# Restricted Randomized Benchmarking with Universal Gates of Fixed Sequence
Length

Mohsen Mehrani1, Kasra Masoudi2, Rawad Mezher3, Elham Kashefi4,5, and Debasis
Sadhukhan5 1 Institute for Quantum Science and Technology, University of
Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada  2
Department of Mathematics, Simon Fraser University, 8888 University Drive,
Burnaby, BC, V5A 1S6, Canada  3 Quandela SAS, 7 Rue Léonard de Vinci, Massy
91300, France  4 Laboratoire d’Informatique de Paris 6, CNRS, Sorbonne
Université, 4 Place Jussieu, Paris 75005, France  5 School of Informatics,
University of Edinburgh, 10 Crichton Street, Edinburgh EH8 9AB, United Kingdom

###### Abstract

The standard randomized benchmarking protocol requires access to often complex
operations that are not always directly accessible. Compiler optimization does
not always ensure equal sequence length of the directly accessible universal
gates for each random operation. We introduce a version of the RB protocol
that creates Haar-randomness using a directly accessible universal gate set of
equal sequence length rather than relying upon a t-design or even an
approximate one. This makes our protocol highly resource efficient and
practical for small qubit numbers. We exemplify our protocol for creating
Haar-randomness in the case of single and two qubits. Benchmarking our result
with the standard RB protocol, allows us to calculate the overestimation of
the average gate fidelity as compared to the standard technique. We augment
our findings with a noise analysis which demonstrates that our method could be
an effective tool for building accurate models of experimental noise.

##  I Introduction

Scalable quantum hardware has witnessed formidable progress over the last
decade led by efforts from academia and major industries [1]. While quantum
processors today are large enough to contain a few hundred qubits, their
usefulness to showcase quantum advantage [2] is severely challenged by its
sensitiveness to environmental noise and imperfect operations [3, 4]. The
recent advancement in quantum technology made quantum computers accessible via
the internet-cloud which allows users to use quantum computers in a hardware-
agnostic way. However, since current hardware suffers from a diverse array of
errors, benchmarking these devices is crucial to understand the efficiency of
quantum processors.

While in the early stages of quantum processors the total number of available
controllable qubits used to be the advertised number to showcase the strength
of a quantum device, it was soon realized that the noise in the device should
also be incorporated within the reported single number metric. Hardware
connectivity should also be an important factor because in hardware with
limited connectivity, one would need extra swap gates to implement two-qubit
gates on distant qubits. In pursuit of such a single metric, concepts like
quantum volume [5, 6, 7] emerged where together with the total size of the
circuit, the depth of the circuits, that can be executed with noise
resilience, was also considered. Such metric however cannot ensure error
resilience of a particular application. This inspired various works [8, 9]
towards better application-oriented performance benchmarks and metrics such as
the algorithmic qubit [10] that uses an algorithm-agnostic approach.

While these kinds of summary metrics are appealing to non-specialists for
portraying the holistic performance of a quantum machine, they fail to capture
the full complexity of errors in the quantum hardware. This is where
component-level specifications come in. Quantum computers suffer from a broad
spectrum of errors, such as state preparation and measurement (SPAM) errors,
gate operation errors, cross-talks, errors due to loss of energy (T1 time) or
phase (T2 time) coherence, etc. Benchmarking errors at such a low level is
important to specialists because it allows them to isolate their sources which
in turn can be comprehended to enhance the device’s performance. Moreover,
such low-level benchmarks must be hardware-agnostic, otherwise understanding
how they interfere and contribute to the single summary metric will become
problematic [11, 12]. Although errors corresponding to single- or two-qubits
can be analyzed in detail using tomographic techniques [13, 14], multiqubit
circuits often face more intricate issues, such as crosstalk [15, 16, 17, 18,
11, 19], for which these approaches are either inadequate or become
prohibitively costly due to exponential increases in sampling complexity.

A well-established method for assessing the average quality of operations in
quantum hardware that is independent of SPAM errors is randomized benchmarking
(RB) [20]. It provides a reliable estimate of the magnitude of the average
error of a random operation robustly in any digital quantum device. In its
original manifestation [20], one is required to access all possible Haar-
random unitary operations that need to be executed over multiple sequence
lengths to amplify small errors in the gate set. The digital quantum computers
that we have today are all universal in the sense that all possible operations
are accessible via some standard universal gate set given that we are allowed
to apply the standard gates as many times as required. The Solovay-Kitaev
theorem [21, 22, 23], when generalized for $n$ qubits, provides an estimate:
an arbitrary $n$ qubit quantum operation $\mathcal{U}(2^{n})$, when decomposed
into single- and two-qubit gates of an universal gateset, would need an
exponentially large ${O}(n^{2}2^{2n}\log(n^{2}2^{2n}/\epsilon))$ number of
such gates to mimic $\mathcal{U}(2^{n})$ upto an accuracy $\epsilon$. Such
exponentially long gate sequences for each Haar-random unitaries within a RB
protocol poses a significant challenge to the near-term quantum hardware as
they can only execute shallow-depth circuits with noise resilience.

However, people soon came up with RB protocols using finite groups [24, 25,
26, 27, 28, 29, 30, 31] such as Clifford groups [24, 32, 33] that can mimic
Haar-random operations. RB based on unitary 2-design generates gate sequences
of different lengths, and at the end, another inverse circuit is applied to
nullify all previous operations. By measuring in the computational basis, the
fidelity between the measured output and the ideal outcome can be computed.
This procedure is repeated for gate sequences of multiple lengths. By plotting
the fidelity against the sequence length, one can extract information about
the average error rate of the quantum device.

In the standard RB, the Haar-random operations should be directly accessible.
However, this is not quite practical. The current quantum devices rely on a
compiler to obtain universal gate decompositions which are directly accessible
to the device. For typical Haar-random operations $\\{U\\}$ , the compiler
optimization does not ensure equal sequence lengths of the universal gate
decompositions. While for a typical $U_{1}$, the compiler could find a shallow
circuit $g_{1}$, it may have to use a comparatively deep circuit $g_{2}$ for
another operation $U_{2}$ (see Fig. 1). Such disparities often result in an
overestimation of the average error rate of the device if one sticks to the
standard RB technique. In this work, we devised a new version of the protocol,
which we call Restricted Randomized Benchmarking where for each Haar-random
operation we use universal gate decomposition of equal sequence length.

In this work, we try to bridge the gap between the theory of RB and its
experimental realization. We explicitly illustrate our results in the case of
single- and two-qubit scenarios. Assuming the directly accessible standard
universal gateset to be
$\\{R_{X}(\pm\frac{\pi}{2},\pm\pi),R_{Z}(\theta),$C-NOT$\\}$ which is inspired
from the native gate-sets of the superconducting hardware, we set out a
protocol to generate Haar-random unitaries using direcly accessible gates of
fixed sequence length. For single qubits, any random operations can be
expressed using 4 sequences of universal gates while for the two-qubits
operations, results from Ref. [34, 35] guarantee that only three C-NOTs
together with a few single qubit rotations are enough. Using these results, we
construct our single- and two-qubit RB gate sequences of equal length for all
applied Haar-random operations. This in turn allows us to calculate the
overestimation of the average gate fidelity as compared to the standard
techniques. We also benchmark our results with the RB protocols based on
2-design which shows a close match. For small qubit numbers, where the
resources required to generate Haar randomness are still manageable, we argue
that the protocol introduced here will lead to a resource-friendly and
compiler-aware technique for performing RB on existing quantum hardware.

