# Theory of mirror benchmarking and demonstration on a quantum computer

Karl Mayer  Alex Hall  Thomas Gatterman  Si Khadir Halit  Kenny Lee  Justin
Bohnet  Dan Gresh  Aaron Hankin  Kevin Gilmore  Justin Gerber  John Gaebler
Honeywell Quantum Solutions

###### Abstract

A new class of protocols called mirror benchmarking was recently proposed to
measure the system-level performance of quantum computers. These protocols
involve circuits with random sequences of gates followed by mirroring, that
is, inverting each gate in the sequence. We give a simple proof that mirror
benchmarking leads to an exponential decay of the survival probability with
sequence length, under the uniform noise assumption, provided the twirling
group forms a 2-design. The decay rate is determined by a quantity that is a
quadratic function of the error channel, and for certain types of errors is
equal to the unitarity. This result yields a new method for estimating the
coherence of noise. We present data from mirror benchmarking experiments run
on the Honeywell System Model H1. This data constitutes a set of performance
curves, indicating the success probability for random circuits as a function
of qubit number and circuit depth.

††preprint: APS/123-QED

##  I Introduction

Quantum computers are expected to someday solve problems that are intractable
for classical computers, but what are the capabilities of quantum computers
today? How should the performance of quantum computers be measured? Perhaps
surprisingly, there is no agreed upon answer to these questions, though a
number of metrics have been proposed. One such metric, the quantum volume
(QV), is given by the maximum number of qubits for which a class of random
circuits produces a distribution that passes a certain statistical test [1].
Other groups have proposed to use textbook algorithms such as Bernstein-
Vazirani or Grover search, which ideally produce deterministic outputs, and
report the algorithmic success probability [2]. Other proposals, inspired by
randomized benchmarking (RB) [3, 4], employ random Clifford circuits of
varying length, which can be inverted in order to measure the survival
probability [5]. There are advantages and disadvantages to each of these
system-level benchmarks, which have been discussed elsewhere [6].

In this paper, we focus on a protocol referred to as mirror benchmarking, of
which different versions were proposed in [7]. In the variation we consider,
the first half of a circuit consists of a sequence of randomly chosen layers,
and the second half applies the inverse of each layer in reverse order, so
that the final state is ideally equal to the initial state. Additionally,
random Pauli gates can be inserted between layers to remove coherent effects,
similar to the randomized compiling procedure [8]. The output of mirror
benchmarking is a plot of the survival probability as a function of the
sequence length. This can be useful to a quantum computer user as it provides
a direct measure of circuit performance for a given qubit number and circuit
depth.

The authors of [7] provide substantial motivation for why mirror benchmarking
is a useful benchmark. They also advocate for the use of mirroring broadly, to
benchmark any circuit of interest, rather than just random circuits as
described here. Restricting our attention to random circuits, our main
contribution is a formula for how the survival probability decays as a
function of sequence length, in analogy to RB protocols. We show that under
assumptions commonly made in RB research, the survival probability decays
exponentially. Under some additional assumptions on the noise, the decay
parameter is the unitarity of the average error per circuit layer. The
unitarity was introduced in [9] and is a measure of the coherence of noise. A
consequence of this result is that besides being a useful benchmark for
system-level performance, mirror benchmarking provides a new method to
estimate the unitarity of noise, for instance of a two-qubit gate. This
requires fewer circuits than the protocol of [9].

We implement mirror benchmarking on the Honeywell System Model H1 quantum
computer. In our implementation, each layer consists of a random single-qubit
Clifford gate on each qubit, followed by a native two-qubit gate applied to
each qubit pair in a random fully-connected pairing of qubits. We run mirror
benchmarking on $n=6,8,10$ qubits. For the largest circuits run in this
experiment, with $n=10$ and sequence length $L=16$, corresponding to a two-
qubit circuit depth of $32$ and $160$ total two-qubit gates, we obtain an
average survival probability of $0.344(18)$.

The outline of this paper is as follows. In Sec. II we give a simple proof
that the survival probability in mirror benchmarking decays with a rate
determined by the unitarity. In Sec. III we address the question of how
accurately the direct sampling of gates used in our experiment approximates a
true uniform sampling over a 2-design. In Sec. IV we describe our
implementation of mirror benchmarking on the Honeywell System Model H1 quantum
computer and present the results. Finally, Sec. V contains a discussion and
conclusion.

