
alternating nearest-neighbor Molmer-Sørensøn entangling gates decorated with
general single qubit unitaries, see Fig. 1 for details. Both the entangling
gates and the single qubit unitaries are parametrized by randomly chosen
angles. Here we simulate random quantum circuits with $L=gQ$, taking $g=1$ and
$g=16$ .

We expect that for such a setup, random quantum circuits will converge with
increasing $L$ to random quantum states sampled according to Haar measure
McClean  _et al._ (2018). Therefore, the setup is general and challenging,
making it relevant for testing error mitigation methods. Furthermore, it is
worth noting that deep circuits implemented for trapped-ion quantum computing
architecture are a favorable setting for VD as this architecture enables all-
to-all connectivity which decreases the depth of the controlled derangement
circuit compiled to native gates Maslov (2017). Furthermore, it is expected
that for sufficiently deep circuits and sufficiently small gate error rates,
the coherent mismatch does not prevent good quality VD-mitigation Koczor
(2021).

###  IV.2 The noise model

We use a noise model of a trapped-ion quantum computer developed in Trout  _et
al._ (2018); Cincio  _et al._ (2021a). The model uses local depolarizing and
dephasing channels as well as imprecision in the rotation angles of the gates.
The parameters used are derived from trapped-ion experiments to simulate
errors of current trapped-ion devices. For a detailed description of the noise
model including gates error rates see Ref. Cincio  _et al._ (2021a). We
neglect measurement error in our simulations as it can be mitigated by
specialized techniques Hamilton  _et al._ (2020); Bravyi  _et al._ (2021).

Figure 3: Convergence of error mitigation methods with increasing $N_{\rm
tot}$. Mean and maximal absolute errors of the expectation value of
$\langle\sigma_{Z}^{1}\rangle$ for random quantum circuits plotted versus the
shot budget. We use $30$ instances of random quantum circuits. In (a) results
for $Q=4,L=4$. In (b) results for $Q=4,L=64$. For $Q=4,L=4$ we find that ZNE
and VD estimates converge first. vnCDR estimates converge slower but
outperform VD and ZNE for $10^{7}-10^{10}$ shot budgets. UNITED convergence is
the slowest but it gives the best results for $N_{\rm tot}=10^{9},10^{10}$.
For $Q=4,L=64$ again ZNE and VD estimates convergence is the fastest but it is
outperformed by vnCDR and UNITED for all considered shot budgets. vnCDR and
UNITED converge slowest but they outperform the other methods for all
considered budgets. We see an improvement of UNITED over vnCDR for $N_{\rm
tot}=10^{9},10^{10}$. Further discussion of VD performance for different shot
budgets is explored in Appendix B.

###  IV.3 The results

First we analyze the case of $L=Q$ ($g=1)$. For this choice we consider
$Q=4,6,8,10$. For each pair of $L$ and $Q$ values we consider $30$ instances
of random quantum circuits. For each random quantum circuit we mitigate the
expectation value of $\sigma_{Z}$ acting on the first qubit of the circuit
$\langle\sigma_{Z}^{1}\rangle$. We perform the mitigation with data obtained
using a finite number of shots. For each mitigated expectation value and each
error mitigation method we assign total number of shots used by the method to
mitigate the expectation value. We call this number the total shot budget
$N_{\rm tot}$. We consider $N_{\rm tot}=10^{5}-10^{10}$.

Different error mitigation methods require estimates of observables from a
different total number of circuits. In the case of ZNE the circuit of interest
needs to be evaluated for several different state preparation noise levels. In
the case of vnCDR, estimates for the observable of interest are required for
many training circuits evaluated at several noise levels. UNITED additionally
requires performing VD mitigation for both the training circuits and the
circuit of interest. Therefore, for the same shot budget, different mitigation
methods have different numbers of shots per observable estimation.
Consequently, for a given total number of shots, the accuracy of individual
observable estimates decreases as the required total number of circuits
increases. We note that for fixed $N_{\rm tot}$ we can distribute shots
between different observable evaluations in many ways. Here for the sake of
simplicity for a given method we assign the same shot numbers for all
observable estimates required to mitigate the observable of interest.