The results obtained allow us to examine the noise characteristics inherent in
the experimental hardware. The standard RB, in its original zeroth-order
version, assumes a gate-independent noise in the hardware to establish an
exponential decay of the average fidelity. However, in practice, the
experimental hardware might contain gate-dependent noise, and therefore the
exponential decay is not guaranteed. We supplement our findings through a
noise analysis of our restricted RB method, demostrating with a typical
example that the exponential decay we observe could be accounted for if the
prevailing noise in the hardware is depolarizing. Although this explanation
may not be exclusive, we believe our method could be an effective tool for
building efficient and quantitatively accurate models of experimental noise in
quantum hardware. An improved noise model is anticipated not only to pave the
way for efficient error detection and mitigation strategies but also to aid in
achieving the long-term goal of quantum error correction in quantum computing.

Figure 1: (upper panel) Randomized benchmarking protocol requires one to
implement any Haar-random operations $U$ from the $n$-qubit operator space
$\mathcal{SU}(2^{n})$. In any digital quantum computer, such operations are
accessible via the universal standard single and two-qubit gates set by the
manufacturer. A quantum device comes with a compiler that decomposes the
$n$-qubit operation $U$ into a sequence of single and two-qubit gates that are
natively accessible in the device. The compiler optimization, however, does
not ensure equal sequence length for all the operations. (lower panel) For a
typical RB step, it implements a Haar-random operations using gateset $g_{1}$
of shallow depth whereas for the next, it uses a much deeper circuit of
gateset $g_{2}$.

##  II Randomised Benchmarking: Preliminaries

An RB protocol typically requires three steps. In the first step, the quantum
register is initialized with a simple quantum state such as
$\rho=|\psi\rangle\langle\psi|$. In the second step, a sequence of random
unitary quantum operations $\\{U_{i}\\}_{i=1,2,\ldots,m}$ of varying length
$m$ is applied which is then finished by an inverse operation
$U_{m+1}^{\text{inv}}$ to nullify all previous random operations. Because of
the presence of the inverse operation at the end, the entire chain of
operations of length $m+1$ turns out to be identity in the noiseless scenario
and therefore we should measure the initial state with unit fidelity. In the
presence of noise, however, the applied gate sequences are not exact. The
noise acting on the gates corrupts their operations which in turn reduce the
final fidelity to something less than one. We repeat this procedure for
different values of $m$ and record the fidelity. In the final step, we extract
the average error rate by fitting the recorded data against sequence length
$m$ which represents the error rate of implementing gate operations and is
free from other sources of errors such as SPAM errors that do not depend on
the length of the sequence.

###  II.1 Haar-Twirl

Let us first consider the scenario for $m=1$. Suppose $U$ represents any Haar-
random quantum operation within the n-qubit unitary space
$U\in\mathcal{U}(2^{n})$, with
${\hat{\bf\Lambda}_{U}}\equiv{\Lambda}_{U}\circ{U}$ describing the noisy
experimental realization of $U$. In general, ${\Lambda}_{U}$ might represent
any completely positive and trace-preserving (CPTP) superoperator that
characterizes the noise associated with the experimental execution of the gate
operation $U$ and can be described by ${\Lambda}_{U}(\rho)=\sum_{k}A_{k}\rho
A_{k}^{\dagger}$ where $A_{k}$ are the Kraus operators. We aim to determine
how different the actual experimental implementation $\hat{\bf\Lambda}_{U}$ is
from the ideal operation $U$. To calculate the same, one can look at the
average gate fidelity,

$\displaystyle F(U,\hat{\bf\Lambda}_{U})=\int d\rho F(U\rho
U^{\dagger},\hat{\bf\Lambda}_{U}(\rho)),$ (1)

where the fidelity is given by
$F(\sigma,\tau)=\Tr[\sqrt{\sigma^{1/2}\tau\sigma^{1/2}}]^{2}$, $\sigma$ and
$\tau$ are density matrices. For pure states, this reduces to
$F({U},\hat{\bf\Lambda}_{U})=\int
d\psi\langle\psi|U^{\dagger}\hat{\bf\Lambda}_{U}(|\psi\rangle\langle\psi|)U|\psi\rangle$
where the average is over all pure states.

The average fidelity of hardware is then calculated by averaging over all the
unitaries, given by

$\displaystyle\bar{F}=\int_{U\in\mathcal{U}(2^{n})}d\mu(U)F({U},\hat{\bf\Lambda}_{U})$
(2)

where the integral is over the Haar measure $d\mu$ [36] on unitary space
$\mathcal{U}(2^{n})$. The average error rate of the device is simply
$1-\bar{F}$.

One can now think of a motion reversal transformation to see the equivalence
$F({U},\hat{\bf\Lambda}_{U})=F(\mathbb{I},{\Lambda}_{U^{\dagger}U})$ where
${\Lambda}_{U^{\dagger}U}$ is the noise channel associated with the motion
reversal experiment
$\hat{\bf\Lambda}_{UU^{\dagger}}\equiv{\Lambda}_{UU^{\dagger}}\circ{UU^{\dagger}}$.
Nielsen [37] pointed out that fidelity of a superoperator $\Lambda$ with
identity is same as the fidelity of its Haar twirl with identity
$F(\int_{U}d\mu(U)U^{\dagger}\Lambda U,\mathbb{I})=F({\Lambda},\mathbb{I})$
which lead us to

$\displaystyle
F({U},\hat{\bf\Lambda}_{U})=F(\mathbb{I},{\Lambda}_{U^{\dagger}U})=F(\mathbb{I},{\Lambda}_{av})$
(3)

where
${\Lambda}_{av}=\int_{U^{\prime}}d\mu(U^{\prime}){U^{\prime}}^{\dagger}\circ{\Lambda}_{U^{\dagger}U}\circ{U^{\prime}}$
is the Haar-twirl of the superoperator ${\Lambda}_{U^{\dagger}U}$. Emerson et
al. [38] came up with this formalism and showed that ${\Lambda}_{av}$ is
unitary invariant i.e.
${\Lambda}_{av}={U}\circ{\Lambda}_{av}\circ{U}^{\dagger}\leavevmode\nobreak\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ \leavevmode\nobreak\
U\in\mathcal{U}(2^{n})$ and have two irreducible representations. Applying
Schur’s lemma on each of them gives the form of the Haar-twirled superoperator

$\displaystyle{\Lambda}_{av}(\rho)=p\rho+(1-p)\Tr(\rho)\frac{\mathbb{I}}{2^{n}}={\Lambda}_{d}(\rho)$
(4)

with $p=\left(\Tr({\Lambda}_{av})-1\right)/(2^{2n}-1)$ which is just a
depolarizing channel.