##  II Mirror benchmarking survival probability

In a mirror benchmarking protocol, a sequence of unitaries
$g_{1},\dots,g_{L}$, followed by their inverses $g_{L}^{-1},\dots,g_{1}^{-1}$,
is applied to a noisy initial state, given by density matrix $\rho$. For the
present discussion, the $g_{i}$ may be individual gates, or more complicated
sequences of gates, but we assume that they are sampled uniformly at random
from a group $G$ that forms a unitary 2-design [10]. Let $\mathcal{E}$ be a
constant error channel associated with each unitary in the sequence. We assume
that $\mathcal{E}$ is unital, that is, $\mathcal{E}(I)=I$, but we discuss how
to relax this assumption in the appendix. We also assume that the error
channel per unitary during the inversion half of the circuit is given by
$\mathcal{E}^{\dagger}$, the dual of $\mathcal{E}$, which is defined by
$\langle\langle\mathcal{E}^{\dagger}(M)|N\rangle\rangle=\langle\langle
M|\mathcal{E}(N)\rangle\rangle$, where $\langle\langle
M|N\rangle\rangle:=\Tr(M^{\dagger}N)$. We postpone a discussion of this
assumption to the end of this section.

At the end of the circuit a measurement is performed that is described by a
noisy POVM with elements $\\{E,I-E\\}$. The average survival probability for a
sequence of length $L$ is then given by

$p(L)=\frac{1}{|G|^{L}}\sum_{g_{1},\dots,g_{L}}\langle\bra{E}\mathcal{E}^{\dagger}g_{1}^{-1}\cdots\mathcal{E}^{\dagger}g_{L}^{-1}\mathcal{E}g_{L}\cdots\mathcal{E}g_{1}\ket{\rho}\rangle,$
(1)

where by abuse of notation we use $g$ to denote both a unitary and its
corresponding superoperator. We can absorb the final $\mathcal{E}^{\dagger}$
into the noisy POVM element $E$, and after defining
$g_{i^{\prime}}=g_{i}\cdots g_{2}g_{1}$ and relabling $g_{i^{\prime}}\mapsto
g_{i}$, the survival probability can be written as

$p(L)=\frac{1}{|G|^{L}}\sum_{g_{1},\dots g_{L}\in
G}\langle\bra{E}(g_{1}^{-1}\mathcal{E}^{\dagger}g_{1})\cdots(g_{L-1}^{-1}\mathcal{E}^{\dagger}g_{L-1})\\\
(g_{L}^{-1}\mathcal{E}g_{L})\cdots(g_{1}^{-1}\mathcal{E}g_{1})\rangle\ket{\rho}\rangle.$
(2)

An important quantity is the twirl of $\mathcal{E}$ over $G$, which is defined
as

$\mathcal{E}_{T}=\frac{1}{|G|}\sum_{g\in G}g^{-1}\mathcal{E}g.$ (3)

A useful fact is that $\mathcal{E}_{T}$ commutes with $g$ for all $g\in G$,
which can be shown as follows:

$\displaystyle g\,\mathcal{E}_{T}$ $\displaystyle=\frac{1}{|G|}\sum_{h\in
G}gh^{-1}\mathcal{E}h$ $\displaystyle=\frac{1}{|G|}\sum_{h\in
G}(hg^{-1})^{-1}\mathcal{E}(hg^{-1})g$
$\displaystyle=\frac{1}{|G|}\sum_{h^{\prime}\in
G}(h^{\prime})^{-1}\mathcal{E}h^{\prime}g$ $\displaystyle=\mathcal{E}_{T}g,$
(4)

where in the third line we set $h^{\prime}=hg^{-1}$, and used the group
transitivity property to change the index of summation from $h$ to
$h^{\prime}$.