In Figure 2 we plot mean and maximal absolute errors of
$\langle\sigma_{Z}^{1}\rangle$. We find that performance of the error
mitigation methods depends primarily on the shot budget. We observe that the
expectation values mitigated by ZNE and VD converge with increasing $N_{\rm
tot}$ fastest. ZNE is the best for the smallest total shot budget $N_{\rm
tot}=10^{5}$. The convergence of vnCDR is slower but it outperforms ZNE and
the other methods for $N_{\rm tot}=10^{6}$ and $N_{\rm tot}=10^{8}$. We find
that UNITED convergence with increasing $N_{\rm tot}$ is the slowest but it
gives the best results for $N_{\rm tot}=10^{10}$. We see that quality of the
mitigated results decreases with increasing $Q$ as the number of noisy gates
increases. Nevertheless even for the largest considered $Q=10$, a factor of
$20$ improvement over the noisy result can be obtained with vnCDR and UNITED.
$Q=4$ results with various $N_{\rm tot}$ are shown in Figure 3(a).

Next we analyze the case of $L=16Q$ ($g=16)$. For this choice we simulate
$Q=4,6,8$. Again, for each pair of $L$ and $Q$ values we consider $30$
instances of random quantum circuits and mitigate
$\langle\sigma_{Z}^{1}\rangle$ for each of them. As in the case of $g=1$ we
perform the mitigation for $N_{\rm tot}=10^{5}-10^{10}$. For $Q=6,L=96$ and
$Q=8,L=128$ we are unable to obtain improvement over the noisy results with
any error mitigation method as the number of the noisy gates is too large. For
smaller systems of $Q=4,L=64$ we find that learning-based error mitigation
methods outperform VD and ZNE even for $N_{\rm tot}=10^{5}$. For $N_{\rm
tot}=10^{7}-10^{10}$ we find that both vnCDR and UNITED give a factor of
$5-10$ improvement over the noisy results. For $N_{\rm tot}=10^{9}-10^{10}$ we
see an improvement of vnCDR over UNITED. Again we find that ZNE and VD
estimates converge fastest with increasing $N_{\rm tot}$. Again UNITED
convergence is the slowest but it also provides the best mitigation quality
for large shot budget. We show $Q=4$ results in detail in Fig. 3(b).

##  V Implementation Details

###  V.1 Zero Noise Extrapolation

In the simulations described above, we use two noise levels $c\in\\{1,2\\}$
and linear extrapolation. We compared this approach to linear extrapolation
for $c\in\\{1,2,3\\}$ and exponential extrapolation for both $c\in\\{1,2\\}$
and $c\in\\{1,2,3\\}$ and found that it is the best one in general.

When scaling the noise in our simulations we use factors $c=2,3$. To implement
$c=2$ we replace $XX(\theta)$ by $XX(\alpha)XX(\theta-\alpha)$ with $\alpha$
chosen randomly. We treat $R_{Z}$ and $R_{Y}$ in the same way. To implement
$c=3$ we replace $XX(\theta)$ by $XX(\theta)XX(\alpha)XX(-\alpha)$ performing
an identity insertion. Again, we treat $R_{Z}$ and $R_{Y}$ in the same way.
Both procedures scale the number of all types of gates in our random quantum
circuits by the noise level.

###  V.2 variable noise Clifford Data Regression

To construct the near-Clifford training circuits we use an algorithm from
Czarnik  _et al._ (2021). We replace $XX$ and $R_{Y}$ by closest to them
Clifford gates of the same type. We also replace a fraction of $R_{Z}$ gates
by closest to them $R_{Z}$ Clifford gates, leaving $N=10$ $R_{Z}$ gates
intact. See details of the algorithm in Czarnik  _et al._ (2021).

When constructing the noise-scaled circuits necessary for vnCDR we use the
same method of noise scaling as used in the ZNE implementation. In our
simulations above we use $c\in\\{1,2,3\\}$. We found that this choice of noise
levels results in better results than using $c\in\\{1,2\\}$.

In order to maximize the information contained within the training set while
minimizing the number of shots used to evaluate their outputs on noisy
hardware we inspect the distribution of the exact values produced by these
circuits. We initially construct $100$ near-Clifford circuits using the method
introduced above. We find that with increasing number of qubits an increased
fraction of their exact expectation values is becoming close to $0$.
Additionally, we find that removing most of such circuits from the training
set is not decreasing the quality of the correction while at the same time
decreasing the shot cost of the mitigation. Therefore, to construct the
training set, we select $N_{t}$ of them with the largest absolute values of
the exact expectation values. We choose $N_{t}=50$. We note that this post-
selection method can be further improved. For example, one can use use Markov
Chain Monte Carlo to construct a training set with desired distribution of
exact or noisy expectation values Czarnik  _et al._ (2020).