Therefore, for a single-layer randomizing benchmarking protocol, the average
fidelity of a device (for all possible operations) depends on a single
parameter. However, when we extend this protocol for a sequence of length $m$,
we must make some assumptions about the noise channel to obtain the single
parameter characterizable form. In general, for a m-sequence RB, the average
fidelity is given by,

$\displaystyle\bar{F}=\int d\mu(U_{1})\ldots d\mu(U_{m})$
$\displaystyle\langle\psi|\hat{\bf\Lambda}_{m}\circ\hat{\bf\Lambda}_{U_{m+1}^{\text{inv}}}\circ\hat{\bf\Lambda}_{{U}_{m}}\circ\hat{\bf\Lambda}_{U_{m-1}}\cdots\hat{\bf\Lambda}_{{U}_{1}}(|\psi\rangle\langle\psi|)|\psi\rangle,$

where $\hat{\bf\Lambda}_{m}$ is the experimental implementation of the state
preparation and final measurement. If we assume that the noise channels
associated with random operations including the inverse operation is time and
gate-independent and represented by ${\Lambda}$, the average fidelity for each
length $m$ become

$\displaystyle\bar{F}=\int d\mu(U_{1})\ldots d\mu(U_{m})$
$\displaystyle\langle\psi|{\Lambda}_{s}\circ{U}_{m+1}^{\text{inv}}\circ{\Lambda}\circ{U}_{m}\circ{\Lambda}\circ{U}_{m-1}\cdots{\Lambda}\circ{U}_{1}(|\psi\rangle\langle\psi|)|\psi\rangle$
$\displaystyle=\langle\psi|{\Lambda}_{s,t}({\Lambda}_{d})^{m}|\psi\rangle,$
(5)

Note that the error channel associated with the inverse operation has also
been absorbed within ${\Lambda}_{s}={\Lambda}_{m}\circ{\Lambda}$ where
${\Lambda}_{m}$ is the superoperator for SPAM errors. After twirling, this
channel becomes ${\Lambda}_{s,t}$. It manifests itself as a depolarizing
channel, as the state measurement is carried out in the basis
$\\{|\psi\rangle,|\psi_{\perp}\rangle\\}$, where off-diagonal elements do not
contribute. ${\Lambda}_{d}$ is the twirled version of the channel ${\Lambda}$
which also has the form of a depolarizing channel characterized by the
depolarizing parameter $p$. Finally, the average fidelity becomes

$\displaystyle\bar{F}=p_{s}p^{m}+(1-p_{s}p^{m})/d=A+Bp^{m}$ (6)

where $p_{s}$ characterizes the twirled depolarizing channel ${\Lambda}_{s,t}$
and the coefficients are $A=1/d$ and $B=(1-{1}/{d})p_{s}.$ This final form of
the average fidelity allows one to extract the parameter $p$ as we plot the
average fidelity with the sequence length $m$. The Haar-twirl plays the
pivotal role here to distill the complex quantum channels into a comprehensive
quantifiable form.

###  II.2 t-design based RB protocol

Since implementing a long sequence of Haar-random unitaries to produce the
Haar-twirl is inefficient, we can bypass the same by mimicking the Haar-twirl
by unitary t-designs. The goal of _unitary t-design_ is to suggest a small set
that duplicates the properties of the distribution over total unitary space
with Haar-measure for any polynomials of degree t [39]. It contains a small
finite set of unitaries $\\{C_{k}\\}_{k=1,\ldots,K}$ with the property

$\displaystyle\frac{1}{K}\sum_{k=1}^{K}P_{t}(C_{k})=\int_{U\in\mathcal{U}(2^{n})}d\mu(U)P_{t}(U),$
(7)

for any polynomial $P_{t}$ of degree t. Therefore, sampling from the same is
equivalent to sampling from the Haar random unitaries up to order $t$. In most
cases of our interest, a 2-design suffices. The 2-design is a uniform
distribution over _Clifford group_ which is generated by
$\expectationvalue{H,S,CNOT}$ (see Appendix B). So, instead of calculating the
average of fidelity at Eq. (2), we can sum over the Clifford members [25].

$\displaystyle\bar{F}=\frac{1}{K}\sum^{K}_{k=1}F(C_{k},\hat{\bf\Lambda}_{C_{k}})$
(8)

This is an extremely efficient approach because any element of the Clifford
group can be generated using only $\mathcal{O}(n^{2}/\log(n))$ single- and
two-qubit gates as compared to the exponential scaling for any arbitrary Haar-
random gate [40].

##  III Restricted Randomized Benchmarking Protocol

Most state-of-the-art quantum hardware relies on its compiler to find the
native gate decomposition of the target unitaries. For example, the random
unitary $U_{i}\in U(2^{n})$ is executed by a gate decomposition of depth d,
given by $g_{i}=\\{X^{1}_{i}\leavevmode\nobreak\ X^{2}_{i}\leavevmode\nobreak\
\cdots X^{d}_{i}\\}$ where each $X$ is the tensor product of single- and two-
qubit native gates acting over n-qubits. When implementing this, the error
channel associated with each native gate does not commute, and therefore we do
not have a simple error channel of the assumed form,
${\Lambda}_{U_{i}}\circ{U}_{i}$, instead it has a more complicated form

$(\Lambda_{X^{1}_{i}}\circ X^{1}_{i})(\Lambda_{X^{2}_{i}}\circ
X^{2}_{i})\leavevmode\nobreak\ \cdots(\Lambda_{X^{d}_{i}}\circ X^{d}_{i})$

for the native gate sequence $g_{i}$ implementing $U_{i}$. As the gates and
the superoperator do not commute in general, one cannot take all native gate
operations $X$’s to the right to get the Haar-random unirary $U_{i}$. The
noisy implementation of the Haar-random $U_{i}$ does not have the form
$\Lambda_{U_{i}}\circ U_{i}$ in this case. Therefore, implementing the Haar
twirl to get the unitary invariant superoperator is not straightforward with
native gate implementation. Moreover, the depth $d$ for any two Haar-random
operations can be different because of the compiler optimization, and
therefore the associated noise channel would contain additional superoperators
even if we assume them to be gate and time-independent. For these reasons, it
is difficult to obtain the Haar-twirled superoperator $\Lambda_{av}$ which is
a unitary invariant that was the prerequisite behind the success of randomized
benhcmarking. The _t-design_ based techniques are not immune to these
shortcomings either as for each of its design elements $C_{k}$ we need to rely
on the native gate sequences for their experimental implementation. Although
they are efficient in terms of sample complexity, the actual runs are not
coming from _t-designs_ anymore, instead they are an approximation of the
device’s noisy native gates. Therefore, although they should mimic the Haar
twirl in practice, their native gate implementation, being a noisy mixure,
does not become unitary invariant.

In this work, we try to address this gap between the theory of RB and its
experimental implementation by ensuring that each implementation of the Haar-
random unitary is consists of equal depth sequences. This allows us to treat
the associated noises in equal footing and leads to a better estimate of the
average fidelity as compared to the standard RB. Notice, however, that the
non-commutativity between the noise channels and native gate operations still
possesses a challenge to the unitary invariance of the Haar-averaged super
operator. In Appendix A, we present a numerical analysis of our considered
twirl considering a typical noise channel composed of both coherent and
incoherent components and discuss the circumstances under which unitary
invariance may hold.