Summing over $g_{L}$ in Eq. (2), and using Eq. (II), the central terms in the
product in Eq. (2) are then

$(g_{L-1}^{-1}\mathcal{E}^{{\dagger}}g_{L-1})\,\mathcal{E}_{T}\,(g_{L-1}^{-1}\mathcal{E}g_{L-1})=g_{L-1}^{-1}\mathcal{E}^{{\dagger}}\mathcal{E}_{T}\mathcal{E}g_{L-1}.$
(5)

Summing the right hand side of Eq. (5) over $g_{L-1}$,

$\frac{1}{|G|}\sum_{g_{L-1}}g_{L-1}^{-1}\mathcal{E}^{{\dagger}}\mathcal{E}_{T}\mathcal{E}g_{L-1}=(\mathcal{E}^{{\dagger}}\mathcal{E}_{T}\mathcal{E})_{T}.$
(6)

Continuing in this way, the general pattern for the survival probability at
sequence length $L$ can be described recursively as follows. Let
$\mathcal{T}_{l}$ for $l=1,2,\dots$ be the sequence of operators defined by
$\mathcal{T}_{1}=\mathcal{E}_{T}$ and
$\mathcal{T}_{l+1}=(\mathcal{E}^{\dagger}\,\mathcal{T}_{l}\,\mathcal{E})_{T}$.
Then

$p(L)=\langle\bra{E}\mathcal{T}_{L}\ket{\rho}\rangle.$ (7)

We will derive an expression for $\mathcal{T}_{L}$. First we recall some facts
about twirls over 2-designs [10]. Let $\mathcal{B}(\mathcal{H})$ denote the
vector space of linear operators on Hilbert space
$\mathcal{H}=\mathbb{C}^{d}$. Let $V_{1}=\mathrm{span}\\{I\\}$ and
$V_{2}=\\{A\in\mathcal{B}(\mathcal{H}):\Tr(A)=0\\}$, and note that
$\mathcal{B}(\mathcal{H})=V_{1}\bigoplus V_{2}$. Let
$\mathcal{M}:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{H})$ and suppose
that $G$ is a 2-design. Then the twirl of $\mathcal{M}$ over $G$ is a linear
combination of two projectors:

$\mathcal{M}_{T}=a\Pi_{1}+b\Pi_{2},$ (8)

where $\Pi_{1}$ and $\Pi_{2}$ are the projectors onto $V_{1}$ and $V_{2}$,
respectively (A proof of this fact that does not rely on representation theory
is found in [11]). If $\mathcal{M}$ is trace-preserving (TP), then
$\mathcal{M}_{T}$ is also TP, and therefore

$\Tr(I)=\Tr(\mathcal{M}_{T}(I))=a\Tr(\Pi_{1}I)=a\Tr(I),$ (9)

which implies that $a=1$. Let $D=\dim(V_{2})=d^{2}-1$. The quantity $b$ is
given by

$\displaystyle b$ $\displaystyle=\frac{1}{D}\Tr(\Pi_{2}\mathcal{M}_{T})$
$\displaystyle=\frac{1}{D}\frac{1}{|G|}\sum_{g\in
G}\Tr(g\Pi_{2}g^{-1}\mathcal{M})$
$\displaystyle=\frac{1}{D}\Tr(\Pi_{2}\mathcal{M}),$ (10)

where in the last equality we used the fact that $\Pi_{2}$ commutes with the
action of $G$. Given an error channel $\mathcal{E}$, define

$f(\mathcal{E})=\frac{1}{D}\Tr(\Pi_{2}\mathcal{E}).$ (11)

The quantity $f(\mathcal{E})$ is related to $F(\mathcal{E})$, the process
fidelity (also called entanglement fidelity) with respect to the identity
[11], according to

$1-f(\mathcal{E})=\frac{d^{2}}{D}\big{(}1-F(\mathcal{E})\big{)}.$ (12)

We will simply write $f$ when the channel $\mathcal{E}$ is clear from context.
By Eqs. (7)-(II), we have

$\mathcal{E}_{T}=\Pi_{1}+f\Pi_{2}.$ (13)

The unitarity of $\mathcal{E}$, which was introduced in [9], is given by

$u=\frac{1}{D}\Tr(\Pi_{2}\mathcal{E}^{\dagger}\Pi_{2}\mathcal{E}).$ (14)

We are now ready to prove a formula for the survival probability.