###  V.3 Virtual Distillation

We implement VD by computing $\frac{{\rm Tr}[\rho^{M}X]}{{\rm Tr}[\rho^{M}]}$
for the noisy state $\rho$ and neglecting errors which occur during the
derangement circuit execution. This assumption enables classical simulation of
VD for a $Q$-qubit system while simulating $Q$ noisy qubits instead of the
$2Q+1$ that would otherwise be required to simulate the derangement. We
present results of VD using $2$ copies as we observed that this typically gave
the best results. We explore the performance of VD for various numbers of
copies in Appendix B.

###  V.4 Unified Data-driven Error Mitigation

To implement UNITED we choose the same near-Clifford circuits as for vnCDR. We
choose $M_{\rm max}=3$ and state preparation noise levels $c\in\\{1,2,3\\}$.
We found that these parameters performed best across the regimes investigated.

###  V.5 Shot budgeting

Each technique described above requires a certain total number of circuits in
order to mitigate the observable of interest. This, therefore, corresponds to
a different number of observable evaluations. For VD, it is necessary to
perform a measurement for $2$ different circuits obtained with the controlled
derangement Koczor (2020). In the case of ZNE, one needs to measure the
expectation value of $n+1$ circuits built from one circuit of interest at
$n+1$ noise levels. For vnCDR, we additionally need to evaluate each training
circuit for $n+1$ noise levels. This means that in total vnCDR requires
$(n+1)(N_{t}+1)$ circuit evaluations. Finally, for UNITED we need to mitigate
the observable estimates using VD with $M_{\rm max}-1$ different copies for
the training circuits, the circuit of interest and for each of $n+1$ noise
levels. Therefore, the number of circuit evaluations required by UNITED is
$(n+1)(N_{t}+1)(2M_{\rm max}-1)$.

In this work we consider several total shot budgets $N_{\rm tot}$. For each
method we distribute the shots across measurements dividing $N_{\rm tot}$ by
the number of measurements. We note that this strategy may not be optimal as
it may be beneficial to distribute the shots unequally between noise levels or
copy numbers. Furthermore, one can fine-tune the values of $n$, $N_{t}$,
$M_{\rm max}$ for given $N_{\rm tot}$. We leave systematic investigation of
more advanced budgeting strategies for a future work.

##  VI Discussion

The ability to efficiently and accurately mitigate hardware errors will be
essential to achieving practical application of near-term quantum computers. A
number of approaches have been proposed to fill this need, each bringing
useful new ideas. In this work, we have discussed some state-of-the-art
methods and examined how the approaches might best be combined. Specifically,
we have considered zero noise extrapolation (ZNE), variable noise Clifford
data regression (vnCDR), and virtual distillation (VD) and proposed two new
methods that represent unifications of the ideas behind these approaches. We
applied these methods to the mitigation of errors in local observables
measured from states prepared with random circuits. As these methods have
different strengths and resource requirements, we have considered their
performance with $10^{5}-10^{10}$ total shots to mitigate the expectation
value of an observable of interest with up to $Q=10$ qubits with depth $L=Q$
as well as up to $Q=8$ for depth $L=16Q$. We used a realistic simulation of a
noisy trapped-ion device, which was chosen as the trapped-ion architecture
enables all-to-all connectivity.

Combining the ideas of near-Clifford circuit data and the multi-copy data from
VD, we formulated Clifford guided VD (CGVD) which uses Clifford data to guide
a Richardson-like extrapolation to the many copy limit. We generalize this
method to perform Clifford guided double Richardson-like extrapolation. We
first extrapolate to the limit of many copies and then we extrapolate to the
limit of zero noise floor. The new method, UNIfied Technique for Error
mitigation with Data (UNITED), can be viewed as unification of CDR, ZNE and VD
methods. Furthermore, vnCDR is a special case of the new general method.