Let us first set out the protocol that implements Haar-random unitaries
$U\in\mathcal{U}(2^{n})$ for $n=1,2$, by using native gate sequences of fixed
depth. We denote the parametric native gates of the device as
$\\{X^{j}(\alpha_{j})\\}_{\vec{\alpha}}$, where $\vec{\alpha}$ is a tuple that
depends on the number of free parameters for an n-qubit unitary. The number of
elements in the tuple in that case is $2^{n}-1$. By establishing a framework
for fixed-length decomposition of any unitary, we set conditions on the
selection of the parameters $\alpha_{j}$ to guarantee Haar randomness.
Therefore, choosing $\\{\alpha_{j}\\}$ from certain marginal distributions, we
generate the Haar-random unitaries to be used for the RB protocol

$\displaystyle\Pi_{j}X^{j}{(\alpha_{j})}=U(\vec{\alpha})$ (9)

Needless to say, the ordering of the sequence is important, otherwise, the
correct representation would not be generated. However, this ordering might be
invariant to unitary matrix choice as there might exist a universal circuit
that can span the entire unitary space through its free parameters. We show
this explicitly for single- and two-qubit cases.

###  III.1 Fixed depth Haar-random single-qubit unitary

The universal gateset of Rigetti QCS 111Rigetti Documentation: pyQuil,
https://pyquil-docs.rigetti.com/en/v2.7.0/apidocs/gates.html are $R_{Z}(\phi)$
with any arbitrary angle, $R_{X}(\theta)$ with angles $\theta=\pm\pi,\pm\pi/2$
and $CZ$ gate. Single-qubit unitaries has three free parameters. For our
construction, we decompose a single-qubit unitary $A(\phi,\theta,\omega)$ as

$\displaystyle A(\phi,\theta,\omega)$
$\displaystyle=\begin{bmatrix}e^{-i(\phi+\omega)/2}\cos(\theta/2)&-e^{i(\phi-\omega)/2}\sin(\theta/2)\\\
e^{-i(\phi-\omega)/2}\sin(\theta/2)&e^{i(\phi+\omega)/2}\cos(\theta/2)\end{bmatrix}$
$\displaystyle=R_{Z}(\omega)\cdot R_{X}(-\pi/2)R_{Z}(\theta)R_{X}(\pi/2)\cdot
R_{Z}(\phi)$ (10)

Figure 2: Native gate construction for arbitrary single-qubit unitary matrices
with three free parameters.

The key idea is to choose the joint probability distribution of the universal
circuit parameters $\\{\theta,\omega,\phi\\}$ in such a way that the resulting
unitaries become Haar random in the unitary space $\mathcal{U}(2)$. The Haar
measure is rotation invariant. So, if one pure state is taken on the Bloch
sphere and mapped by Haar-random matrices, the final state is expected to be
equally likely any other point on the sphere surface. So, we expect the Haar
measure to have a uniform surface density.

Assuming a uniform density of the surface of the sphere, the correct
distribution over $\\{\phi,\theta,\omega\\}$ can be found. $R_{Z}(\omega)$ and
$R_{Z}(\phi)$ are simply a phase shift. As the distribution is symmetric to
rotation along the z-axis, we expect a uniform distribution over these two
angles. However, $R_{Y}(\theta)=R_{X}(-\pi/2)R_{Z}(\theta)R_{X}(\pi/2)$ is a
beamsplitter. To sample $\theta$ uniformly at random in this context, we must
sample from the distribution Prob($\theta$)=$\sin\theta$. By this intuition,
we can easily deduce that the marginal distributions:

$\displaystyle d\mu_{H}(U)$ $\displaystyle=\sin\theta d\theta d\omega d\phi$
(11) $\displaystyle\rho(\phi,\omega)$ $\displaystyle=\frac{1}{2\pi}$ (12)
$\displaystyle\rho(\theta)$ $\displaystyle=\frac{\sin(\theta)}{2}$ (13)

So if the triplets $\\{\omega,\theta,\phi\\}$ are sampled by using these
marginal distributions, it is guaranteed that the final circuit will be a
Haar-random circuit. We verify the same by plotting the state $A|0\rangle$ on
the Bloch sphere. Beginning with a constant initial state and subjecting it to
a Haar random unitary, we anticipate that the resultant states $A|0\rangle$
will uniformly populate the Bloch sphere, which we confirm in Fig. 3.

Figure 3: Uniformly distributed single-qubit states $A|0\rangle$ on the Bloch
sphere. The uniformly distributed random states ensure the Haar randomness of
the constructed single-qubit unitaries.

Now that we have constructed a fixed-depth Haar-random single-qubit circuit
for a single layer $m=1$, we can go for the RB protocol with different $m$.
The fixed-depth circuits come into practice while generating random sequences
of Haar random unitaries for the RB protocol. As we indicated earlier, Haar-
random unitary transformations are implemented in a row plus the inverse of
the total unitary transformation of the circuit. In the noiseless case, we are
assured that the fidelity of the output state and the initial state would be
maximally one. However, any quantum device encounters in-between noise which
reduces the fidelity average.

In Fig. 4, the average fidelity (F) versus sequence length (m) can be
observed. For simplicity, we use $F$ to denote average fidelity instead of
$\bar{F}$ in the remainder of the paper. Motivated by Eq. (6), we fit the same
to an exponential curve, which shows an excellent match. Moreover, the
obtained fidelity using restricted RB matches up to 3 decimal points to the
obtained result using the t-design version. This is expected for single-qubit
because the compiler optimization for the single-qubit unitaries is always of
fixed depth. However, when comparing with the calibration data (see Table 1),
we see that the reported data is a little overestimated than our findings.

Figure 4: Comparative Analysis of RB schemes on single-qubit systems. - Here
we compare two RB schemes - restricted RB and two-designs RB - applied on the
Zeroth qubits of the Aspen-M-3 processor of Rigetti. Detailed experimental
procedures are discussed in the results section. Error bars represent the
standard deviation from 200 generated circuits with 800 shots each (refer to
Table 1 for detailed results). The purple curve illustrates the extrapolated
exponential fit for the two-designs RB, given by the equation $F=0.422\times
0.987^{m}+0.534$. Similarly, the red curve represents the fit for the
restricted RB, described by $F=0.425\times 0.987^{m}+0.508$.

###  III.2 Fixed depth Haar-random two-qubit unitary

Let us now proceed towards the main result of our work. For any multi-quit
random unitaries, the compiler optimization would produce native gate
sequences of different depths that tries to minize the number of gate used. To
the best of our knowledge, there exists no protocol that can guarantee a fixed
depth (and composed of 1 and 2-qubit gates) circuit for generating any n-qubit
random operation $U\in\mathcal{U}(2^{n})$, although interesting progress in
this direction has been made in Ref. [42]). Fortunately, in the case of two
qubits, Refs. [35, 34] guarantee that any two-qubit random operation can be
produced using only three CNOT gates. Motivated by these results, we now give
a protocol to generate Haar-random two-qubit unitary using a fixed-depth
native gate sequence which contains $\\{R_{Z}(\phi),R_{X}(\theta),CZ\\}$
gates. The 15 free parameters are chosen as in Fig. 5.