######  Lemma 1.

Let $\mathcal{T}_{l}$ for $l=1,2,\dots$ be a sequence of operators defined by
$\mathcal{T}_{1}=\mathcal{E}_{T}$, and
$\mathcal{T}_{l+1}=(\mathcal{E}^{\dagger}\mathcal{T}_{l}\mathcal{E})_{T}$.
Then for all $l$,

$\mathcal{T}_{l}=\Pi_{1}+fu^{l-1}\Pi_{2}.$

###### Proof.

We proceed by induction. The base case is given by Eq. (13). Suppose the
statement is true for a particular $l$. Then

$\displaystyle\mathcal{T}_{l+1}$
$\displaystyle=(\mathcal{E}^{\dagger}(\Pi_{1}+fu^{l-1}\Pi_{2})\mathcal{E})_{T}$
$\displaystyle=\Pi_{1}+\frac{1}{D}\Tr\big(\Pi_{2}(\mathcal{E}^{\dagger}(\Pi_{1}+fu^{l-1}\Pi_{2})\mathcal{E})\big{missing})\Pi_{2},$
(15)

where in the second equality we used Eqs. (8) and (II) with
$\mathcal{M}=\mathcal{E}^{\dagger}(\Pi_{1}+fu^{l-1}\Pi_{2})\mathcal{E}$. Since
$\mathcal{E}$ is trace preserving, $\Pi_{1}\mathcal{E}\Pi_{2}=0$, from which
it follows that

$\Tr(\Pi_{2}\mathcal{E}^{\dagger}\Pi_{1}\mathcal{E})=\Tr(\mathcal{E}^{\dagger}\Pi_{1}\mathcal{E}\Pi_{2})=0.$
(16)

Therefore,

$\displaystyle\mathcal{T}_{l+1}$
$\displaystyle=\Pi_{1}+\frac{1}{D}fu^{l-1}\Tr(\Pi_{2}\mathcal{E}^{\dagger}\Pi_{2}\mathcal{E})\Pi_{2}$
$\displaystyle=\Pi_{1}+fu^{l}\Pi_{2}.$ (17)

∎

######  Theorem 1.

The survival probability for mirror benchmarking is given by

$p(L)=Afu^{L-1}+B,$ (18)

for constants $A$ and $B$ that depend only on the state prep and measurement.

###### Proof.

This follows directly from Lemma 1 and Eq. (7) with
$A=\langle\bra{E}\Pi_{2}\ket{\rho}\rangle$ and
$B=\langle\bra{E}\Pi_{1}\ket{\rho}\rangle$. ∎

A few comments are in order. The assumption that the error channel during the
inversion half of the circuit is $\mathcal{E}^{{\dagger}}$ implies that a
unitary error $U$ on $g_{i}$ will correspond to error $U^{-1}$ on
$g_{i}^{-1}$. This may not always hold but often can be enforced by a suitable
compilation procedure. For example, if $g_{i}=e^{-iZZ\theta}$ is subject to a
control error in $\theta$, then $g_{i}^{-1}$ may be implemented by
$Pe^{-iZZ\theta}P$, for any Pauli $P$ that anticommutes with $ZZ$. If we relax
this assumption, and allow for an arbitrary error channel $\mathcal{E}_{inv}$
on the inverse gates, then the unitarity $u$ in Th. 1 is replaced with
$\frac{1}{D}\Tr(\Pi_{2}\mathcal{E}_{inv}\Pi_{2}\mathcal{E})$.

In the limit of perfectly coherent errors, that is $u=1$, the survival
probability does not decay at all, since the coherent errors during the second
half of the circuit under these assumptions exactly cancel those from the
first half. At the other extreme, when the error is a depolarizing channel,
$\mathcal{E}=\Pi_{1}+f\Pi_{2}$ and therefore $u=f^{2}$ and $p(L)=Af^{2L-1}+B$.
It is only in this special case that the decay parameter also extracts the
fidelity of $\mathcal{E}$. If $\mathcal{E}$ is a stochastic Pauli channel,
which we assume in our experimental implementation by the use of Pauli
randomization in the protocol, then $\mathcal{E}=\mathcal{E}^{{\dagger}}$ is
diagonal in the Pauli basis of the superoperator representation. In this case,
the unitarity is no longer a measure of coherent errors, but is still defined
by Eq. (14). The process fidelity then is bounded in terms of the unitarity
according to

$\frac{1}{d^{2}}\big{(}1+D\,u\big{)}\leq
F(\mathcal{E})\leq\frac{1}{d^{2}}\big{(}1+D\sqrt{u}\big{)}.$ (19)

A derivation of this bound is given in the appendix.