We find that when at $N_{\rm tot}=10^{10}$, UNITED is the optimal technique
for $L=Q$ depth circuits of up to 8 qubits by a factor of 1.5 compared to
vnCDR on average. We also find that vnCDR is preferable at budgets $N_{\rm
tot}=10^{6},10^{8}$. For $N_{\rm tot}=10^{5}$, ZNE is optimal for circuits
with depths $L=Q$ while vnCDR is optimal for more noisy $L=16Q$. We observe
that increasing the shot budget improves the error mitigated expectation value
only until a performance ceiling is reached. As we see, different techniques
have different performance ceilings, and we find that unified methods are able
to draw from more data than VD and ZNE to reach better corrections when more
resources are available.

As the available quantum devices continue to grow in qubit number and reduce
the rate of hardware errors, practical applications of quantum computing grow
ever closer. Due to the high number of qubits that are required for full error
correction, it seems likely that error mitigation will play a pivotal part in
those early applications. This work provides a useful reference for those who
are seeking such applications to determine which combination of strategies is
best for their use case.

A natural question one might ask is if there are other forms of data that
would be useful to include in an error mitigation approach. Error mitigation
for deep quantum circuits is challenging, but perhaps some other source of
data (other than those we considered) would extend the depth for which error
mitigation is practical. Further unification with other error mitigation
techniques such as noise-resilient quantum compiling Cincio  _et al._ (2018,
2021b); Murali  _et al._ (2019); Khatri  _et al._ (2019); Sharma  _et al._
(2020), verified phase estimation O’Brien  _et al._ (2021) and specific
approaches leveraging symmetries and/or post-selection McArdle  _et al._
(2019); Bonet-Monroig  _et al._ (2018); Otten  _et al._ (2019); Cai (2021b) is
likely to be a promising research direction.

Another interesting direction for future work is to consider how best to
determine when to measure new expectation values (i.e. more near-Clifford
circuits, more noise levels, more copies of the state, etc.) versus expending
more shots per expectation value in order to get the best mitigation possible
with fixed resources. Finally, we note that investigation of effects of noisy
controlled derangement on VD and UNITED performance will be necessary for full
understanding of these methods’ potential.

##  VII Acknowledgements

This work was supported by the Quantum Science Center (QSC), a National
Quantum Information Science Research Center of the U.S. Department of Energy
(DOE). DB was supported by the U.S. DOE through a quantum computing program
sponsored by the Los Alamos National Laboratory (LANL) Information Science &
Technology Institute. M.H.G was supported by “la Caixa” Foundation (ID
100010434), Grant No. LCF/BQ/DI19/11730056. Piotr C. and AA were supported by
the Laboratory Directed Research and Development (LDRD) program of LANL under
project numbers 20190659PRD4 (Piotr C.) and 20210116DR (AA and Piotr. C). PJC
also acknowledges initial support from the LANL ASC Beyond Moore’s Law
project. LC was also initially supported by the U.S. DOE, Office of Science,
Office of Advanced Scientific Computing Research, under the Quantum Computing
Application Teams program.

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##  Appendix A Correcting for global depolarizing noise with CGVD or UNITED

Here we motivate the UNITED ansatz by considering its performance on
mitigating the action of a global depolarizing noise channel on some
observable of interest. The ansatze used in CDR and vnCDR have been shown to
perfectly correct global depolaraizing noise and this fact is used to motivate
the form of the CGVD ansatz. We note in this setting there is no limit set by
coherent mismatch when using VD, so one expects a combination of VD and CDR to
perform well. First, we consider CGVD since the treatment is the same as in
the UNITED case and can be generalized very simply.

Consider a global depolarizing noise channel that acts on all $Q$ qubits. For
some state $\rho=\ket{\psi}\bra{\psi}$ produced by quantum circuit $U$, this
channel takes the form:

$\rho\longrightarrow(1-p)\rho+p\frac{\mathbb{1}}{d}.$ (21)

Consider the effect of the above channel on some observable of interest $X$

$\displaystyle\hat{X}=(1-p)X^{\rm exact}+p\frac{{\rm Tr}(X)}{d},$ (22)

where $X^{\rm exact}={\rm Tr}{(\rho X)}$. We consider the CGVD ansatz that
combines observables evaluated at several numbers of copies in the VD circuit
structure. For simplicity we omit the small changes in the rate of errors for
different numbers of copies due to the swap noise. We can rewrite (14) as
follows

$\displaystyle\hat{X}_{\rm CGVD}=\sum_{m=1}^{M}a_{m}\hat{X}_{m},$ (23)

where $\hat{X}_{\rm CGVD}$ is the estimate of the observable of interest,
$a_{m}$ are the parameters fitted using least squares regression on the data
generated by the near-Clifford training circuits mitigated using VD with up to
$M$ copies.