Figure 5: Construction of any two-qubit unitary $U$ with 15 control
parameters. $A,B,C$ and $D$ are single-qubit gates while $V$ is entangling
two-qubit gate with 3 free parameters. Refer Fig. 6 for the native gate
implementation of $V$.

The following algorithm produced Haar-random two-qubit unitaries:

Algorithm: 1. Generate angles $\phi_{1},\phi_{2},\phi_{3},\phi_{4}$ with the
probability distribution: $\displaystyle
d\mu_{H}=\frac{1}{(2\pi)^{4}4!}\Pi_{i<j}\absolutevalue{e^{i\phi_{i}}-e^{i\phi_{j}}}^{2}d\phi_{1}d\phi_{2}d\phi_{3}d\phi_{4},$
2. Find $\alpha,\beta,\delta$ by the pair-wise average of any two
$\\{\phi\\}_{j}$. 3. Tensor product two single-qubit Haar-random unitary
$A\otimes B$. 4. Tensor product two single-qubit Haar-random unitary $C\otimes
D$. 5. Introduce $U=(C\otimes D)V(A\otimes B)$ as the two-qubit Haar-random
circuit.

We first determine the conditional probability distributions of the 15
parameters in the decomposed circuit as shown in Fig. 5 and then verify that
the construction indeed produces random unitary matrix from the Haar
distribution.

####  III.2.1 Magic basis definition

The decomposition algorithm starts with transforming $U$ into magic basis
$u=\Lambda^{\dagger}U\Lambda$ for which $\Lambda$ is the magic matrix:

$\displaystyle\Lambda=\frac{1}{\sqrt{2}}\begin{pmatrix}1&i&0&0\\\ 0&0&i&1\\\
0&0&i&-1\\\ 1&-i&0&0\\\ \end{pmatrix}$ (14)

Since a change in basis does not contract or stretch the distribution, $u$ can
be considered as a Haar-random matrix. From now onwards, when we write
matrices with small letters mean it is in magic basis, like
$v=\Lambda^{\dagger}V\Lambda$, where V is the intermediate circuit standing
for all multi-qubit operations (see Fig. 6).

Figure 6: The construction of the entangling part $V(\alpha,\beta,\delta)$ for
generating any two-qubits unitary operator as shown in Fig. 5 using three CZ
gates.

####  III.2.2 Conditional distribution of $\alpha,\beta,\delta$

Three control parameters $\alpha,\beta,\delta$ depend on the eigenvalues of
the matrix $uu^{T}$. Since $uu^{T}$ is unitary, it must have phase eigenvalues
$\lambda_{j}=e^{i\phi_{j}}\,$ for $\,j\,\in\\{1,2,3,4\\}$, and they can be
found by the following equations [43]:

$\displaystyle\begin{cases}\alpha&=\frac{\phi_{1}+\phi_{2}}{2}\\\
\beta&=\frac{\phi_{1}+\phi_{3}}{2}\\\
\delta&=\frac{\phi_{2}+\phi_{3}}{2}\end{cases}$ (15)

To determine the probability distribution of $\alpha$, $\beta$, and $\delta$,
we first need to determine the probability distribution over $\phi_{j}$.
According to Mezzadri [44], the Haar measure induces a uniform distribution
over eigenvalue phases. However, these phases are not conditionally
independent. Therefore, it is necessary to find the joint probability
distribution of all the eigenvalues. It has been shown that the joint
probability distribution of eigenvalues for a $n$-dimensional Haar random
unitary can be found using Weyl’s formula [45]

$\displaystyle
d\mu_{H}=\frac{1}{(2\pi)^{n}n!}\Pi_{i<j}\absolutevalue{e^{i\phi_{i}}-e^{i\phi_{j}}}^{2}d^{n}\phi$
(16)

Considering the later distribution for $u$ eigenvalues, the joint probability
distribution of the same can be found for $uu^{T}$. In Sec. III.2.3 we show
how they come from the Haar measure. Sampling $\\{\phi_{j}\\}^{n=4}_{j=1}$
according to Eq. (16) for $n=4$

$\displaystyle
d\mu_{H}=\frac{1}{(2\pi)^{4}4!}\Pi_{i<j}\absolutevalue{e^{i\phi_{i}}-e^{i\phi_{j}}}^{2}d\phi_{1}d\phi_{2}d\phi_{3}d\phi_{4}$
(17)

and then using the Eq. (15), we get the parameter set
$\\{\alpha,\beta,\delta\\}$ for construction of the entangling part $V$ as
shown in Fig. 6. The complete construction of the Haar-random unitary $U$ is
completed after generating single-qubit rotations $A\otimes B$ and $C\otimes
D$ [43]. It is essential to select each according to a single-qubit Haar-
random distribution. Since the Haar measure is a rotation-invariant
distribution, each of these parts is simply a single-qubit rotation.

####  III.2.3 $uu^{T}$ eigenvalues distribution

According to the standard unitary decomposition [46], a unitary matrix U can
be expressed as $U=Qe^{iT}$, where $Q$ is an orthogonal matrix and $T$ is a
symmetric matrix. This means that for a complex symmetric unitary matrix like
$UU^{T}$, we can simplify it to $UU^{T}=e^{2iT}$. We can find a matrix $S$ for
any complex symmetric unitary matrix $W$ such that $W=e^{2iS}$. Moreover, we
can obtain $S$ by using the formula $S=\frac{1}{2i}log(W)$. By combining any
member of $O(n)$, say $K$, with $e^{2iS}$, we can create a unitary matrix
$J=Ke^{2iS}$, where $JJ^{T}=W$. The set of possible unitary matrices $J$ for a
given $W$ is the same as that for $O(n)$ and is therefore invariant to $W$.
Although the function $f(u)=uu^{T}$ is not one-to-one, it is a many-to-one
function, where the number of pre-images for any given image is infinite. This
means that there are infinitely many unitary matrices that could have produced
any given complex symmetric unitary matrix. The size of the possible matrix
$u$ is constant across the image of the function. In summary, the many-to-one
property of this function implies that if we had a Haar distribution in the
domain $U(n)$, then we would have the same distribution in the image of the
function.

Figure 7: Probability distribution of the eigenvalue spacing of the two-qubit
unitaries produced using our algorithm and that of the circular unitary
ensemble (CUE).

####  III.2.4 Verification

It is not easy to verify Haar uniformity by visualizing it on the Bloch sphere
like the single-qubit case. Nevertheless, multiple verification methods exist
that can confirm the resulting distribution adheres to a Haar distribution.
One such test is to check the eigenvalue phase spacing of the unitary matrices
generated using our protocol. The distribution of the same closely matches the
circular unitary ensemble (CUE), which can confirm that the unitaries produced
using our protocol are Haar-uniform. We first generate a set of two-qubit
unitaries following our algorithm and calculate their eigenvalues. We plot the
probabilities of the eigenvalue spacings of our generated unitaries in Fig. 7.
The probability distribution exhibits a close resemblance to that of the CUE.