##  III Direct circuit sampling

Theorem 1 of the previous section assumed that the group $G$ is a 2-design.
However, in the experiment described in the next section, each layer unitary
$g_{i}$ is sampled from a generating set for the $n$-qubit Clifford group
$C_{n}$, rather than from the full group. This is done to avoid the need to
compile a random Clifford group element, and so that the sequence lengths of
the decay curve more directly correspond to two-qubit gate depth. Thus, our
implementation of mirror benchmarking is analogous to the direct benchmarking
protocol of [5]. Here, we justify the claim that Th. 1 should still
approximately apply.

The problem of RB using a generating set for the Clifford group was studied in
[12] as a Markov process on the Cayley graph of the Clifford group. The $i$-th
element of the sequence, $g_{i}$, approximately samples from the group if an
initial distribution over the generating set diffuses to an approximately
uniform distribution over the full graph. We take a different approach, and
simulate the distribution of unitaries $U^{(L)}=g_{L}\cdots g_{2}g_{1}$
generated by sequences of length $L$, with the gate set used in our experiment
and described in Sec. IV. A set of unitaries $\\{U_{i}\\}_{i}$ is a $t$-design
if and only if

$\Phi_{t}:=\mathbb{E}_{i}\absolutevalue{\mathrm{Tr}(U_{i})}^{2t}=t!\,,$ (20)

where the quantity on the left hand side is called the $t$-th frame potential
[13]. We numerically estimate the $2^{\mathrm{nd}}$ frame potential $\Phi_{2}$
for the set $\\{U^{(L)}_{i}\\}_{i}$ generated by mirror benchmarking circuits
of sequence length $L=2,4,\dots,16$ and qubit numbers $n=4,6,8$. The results
are shown in Fig. 1. For each data point, we generated 10,000 random circuits
and computed the quantity in Eq. (20), using the standard error for the error
bars.

The plot shows convergence to the 2-design value, with faster convergence for
smaller qubit number. For short sequence lengths, typically $L<n$, the gate
set for direct mirror benchmarking circuits fails to approximate a 2-design,
and therefore fitting $p(L)$ to a single exponential decay should be
understood as a heuristic benchmark, rather than an unbiased estimate of the
unitarity. An interesting and probably challenging open problem would be to
bound the error in $p(L)$ given by Th. 1 as a function of the deviation from
2-design as measured by $\Phi_{2}$.

Figure 1: Plot of $\Phi_{2}$ defined by Eq. (20) versus sequence length $L$
for $n=4,6,8$. For a $2$-design, $\Phi_{2}=2$. Each data point is estimated as
the average over 10,000 random unitaries generated by the direct gate set
described in Sec. IV.

##  IV Experiment

We implement mirror benchmarking on the Honeywell System Model H1 quantum
computer. This system contains 10 trapped-ion qubits in a linear QCCD
architecture described in [14, 15]. The native two-qubit gate is the phase-
insensitive Mølmer-Sørenson gate [16, 17], given by

$U_{zz}=e^{-iZZ\pi/4}.$ (21)

A circuit diagram of our implementation is shown in Fig. 2. Each $g_{i}$ is a
circuit layer with a random single-qubit Clifford gate applied to each qubit
followed by a native two-qubit gate applied to each qubit pair in a random
fully-connected pairing of qubits. Finally, we include Pauli randomization to
prevent coherent errors from systematically producing higher survival
probabilities. Random Pauli gates that multiply to the identity are inserted
between each layer. The second Pauli gate in each product is pushed through
the subsequent layer using the commutation relations and combined with the
next single-qubit gate.