We can express an observable mitigated with VD using $m$ copies as

$\hat{X}_{m}=\frac{{\rm Tr}{\big{(}\rho^{m}X}\big{)}}{{\rm
Tr}{\big{(}\rho^{m}}\big{)}}.$ (24)

We can evaluate $\hat{X}_{m}$, assuming $\rho$ has been acted upon by a global
depolarizing channel as defined in (21). First focusing on the denominator,

$\displaystyle{\rm Tr}{\big{(}\rho^{m}\big{)}}$ $\displaystyle={\rm
Tr}{\bigg{(}\sum_{k=0}^{m}\begin{pmatrix}m\\\
k\end{pmatrix}\big{(}(1-p)\rho\big{)}^{(m-k)}\bigg{(}\frac{p}{d}\bigg{)}^{k}\bigg{)}}$
(25) $\displaystyle=\sum_{k=0}^{m}\begin{pmatrix}m\\\
k\end{pmatrix}(1-p)^{(m-k)}{\rm
Tr}{(\rho}^{m-k})\bigg{(}\frac{p}{d}\bigg{)}^{k}\ .$ (26)

Noting that when $m=k$ we are tracing over the identity, we obtain:

$\displaystyle{\rm
Tr}{\big{(}\rho^{m}\big{)}}=\bigg{(}\frac{1+(d-1)(1-p)}{d}\bigg{)}^{m}+(d-1)\bigg{(}\frac{p}{d}\bigg{)}^{m}\
.$ (27)

where $d=2^{Q}$ is the dimension of the Hilbert space. Now evaluating the
numerator of (24) we obtain

$\displaystyle\begin{split}{\rm Tr}{\big{(}\rho^{m}X\big{)}}={\rm Tr}{(\rho
X)}\bigg{(}\frac{1+(d-1)(1-p)}{d}\bigg{)}^{m}\\\
+\bigg{(}\frac{p}{d}\bigg{)}^{m}\bigg{(}{\rm Tr}{(X)}-{\rm Tr}{(\rho
X)}\bigg{)}\ .\end{split}$ (28)

Combining the above we arrive at the following expression,

$\displaystyle X_{m}=f_{m}{\rm Tr}{(\rho X)}+\frac{1-f_{m}}{d}{\rm Tr}{(X)}\
,$ (29)

where

$\displaystyle f_{m}=1-\frac{dp^{m}}{(d-1)p^{m}+(d-p(d-1))^{m}}$ (30)

Taking $m=1$ leads to $f_{1}=1-p$ which gives the expected result. Using (23)
we can determine the criteria for this ansatz to give a perfect mitigation,

$\hat{X}_{\rm CGVD}=\sum_{m=1}^{M}a_{m}\big{(}f_{m}{\rm Tr}{(\rho
X)}+\frac{1-f_{m}}{d}{\rm Tr}{(X)}\big{)}.$ (31)

Therefore, we require

$\sum_{m}^{M}a_{m}f_{m}=1,\quad{\rm Tr}{(X)}=0.$ (32)

A global depolarizing noise channel will have the same effect on the training
circuits as it will on the circuit of interest. As we have access to the exact
observable for these circuits, the coefficients required to lead to a perfect
mitigation can be learned. As such, the ansatz used in CGVD can be used to
perfectly mitigated global depolarizing noise using a near-Clifford training
set.

It is straightforward to extend this analysis to involve the ansatz including
multiple noise levels. Notice that under the action of global depolarizing
noise defined in (21), if this channel acts $j$ times throughout the dynamics
we can write

$\rho_{j}\longrightarrow(1-p^{j})\rho+p^{j}\frac{\mathbb{1}}{d}.$ (33)

Now lets consider UNITED ansatz that combines observables mitigated with VD
suing various numbers of copies at various noise levels. This ansatz can be
written as follows

$\displaystyle\hat{X}_{\rm
UNITED}=\sum_{m=1}^{M}\sum_{j=1}^{n}a_{m,j}\hat{X}_{m,j},$ (34)

where $\hat{X}_{\rm UNITED}$ is the mitigates observable, $a_{m,j}$ are the
parameters fitted using least squares regression on the data generated by the
near-Clifford VD training circuits with up to $M$ copies, evaluated with $n$
different noise levels.