Randomized Benchmarking for Aspen-M-3
---
Protocol name |  Average Fidelity |  Number of different gates sequences |  Number of shots per sequence
Reported calibration data ($0^{\text{th}}$ qubit, f1QRB) |  $99.73\pm 0.01$ |  |
1Q t-design with compiler |  0.987 |  200 |  800
1Q with conditional probability over native gate set |  0.987 |  200 |  800
Reported calibration data ($41^{\text{th}}$ and $42^{\text{th}}$ qubit, fCZ) |  $(98.68\pm 0.38)^{3}$ |  |
2Q t-design with compiler |  0.932 |  200 |  800
2Q with conditional probability over native gate set |  0.917 |  200 |  800
Table 1: Records of different Randomized Benchmarking method. For single-qubit
scenarios, the 0th qubit of the Aspen-M-3 device was used, and for two-qubit
scenarios, the couple of $41^{\text{th}}$ and $42^{\text{th}}$ qubit were used
due to their maximum entanglement efficiency at the time of calibration.

####  III.2.5 Restricted RB results

Using the fixed-depth gate sequence for each random operation, we ensure that
the noise channels associated with each random operation are uniformly
treated, which when compared with the compiler-optimized algorithm shows the
overestimation of the latter. For the t design based RB, we randomly sample
from the set of two-qubit Clifford operators [25]. To create an RB sequence of
length m, we uniformly sample from the Clifford set each time which was later
implemented using a fixed depth gate sequence. The inverse operations are
first computed from the previous $m$ chosen Clifford element that is executed
using our given algorithm. We benchmark them against the calibration data. Due
to the unavailability of 2-qubit randomized benchmarking data, we use an
approximate way of calculating the same. We assume that the single-qubit
operations are perfect and there is no associated error to its implementation.
Therefore, the only noisy operations are two-qubit ones. As Refs. [35, 34]
guarantees that only three two-qubit gates are enough to implement any
arbitrary two-qubit operations, we can take two-qubit gate fidelity to the
power three as an approximate estimate of average two-qubit fidelity from the
calibration data. For Rigetti hardware, the average CZ gate-fidelity is given
by 0.9862 (see Table 1). Therefore the RB estimate from the calibration data
would be $0.9862^{3}\approx 959(168)$.

In Fig. 8, we plot the average fidelity against sequence length m for the
restricted RB and the t-design RB protocol. Fitting the data to Eq. (6),
results in the average fidelity 0.932 and 0.917 for the t-design and the
restricted RB fit respectively. Comparing them with the calibration data, we
can safely conclude the calibration data is clearly an overestimation of the
actual two-qubit gate fidelity. It might be interesting to check-out the
possibility of finding a one-to-one correspondence between the calibration
data and the overestimation so that the actual performance of device can be
inferred directly from the calibration data of the given native gaterset,
however, that exceeds the current scopes of this study.

Figure 8: Comparative analysis of two-qubit RB. - We present a comparison
between two versions of Randomized Benchmarking (RB) - restricted RB and two-
designs RB - applied to qubits 41st and 42nd of Rigetti’s Aspen-M-3 quantum
processor. Error bars represent the standard deviation from 200 generated
circuits with 800 shots each (refer to Table 1 for detailed results). The
purple curve illustrates the extrapolated exponential fit for the two-designs
RB, given by the equation $F=0.710\times 0.932^{m}+0.166$. Similarly, the red
curve represents the fit for the restricted RB, described by $F=0.662\times
0.917^{m}+0.221$.

As mentioned earlier, the noise maps associated with the native gate
implementations generally do not commute with the gate operations.
Consequently, the conventional RB theory, which predicts an exponential decay
of average fidelity due to the unitary invariance of the Haar-twirl, does not
apply in this scenario. However, in our observed result we do observe an
exponential decay of the average fidelity. While some studies [47, 48, 49, 50]
have observed nonexponential decay potentially due to deviation from assumed
noise characteristics, instances of exponential decay can still occur under
different conditions [51, 52, 53]. In Appendix A, we present an analysis of
our considered Haar twirl assuming that our noise channel is composed of both
coherent and incoherent components which we take to be amplitude-damping and
depolarizing channels. We show that the effective noise channel corrsponding
to the Haar-averaged unitaries achieves gate independence when the strength of
the depolarizing noise is greater than that of the depolarizing noise. We
speculate that this could be one of the explanations of our observed
exponential decay.

An natural question about our method would the scalability of our protocol to
n-qubits. We note here that generalization of our protocol beyond two qubits
is not straightforward since the optimal constrction of an n-qubit unitary
using any universal gateset is not known beyond two qubits. However, it is
possible to ensure the fixed sequence length construction for any n-qubit
unitary [54, 55, 56], but the optimality of the same is not guaranteed.
Therefore, our approach can be adapted to any number of qubits using a fixed
sequence length, though it might not be an optimal construction.

##  IV Summary

In this work, we introduced a new approach to randomized benchmarking by using
fixed-length native gate sequences. Conventional RB methods, including the
efficient t-design based ones, often overestimate the actual performance of
the device. We show this rigorously and quantify the same for some exemplary
scenarios.

Our first contribution is the protocol to generate Haar uniform unitary for
single and two-qubit operations by accomplishing a suitable choice of marginal
distribution of native gate parameters. We have corroborated the same with
different verification tests which asserts that the protocols can indeed
generate fixed-depth decomposition of Haar-uniform operations for single and
two-qubit scenarios. We demonstrate how Haar randomness can be practically
generated, making it a resource-friendly and compiler-aware option for
implementing RB on current quantum hardware. Secondly, we find the average
gate fidelity using two known methods and compare the same with our restricted
RB technique. We show that the inference from the calibration data about the
RB fidelity is not an accurate estimate of the device noise. The standard and
t-design based methods also overestimate the device performance when compiler
optimization is enabled. This method allows for a true assessment of the
hardware’s performance, rather than simply trusting its calibration data or
the advertised single-number metric. Our restricted RB protocol aims to
reconcile the theoretical aspects of RB with its practical implementation,
emphasizing its effectiveness and appropriateness for systems with fewer
qubits.

Our observed results are robust and consistent in various trials with
different sampling sequences. This implies that the method used for the
sampling is reliable and does not affect the outcome of the Haar-average under
different sampling conditions. This stability is crucial for validating the
effectiveness of the protocols developed for generating Haar-uniform
operations. We supplement our results with a noise analysis to elucidate on
the hardware’s noise profile. One plausible explanation for our observed
results could be attributed to the predominant depolarizing noise within the
hardware, particularly when considering a smaller number of qubits. Such an
approach could be useful for building better noise models for the hardware,
subsequently facilitating the development of better error mitigation
strategies.

###### Acknowledgements.

DS thanks Eliott Mamon and Marine Demarty for illuminating discussions. This
work was supported in part by the Innovate UK grant Commercializing Quantum
Technologies (Application No. 44167).