Figure 2:  Mirror benchmarking circuit diagram. a) General form for mirror
benchmarking with $L$ layers, shown for $n=6$. b) The $i$-th layer unitary
used in our implmentation. Random single-qubit Clifford gates $C_{i,j}$ are
applied to qubit $j$. A random permutation $\sigma_{i}\in S_{n}$ is applied,
and the native $U_{ZZ}$ gate is then applied to each qubit pair. Random Paulis
$P_{i,j}$ are inserted before and pushed through the $U_{ZZ}$ gates to leave
the circuit unitary unchanged. The Paulis $P^{\prime}_{i,j}$ are combined with
the subsequent round of Cliffords before execution on actual hardware.

The results of our experiment are shown in Fig. 3. The figure shows the
average $p(L)$ versus $L=4,8,12,16$ and with qubit number $n=6,8,10$. We only
use an even number of qubits, since increasing to odd $n+1$ does not increase
the number of two-qubit gates in our circuits, which are the leading source of
circuit error. For each sequence length, we sample 10 circuits, and repeat
each circuit for 100 shots. We insert random Pauli gates before the
measurement, so that the constant $B$ can be fixed to $1/2^{n}$, as described
in [18]. The error bars are computed by a bootstrapping procedure adapted from
[12]. For each sequence length, the random circuits are resampled with
replacement and then a parametric bootstrap is performed on the observed
probability distribution for each circuit. The unitarity estimate at $n=10$ is
$u=0.938(4)$, corresponding to process fidelity bounds of $0.938(4)\leq
F(\mathcal{E})\leq 0.969(2)$ per random fully-connected layer with $5$ two-
qubit gates.

$n$ | $6$ | $8$ | $10$
---|---|---|---
Unitarity | $0.962(3)$ | $0.949(5)$ | $0.938(4)$
$F_{lower}$ | $0.962(3)$ | $0.949(5)$ | $0.938(4)$
$F_{upper}$ | $0.981(2)$ | $0.974(3)$ | $0.969(2)$

Figure 3: Mirror benchmarking data plotting $p(L)$ versus $L$ for qubit
numbers $n=6,8,10$, with 10 circuits per sequence length, and 100 shots per
circuit. The data was taken on the Honeywell System Model H1. The table gives
the unitarity estimate obtained by best-fitting to $p(L)=Au^{L-1}+1/2^{n}$, as
well as the lower and upper bounds on the process fidelity given by Eq. (19).

To study the accuracy and precision of the unitarity estimate, we simulate 100
different mirror benchmarking experiments for $n=6,8,10$ with depolarizing
error on the two-qubit gates. For each simulated experiment, the depolarizing
parameter was chosen uniformly at random between $0.0$ and $1.0\times
10^{-2}$. Figure 4 shows a scatter plot of the estimated unitarity versus
depolarizing parameter, with the true unitarity plotted for comparison. The
true unitarity for circuit layers consisting of $n/2$ two-qubit gates with
uniform depolarizing error is given by a formula that we derive in the
appendix. For a single point, the estimation error is the difference between
the estimated and true unitarities. The mean of the estimation errors of the
simulated data points is less than $2\times 10^{-4}$, and the standard
deviations are $1.5,2.2,2.6\times 10^{-3}$ for $n=6,8,10$, respectively. This
indicates that for this error model, mirror benchmarking gives an unbiased
estimate of the unitarity, to within the numerical precision of the
simulation.

Unitarity Estimation Error  $n$ $6$ $8$ $10$ Mean $1.2\times 10^{-4}$
$1.2\times 10^{-5}$ $1.8\times 10^{-4}$ Std. $1.5\times 10^{-3}$ $2.2\times
10^{-3}$ $2.6\times 10^{-3}$

Figure 4:  Scatter plot of unitarity estimates from 100 simulated mirror
benchmarking experiments versus two-qubit depolarizing parameter. The solid
lines plot the true unitarity. Each experiment used sequence lengths
$L=4,8,12,16$ and $10$ circuits per sequence length.