Working through the same analysis with this ansatz we find that for the
effects of global depolarizing noise the be perfectly mitigated we require

$\sum_{m}^{M}\sum_{j}^{n}a_{m,j}f_{m,j}=1,\quad{\rm Tr}{(X)}=0.$ (35)

where

$f_{m,j}=1-\frac{dp^{mj}}{(d-1)p^{mj}+(d-p^{j}(d-1))^{m}}.$ (36)

Therefore, we can conclude that both CGVD and UNITED can perfectly mitigate
the effect of global depolarizing noise on some observable of interest.

Figure 4: Convergence of VD estimates with increasing $M$. In (a) results for
$N_{\rm tot}=10^{5}$ and in (b) results for $N_{\rm tot}=10^{10}$ plotted
versus number of VD copies $M$. For reference purposes we include single copy
noisy results as $M=1$. The plots show that for $g=1$ $M=2$ is optimal while
for $Q=4$, $g=16$ the optimal $M$ increases with increasing $N_{\rm tot}$. The
left plots show the mean values while the right plots show the maximal value.

The assumption ${\rm Tr}{(X)}=0$ is not restrictive due to the common
decomposition of operators into Pauli terms (which have zero trace) and the
identity. However, we can consider an ansatz which includes an additional
constant term to be fitted. In which case (23) becomes

$\displaystyle\hat{X}_{\rm CGVD}=\sum_{m=1}^{M}a_{m}\hat{X}_{m}+b,$ (37)

Thus, (31) becomes

$\hat{X}_{\rm CGVD}=\sum_{m=1}^{M}a_{m}\big{(}f_{m}{\rm Tr}{(\rho
X)}+\frac{1-f_{m}}{d}{\rm Tr}{(X)}\big{)}+b.$ (38)

Therefore, including a constant term in the ansatz, to perfectly mitigate the
effect of depolarizing noise we require

$\sum_{m}^{M}a_{m}f_{m}=1,\quad\sum_{m}^{M}\frac{1-a_{m}}{d}{\rm Tr}{(X)}=b\
.$ (39)

Including a constant term in the UNITED ansatz leads to a similar conclusion.
Therefore, one can see global depolarizing noise can be perfectly mitigated
using near-Clifford circuit to train the parameters of the CGVD and UNITED
ansatze.

##  Appendix B Optimal parameters for virtual distillation

Here we systematically explore convergence of VD estimates with increasing
number of copies $M$. In Figure 4 we plot the mean and maximal errors of VD
$\langle\sigma_{Z}^{1}\rangle$ estimate for $30$ random quantum circuits and
$M=2-6$. In this figure we show results for smallest $N_{\rm tot}=10^{5}$ and
the largest $N_{\rm tot}=10^{10}$ considered here. We find that for $Q=4-10$
and $g=1$ $M=2$ gives typically optimal results for both shot budgets. We note
that for $Q=4-10$, $g=1$ and $N_{\rm tot}=10^{10}$ $M=2-5$ VD give very
similar results indicating that $M=2$ is large enough to reach the noise
floor. Our findings agree with Koczor (2021), which argues that with an
exception of highly noisy circuits $M\leq 3$ is sufficient to reach the noise
floor. For more noisy $Q=4,g=16$ circuits we find that the optimal $M$ depends
strongly on $N_{\rm tot}$. For $N_{\rm tot}=10^{5}$ $M=2$ is optimal but it
gives on average a small factor $1.2$ improvement over the noisy results. For
$N_{\rm tot}=10^{10}$ $M=4$ is optimal giving a factor of $2.5$ improvement.
Based on the obtained results we use $M=2$ in the main text as it is optimal
for most circuits. We note that although with $M=4$ VD performance can be
improved for $Q=4,g=16$ and large $N_{\rm tot}$ this improvement is not large
enough to outperform vnCDR and UNITED.