##  Appendix A Gate independence of the effective twirl

In the original RB protocol, when constructing the Haar-twirl of the noise
channel, we assume that each random operation $U$ is associated with a noise
channel $\Lambda_{U}$. However, in the restricted RB protocol, each random
operation is not a single operation, instead, they are made of device
parametric native gates as $\\{X^{(\alpha)}\\}_{i,\alpha}$ where $\alpha$ is
the gate parameter and $i$ is the sequence depth. Therefore, there are
associated noise channels to each of the native gate operations and it is not
straightforward to see that averaging over Haar-random operation will create a
unitary invariant Haar-twirled channel.

In the following, we show that even in the case of restricted RB, it is
possible to construct an effective noise channel where each Haar-random
operations have an effective gate-dependent noise channel which consists of
the the native gates and their noise channels.

###  A.1 Effective noise channel

(a)

(b)

Figure 9: Effective noise channel for a single gate $R$ following a noise
channel. Here, $R$ is an arbitrary native gate, and $\Lambda$ and
$\tilde{\Lambda}$ are different noise channels in a way that two ends of both
circuits have the same density of quantum states.

(a)

(b)

(c)

(d)

Figure 10: Construction of effective noise channel for a multilayer circuit
for a Haar-random operation $U$ consisting native gates
$\\{R_{1},R_{2},\ldots,R_{k}\\}$ and their associated noise channels
$\\{\Lambda_{1},\Lambda_{2},\ldots,\Lambda_{k}\\}$. In practice, each gate
will have its own error channel as shown in (a) which is in general non-
commuting. In (b) and (c), we show a systematic iterative construction of the
effective error channel to the target operation. In (d), the blue shaded
region is the effective noise channel corresponding to the Haar-random
operation $U$ whose native gate implementation is the green shaded region
consisting of native gates.

The algorithm in the method describes a circuit composed of a sequence of
native gates. Since none of them are perfect, each of native gate will undergo
a distinct local noise channel. However, it can be shown that we can think of
an effective noise channel for the total circuit.

Let us think of a native gate $R$ implemented on an input state $\rho_{in}$.
Providing after $R$ the state goes into some noise channel (Fig. 9(a)), we can
write the output state $\rho_{out}$ as the following:

$\displaystyle\rho_{out}=\sum_{l}B_{l}R\rho_{in}R^{\dagger}B^{\dagger}_{l}$
(18)

Where $B_{l}$ matrices represent the corresponding Kraus operators of the
noise channel. We can think of the total map as a transformation below:

$\rho_{in}\longrightarrow\rho_{out}$

Generally, $B_{l}$ operators do not commute with $R$. However, we can find
another noise channel starting from $\rho_{in}$ and mapping to
$R^{\dagger}\rho_{out}R$.

$\displaystyle\rho_{in}$ $\displaystyle\longrightarrow R^{\dagger}\rho_{out}R$
(19)

Since this is a completely positive map, Choi-Kraus theorem [57] states that
we can always find corresponding Kraus operators $A_{k}$ such that:

$\displaystyle\rho_{in}$ $\displaystyle\rightarrow R^{\dagger}\rho_{out}R$
(20) $\displaystyle R^{\dagger}\rho_{out}R$
$\displaystyle=\sum_{k}A_{k}\rho_{in}A^{\dagger}_{k}$ (21)

Where $A_{k}$s are the Kraus operators of the new channel (refer Fig. 9(b)).
equivalently:

$\displaystyle\rho_{out}=\sum_{k}RA_{k}\rho_{in}A^{\dagger}_{k}R^{\dagger}$
(22)

As long as the input and output states are positive, we can always find
correct Kraus operators satisfying Eq. (22) and (18). Particularly, for this
single-layer circuit, $A_{k}$s can be found by this definition:

$\displaystyle A_{k}=R^{\dagger}B_{k}R$ (23)

The above mentioned proof has a very practical implication. For a more
generalized case in which we have a series of native gates
$\\{R_{i}\\}^{k}_{i=1}$trying to make a Haar-random unitary circuit (Fig.
10(a))

$\displaystyle\rho_{out}$
$\displaystyle=\sum_{i_{1},i_{2},...,i_{k}}B_{i_{k}}R_{k}...B_{i_{2}}R_{2}B_{i_{1}}R_{1}\rho_{in}R^{\dagger}_{1}B^{\dagger}_{i_{1}}R^{\dagger}_{2}B^{\dagger}_{i_{2}}...R^{\dagger}_{k}B^{\dagger}_{i_{k}}$
(24) with $\displaystyle U=\Pi^{k}_{i=1}R_{i}$ (25)

where $\Lambda_{k}$s represent the in-between layer noise channels.
Eventually, we can continue the induction similarly till the kth-step (Fig.
10(d)) such that

$\displaystyle\rho_{out}=$
$\displaystyle\sum_{i_{1},i_{2},...,i_{k}}A_{i_{k}}...A_{i_{2}}A_{i_{1}}R_{k}...R_{2}R_{1}\rho_{in}R^{\dagger}_{1}R^{\dagger}_{2}...R^{\dagger}_{k}A^{\dagger}_{i_{1}}A^{\dagger}_{i_{2}}...A^{\dagger}_{i_{k}}$
(26)

Effectively, we can introduce $K_{n}\equiv A_{i_{k}}...A_{i_{2}}A_{i_{1}}$ as
the Kraus operator of an effective noise channel for the total circuit
depending on circuits gates (refer Fig. 10(d))

$\displaystyle\rho_{out}=\sum_{n}K_{n}U\rho_{in}U^{\dagger}K^{\dagger}_{n}.$
(27)

###  A.2 Numerical evidence of exponential decay of average fidelity for
weekly gate-dependent effective noise channel

For the standard RB, we assume that the noise associated with each Haar-random
unitary is gate-independent which ultimately leads to the exponential decay of
average fidelity as shown in Eq. (6). However, in our protocol of restricted
RB, even if we assume that the noise channel for each native gate is gate-
independent, the effective noise channel (refer Fig. 10(d) ) corresponding to
a single Haar-random unitary is still gate-dependent as the noise channels, in
general, do not commute with the gates. So, the proof for the unitary
invariance of the Haar-twirled superoperator and subsequently the exponential
decay of the average fidelity is not straightforward to see for restricted RB
protocol.

However, as can be seen for single- and two-qubit restricted RB results (refer
Fig. 4 and 8 respectively), the average fidelities do decay exponentially in
our demonstration with Rigeeti ASPEN-M-3 hardware. Unfortunately, no precise
error channel is known for individual qubits and native gate operations on
them for the Rigetti hardware. However, assuming some noise model, it is
possible to extract the noise parameters from the calibration data. The
calibration data of the Rigetti ASPEN-M-3 hardware from the day of our
experiments is shown in Table 2.

Following the footprints of Ref. [58, 59], we assume that the experimental
noise in the hardware is mainly depolarizing ($\Lambda_{d}$). In addition, to
model the energy decay, we assume that there is also an associated amplitude
damping channel, $\Lambda_{AD}$. In the following, assuming a typical noise
model, we try to supplement our experimental result of exponential decay of
average fidelity for our restricted RB protocol. The assumed noise model for
each gate is therefore a mixture of depolarising and amplitude-damping noise
channels.