##  V Discussion

We have shown that for mirror benchmarking with random circuits, under some
assumptions the survival probability decays exponentially according to the
unitarity of the average error channel per layer. This justifies the use of
mirror benchmarking as a system-level test of quantum computer performance and
leads to a number of applications. It is well known that coherent errors can
cause large quantum circuits to perform worse than would be predicted from the
fidelities of single and two-qubits gates alone [19]. A mirror benchmarking
experiment could be performed with and without the use of randomized
compiling. If the noise is partially coherent, then the survival probability
and extracted unitarity will be higher without randomized compiling,
indicating that randomized compiling should be used for real quantum circuits
of interest. Apart from providing a system-level benchmark, mirror
benchmarking also reduces to a protocol for estimating the unitarity of gate
errors, when performed on just one or two qubits. This is an alternative to
the protocol in [9], which relies on repeating circuits over multiple
measurement settings. We leave an exploration of the ability for mirror
benchmarking to diagnose coherent errors to a future work.

We conclude by pointing out some limitations of our present work and open
problems. First, we proved our main result under the uniform noise assumption,
that is, that $\mathcal{E}$ is the same error channel for every layer $g_{i}$.
This assumption is unrealistic but is commonly made in the research on RB, and
often works well in practice. We speculate that this assumption can be
relaxed, as was shown for RB in [20]. Also, we do not present a statistical
analysis for the unitarity estimate (i.e. a confidence interval as a function
of sample and circuit complexity) in this work. We believe that methods from
[21] and [18] can be adapted to our setting, but this is an open problem.
Finally, we remark that the layer unitaries in mirror benchmarking in general
can be chosen to be whatever one wants. In particular, one could choose Haar
random elements of $SU(4)$ for each qubit pair to define a mirrored version of
the quantum volume (QV) test [1]. This would solve the problem that QV
requires classical simulation and is therefore fundamentally unscalable. We
wonder what the relationship is between the survival probability and the heavy
outcome probability in ensembles of mirror benchmarking and QV circuits,
respectively, and whether mirror benchmarking can be used to extrapolate QV
beyond the regime of classical simulatability.

###### Acknowledgements.

We acknowledge Timothy Proctor for a helpful discussion. We thank Charlie
Baldwin and David Hayes and for comments and suggestions. We thank the entire
experimental team at Honeywell Quantum Solutions for making this work
possible.

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##  Appendix A Non-unital error channels

In Sec. II, we assumed that $\mathcal{E}$ is unital, but we can relax this
assumption and recover the result of Th. 1 as follows. As a CPTP map,
$\mathcal{E}$ can be written as
$\mathcal{E}=\Pi_{1}+\mathcal{E}_{n}+\mathcal{E}_{u}$, where
$\mathcal{E}_{n}=\Pi_{2}\mathcal{E}\Pi_{1}$ and
$\mathcal{E}_{u}=\Pi_{2}\mathcal{E}\Pi_{2}$ are the non-unital and unital
parts of $\mathcal{E}$, respectively. If $\mathcal{E}$ has a non-unital part,
then $\mathcal{E}^{\dagger}$ is not TP. In this case, we make the assumption
that the error channel following each inverse layer $g_{i}^{-1}$ is given by
$\mathcal{E}^{\prime}:=\Pi_{1}+\mathcal{E}_{n}+\mathcal{E}_{u}^{\dagger}$.
That is, the error channel has the same non-unital part but has a unital part
that is dual to the unital part of $\mathcal{E}$. The motivation for this
assumption is that we expect control errors on gates (such as over-rotations)
to be inverse to the control errors on the corresponding inverse gates. Non-
unital errors (such as amplitude damping channels), however, are expected to
be constant between the front half and back half of the mirrored circuit. By
replacing $\mathcal{E}^{{\dagger}}$ with $\mathcal{E}^{\prime}$ in Sec. II we
recover Th. 1.

##  Appendix B Bounds on process fidelity from unitary

Let $\mathcal{E}_{ij}=\frac{1}{d}\Tr(P_{i}\mathcal{E}(P_{j}))$ be the matrix
elements of the error channel in the Pauli basis of the superoperator
representation, with the convention that $P_{0}=I$. The unitarity is given by

$u=\frac{1}{D}\sum_{i,j>0}\mathcal{E}_{ij}^{2}\geq\frac{1}{D}\sum_{i>0}\mathcal{E}_{ii}^{2}.$
(22)

The latter inequality is saturated when $\mathcal{E}$ is a stochastic Pauli
channel. Meanwhile, the process fidelity is

$F(\mathcal{E})=\frac{1}{d^{2}}\big{(}1+\sum_{i>0}\mathcal{E}_{ii}\big{)}.$
(23)

From the previous two equations, upper and lower bounds on $F(\mathcal{E})$
for stochastic Pauli channels can be obtained. On one hand, since each
$\mathcal{E}_{ii}\in[0,1]$,

$\sum_{i>0}\mathcal{E}_{ii}^{2}\leq\sum_{i>0}\mathcal{E}_{ii}.$ (24)

On the other hand, minimizing $\sum_{i>0}\mathcal{E}_{ii}^{2}$ under the
constraint that $\sum_{i>0}\mathcal{E}_{ii}=c$ gives $\mathcal{E}_{ii}=c/D$
for all $i$, at which point $\sum_{i>0}\mathcal{E}_{ii}^{2}=c^{2}/D$.
Therefore,

$\frac{1}{D}\big{(}\sum_{i>0}\mathcal{E}_{ii}\big{)}^{2}\leq\sum_{i>0}\mathcal{E}_{ii}^{2}.$
(25)

Using these two inequalities and substituting the expressions for $u$ and
$F(\mathcal{E})$, after a little algebra we obtain

$\frac{1}{d^{2}}\big{(}1+D\,u\big{)}\leq
F(\mathcal{E})\leq\frac{1}{d^{2}}\big{(}1+D\sqrt{u}\big{)}.$ (26)

##  Appendix C Unitarity of tensor-product of depolarizing channels

We derive a formula for the unitarity of $n/2$ parallel two-qubit depolarizing
channels, with $n$ even. For a system of dimension $d$, a depolarizing channel
with parameter $p$ acts on a linear operator $X$ as

$\mathcal{D}(X)=(1-p)X+p\Tr(X)I/d.$ (27)

Now fix $d=4$, corresponding to a two-qubit system, and let $N=n/2$. We are
interested in the tensor product channel $\mathcal{D}^{\otimes N}$. For
$n$-qubit Pauli operator $P=P_{1}P_{2}\cdots P_{N}$, where $P_{i}$ is a two-
qubit Pauli operator acting on the $i$-th qubit pair, we have

$\displaystyle\mathcal{D}^{\otimes N}(P_{1}\cdots P_{N})$
$\displaystyle=\bigotimes_{i=1}^{N}\bigg{(}(1-p)P_{i}+\frac{p}{4}\Tr(P_{i})I\bigg{)}$
$\displaystyle=\Big{(}(1-p)^{N}+(1-p)^{N-1}p\sum_{i}\delta_{P_{i},I}+(1-p)^{N-2}p^{2}\sum_{i<j}\delta_{P_{i},I}\delta_{P_{j},I}+\dots\Big{)}P_{1}\cdots
P_{N}$
$\displaystyle=\Big{(}\sum_{j=0}^{N-w(P)}\binom{N-w(P)}{j}(1-p)^{N-j}p^{j}\Big{)}P_{1}\cdots
P_{N},$ (28)

where in the second line we used $\Tr(P_{i})=4\delta_{P_{i},I}$, and where
$w(P)=|\\{i:P_{i}\neq I\\}|$ is the weight of $P$ with respect to the $N$ two-
qubit subsystems. The unitarity is given by

$u(\mathcal{D}^{\otimes N})=\frac{1}{(4^{N}-1)4^{N}}\sum_{P\neq
I}\Tr\big(P\mathcal{D}^{\otimes N}(P)\big{missing})^{2}.$ (29)

We compute the sum over $P$ by first summing over the weights of $P$. For a
given $w$, there are $15^{w}\binom{N}{w}$ Paulis of weight $w$, since there
are $\binom{N}{w}$ choices of $w$ subsystems, and $(4^{2}-1)=15$ different
Paulis per subsystem satisfying $P_{i}\neq I$. Therefore,

$u(\mathcal{D}^{\otimes
N})=\frac{1}{4^{N}-1}\sum_{w=1}^{N}15^{w}\binom{N}{w}\Bigg{(}\sum_{j=0}^{N-w}\binom{N-w}{j}(1-p)^{N-j}p^{j}\Bigg{)}^{2}.$
(30)

This is the equation for the true unitarity that is plotted in Fig. 4.