$\displaystyle\Lambda=\Lambda_{d}\circ\Lambda_{AD}$ (28)

where $\Lambda_{d}$ and $\Lambda_{AD}$ are described by

$\displaystyle\Lambda_{d}$
$\displaystyle(\rho)=\lambda\rho+(1-\lambda)\Tr(\rho)\frac{\mathbb{I}}{2^{n}}$
(29) $\displaystyle\Lambda_{AD}$
$\displaystyle\Big{(}\begin{bmatrix}\rho_{00}&\rho_{01}\\\
\rho_{10}&\rho_{11}\end{bmatrix}\Big{)}=\begin{bmatrix}\rho_{00}+\epsilon\rho_{11}&\sqrt{1-\epsilon}\rho_{01}\\\
\sqrt{1-\epsilon}\rho_{10}&(1-\epsilon)\rho_{11}\end{bmatrix}$ (30)

respectively. The parameter values of the assumed model can be identified by
matching the observed average fidelities from the calibration data. For the
single-qubit local depolarizing model $\Lambda_{d}$, the depolarising noise
parameter turns out to be

$\lambda=2(1-F)$

where $F$ is the f1Q sim. RB calibration data [60]. As can be verified from
the calibration data in Table 2, the single-qubit gate fidelities are
typically one order of magnitude higher than the two-qubit CZ gate fidelity.
As fidelity loss due to single-qubit gate operations is insignificant as
compared to the two-qubit gate operations, we further assume that the single-
qubit gates are perfect and there is no associated noise channel. The noise in
the RB experiment is solely contributed by the two-qubit gate channel that
contains depolarizing and amplitude-damping channels of the following form

$\displaystyle\Lambda^{2Q}=(\Lambda_{d}^{1}\circ\Lambda_{AD}^{1})\otimes(\Lambda_{d}^{2}\circ\Lambda_{AD}^{2})$
(31)

We now show by taking this fairly simple yet realistic noise model that the
effective noise channel in the case of restricted RB protocol is indeed weakly
gate-dependent which is a prerequisite of the twirling and the exponential
decay of the RB fidelity.

Qubit | T1 ($\mu$s) | T2 ($\mu$s) | f1QRB | f1Q sim. RB | fActiveReset | fRO | Date
---|---|---|---|---|---|---|---
0 | 10.02 | 18.16 | 99.73% $\pm$ 0.01% | 99.56% $\pm$ 0.02% | 97.5% | 91.9% | 21st July 2023

(a) Two-qubits device calibration Pair | fXY | fXY std err | fCZ | fCZ std err | Avg T1 (µs) | Avg T2 (µs) | Avg fActiveReset | Avg fRO | Date
---|---|---|---|---|---|---|---|---|---
41-42 | 0.9688 | 0.004721 | 0.9868 | 0.003863 | 32.4 | 50.6 | 0.994 | 0.9105 | 19th July 2023

Table 2: Calibration data of the considered qubits of Rigetti ASPEN-M-3
hardware on the day of the experiment.

In Ref. [53], it was shown that in the case of gate-dependent noise, one can
expect an exponential decay fidelity in RB provided that the gate dependency
of the associated noise channel is weak. We here provide a quantification of
this weakness by the diamond norm distance between two noise channels. We
provide numerical evidence to show that when the value of noise parameters is
small, the effective noise channel in restricted RB is independent.

Figure 11:  The average diamond norm distance between pairs of effective noise
channels, corresponding to two independent Haar-random unitaries, was
calculated. This average was derived from 10 pairs of independent Haar-random
unitaries, a number sufficient to reduce the standard error for each cell by
one order of magnitude. The ordinate and abscissa are respectively the
depolarising and amplitude damping noise strength as defined in Eq. (29-30).
The color-coded value of the average diamond norm distance is presented in the
legend on the right. The effective noise channel is gate independent when the
diamond norm distance is zero which is the case when $\epsilon=\lambda=0$ i.e.
in the absence of any noise. With increasing the value of the amplitude
damping parameter $\epsilon$, the effective noise channel started to become
gate-dependent. However, the weak gate dependency appears to be much more
robust against depolarising noise strength $\lambda$.

The diamond norm distance is a well-established measure of channel
discrimination [61]. If $\Lambda_{a}$ and $\Lambda_{b}$ are two channels
acting on two-qubit operations, the diamond norm distance is defined as

$\displaystyle d_{\diamond}(\Lambda_{a},\Lambda_{b})$
$\displaystyle=||\Lambda_{a}-\Lambda_{b}||_{\diamond}$ (32) $\displaystyle=$
$\displaystyle\max_{\rho}||(\mathbb{I}_{4}\otimes\Lambda_{a})\rho-(\mathbb{I}_{4}\otimes\Lambda_{b})\rho||_{1}$

where the maximization is done over all density matrix $\rho$ of dimension
$2^{4}$. In restricted RB, $\Lambda_{a}$ and $\Lambda_{b}$ are two effective
noise channels corresponding to two independent Haar-random unitaries. Note
that the effective noise channels are explicitly dependent on the native gates
which constitute the Haar-random unitary. We will now show that even though
the individual effective the error channels are gate-dependent, their averages
over the Haar measure can be weakly gate-dependent under certain conditions.
In Fig. 11, we plot the average diamond distance $\bar{d_{\diamond}}$ against
the depolarising and amplitude-damping noise strength. The average diamond
norm distance $\bar{d_{\diamond}}$ can be written as

$\displaystyle\bar{d_{\diamond}}=\frac{1}{n_{p}}\sum_{i}d_{\diamond}(\Lambda_{a_{i}},\Lambda_{b_{i}}),$
(33)

where $(\Lambda_{a_{i}},\Lambda_{b_{i}})$ represent pairs of random effective
noise channels generated according to Haar measure, and $n_{p}$ denotes the
total number of sampling pairs. The average is taken by creating many
independent Haar-random unitaries and their respective effective error
channel.

As can be seen from Fig. 11, the effective noise channel is becoming
increasingly gate-dependent as we increase the amplitude damping noise
parameter $\epsilon$. However, the depolarising noise is comparatively less
fatal. Even with a moderate value of depolarising noise parameter $\lambda$,
the effective noise channel remains weakly gate-dependent. Therefore, as long
as the principal source of our experimental noise is depolarising, we should
expect an exponential decay of average fidelity as observed in Fig. 4 and 8.
The analytical proof for gate dependent noise twirling [53] holds in this case
as well since the Haar measure is a 2-design. Following a similar treatment,
one can show that the Haar-twirl in this case becomes a linear map composed of
a depolarizing channel and a channel that reduces the trace. Therefore, even
when the noise superoperator is both gate-dependent and trace-preserving, the
average fidelity, when averaged over the Haar measure, corresponds to an
exponential decay with an additional single perturbation term.

##  Appendix B Generators of Clifford members

Three fundamental gates, namely the Hadamard, the phase gate S, and the CNOT,
generate the Clifford group. Their matrix form in the computational basis is
given by

$\displaystyle H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\
-1&1\end{bmatrix},\hskip 28.45274ptS=\begin{bmatrix}1&0\\\
0&e^{i\pi/2}\end{bmatrix},\hskip 28.45274pt$ (34) $\displaystyle
CNOT=\begin{bmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\
\end{bmatrix}.$ (35)

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