# Noise tailoring for Robust Amplitude Estimation

Archismita Dalal<EMAIL_ADDRESS>Amara Katabarwa
<EMAIL_ADDRESS>Zapata Computing, Boston, MA 02110 USA

###### Abstract

A universal fault-tolerant quantum computer holds the promise to speed up
computational problems that are otherwise intractable on classical computers;
however, for the next decade or so, our access is restricted to noisy
intermediate-scale quantum (NISQ) computers and, perhaps, early fault tolerant
(EFT) quantum computers. This motivates the development of many near-term
quantum algorithms including robust amplitude estimation (RAE), which is a
quantum-enhanced algorithm for estimating expectation values. One obstacle to
using RAE has been a paucity of ways of getting realistic error models
incorporated into this algorithm. So far the impact of device noise on RAE is
incorporated into one of its subroutines as an exponential decay model, which
is unrealistic for NISQ devices and, maybe, for EFT devices; this hinders the
performance of RAE. Rather than trying to explicitly model realistic noise
effects, which may be infeasible, we circumvent this obstacle by tailoring
device noise to generate an effective noise model, whose impact on RAE closely
resembles that of the exponential decay model. Using noisy simulations, we
show that our noise-tailored RAE algorithm is able to regain improvements in
both bias and precision that are expected for RAE. Additionally, on IBM’s
quantum computer our algorithm demonstrates advantage over the standard
estimation technique in reducing bias. Thus, our work extends the feasibility
of RAE on NISQ computers, consequently bringing us one step closer towards
achieving quantum advantage using these devices.

##  I Introduction

Amazing progress and development of quantum processors and the paradigm of
hybrid quantum-classical algorithms have worked in tandem to spark the age of
near-term quantum computing. The two major events that acted as catalysts for
this noisy intermediate-scale quantum (NISQ) era are the invention of
variational quantum eigensolver (VQE) Peruzzo  _et al._ (2014) and publicly-
available access to IBM’s quantum devices through their cloud platform ibm
(2016). In addition to VQE for quantum chemistry, with promising experimental
demonstrations Arute  _et al._ (2020); Kandala  _et al._ (2017); O’Malley  _et
al._ (2016); Hempel  _et al._ (2018), variational quantum algorithms have
applications in diverse fields including discrete optimization Farhi  _et al._
(2014); Bravyi  _et al._ (2022); Egger  _et al._ (2021); Akshay  _et al._
(2019), quantum machine learning and generative modeling Schuld  _et al._
(2020, 2021); Dallaire-Demers and Killoran (2018); Cao and Wang (2021); Liu
and Wang (2018); Benedetti  _et al._ (2019); Perdomo-Ortiz  _et al._ (2018);
Alcazar  _et al._ (2020); Rudolph  _et al._ (2022) and quantum error
correction Johnson  _et al._ (2017).

Extensive studies to elucidate the viability of quantum advantage in near-term
quantum computing have produced mixed results so far Rudolph  _et al._ (2022);
Huang  _et al._ (2022, 2021); Alcazar  _et al._ (2022); Somma and Subaşı
(2021); Jaderberg  _et al._ (2022); Wecker  _et al._ (2015). A recent analysis
argued that VQE will fail to provide quantum advantage over state-of-the-art
quantum chemistry algorithms for estimating the ground-state energy of an
industry-scale problem Hamiltonian Gonthier  _et al._ (2020). This drawback of
VQE arises because evaluating even a single energy within chemical accuracy
requires a large number of samples, which was estimated to take up to days for
some realistic molecules. Specifically, the number of samples required for a
desired accuracy of $\epsilon$ scales as
$\mathcal{O}(\frac{1}{\epsilon^{2}})$, leading to huge run-times for VQE. This
lesser-acknowledged problem of infeasible runtimes, which was dubbed the
“measurement problem” Gonthier  _et al._ (2020), poses a major roadblock for
hybrid algorithms, as well as many non-variational algorithms, towards
achieving quantum advantage on devices in the NISQ era and beyond.

In an effort to deal with the measurement problem on NISQ devices, recent
works apply techniques of quantum amplification and quantum estimation
borrowed from the field of fault-tolerant quantum computing Wang  _et al._
(2019, 2021); Giurgica-Tiron  _et al._ (2022a); Tanaka  _et al._ (2021). As a
first step towards this effort, Ref. Wang  _et al._ (2019) achieves a
reduction in the desired number of samples for VQE by improving the sample
scaling from $\mathcal{O}(\frac{1}{\epsilon^{2}})$ to
$\mathcal{O}(\frac{2}{1-\alpha}(\frac{1}{\epsilon^{2(1-\alpha)}}-1))$, where
$\alpha\in(0,1]$. More generally, these algorithms employ short-depth versions
of quantum amplitude estimation Brassard  _et al._ (2002); Abrams and Williams
(1999) to improve scaling and reduce runtimes for VQE, thus yielding
interpolating scaling between the standard and the quantum phase estimation
sampling rates of $\mathcal{O}(\frac{1}{\epsilon^{2}})$ and
$\mathcal{O}(\frac{1}{\epsilon})$, respectively.

Our work focuses on the near-term implementation of the quantum estimation
algorithm Wang  _et al._ (2021). An essential component of this algorithm is
the likelihood function, which is modelled as an exponentially-decaying
periodic function with a decay parameter $f$, over measurement data. This
“fidelity” parameter depends on the two-qubit gate fidelity ($f_{2Q}$), the
number of two-qubit gates ($D$) and the number of qubits ($n$), thus
incorporating noise effects into algorithm design and making this algorithm
robust. A detailed analysis based on this model derives the total run-time

$\displaystyle t=$ $\displaystyle
e\frac{f_{2Q}^{nD/2}}{2\bar{p}^{2}}\Bigg{[}\frac{nD\ln(1/f_{2Q})}{2\epsilon^{2}}+\frac{1}{\sqrt{3}\epsilon}$
$\displaystyle+\sqrt{\left(\frac{nD\ln(1/f_{2Q})}{2\epsilon^{2}}\right)^{2}+\left(\frac{2\sqrt{2}}{\epsilon}\right)^{2}}\Bigg{]},$
(1)

where $\bar{p}$ incorporates state preparation and measurement errors. This
expression highlights the interplay between fidelity (noise) and depth
(quantum coherence) towards reducing runtimes and improving scaling of
measurements with respect to accuracy.

In practice, using quantum coherence to improve sampling efficiency requires
running deeper quantum circuits, but their efficiency might be curbed by noise
Katabarwa  _et al._ (2021); Rao  _et al._ (2020); Giurgica-Tiron  _et al._
(2022b); Herbert  _et al._ (2021). Additionally, more complex errors
originating from unpredictable sources, including miscalibration of gates and
undesired correlations between qubits, in NISQ devices can deviate the
likelihood function from the assumed exponential decay model. To this end,
there have been two techniques so far for improving the performance of
amplitude estimation in such noisy scenarios; see Fig. 1. The first one is to
experimentally approximate the decay parameter and insert it into the
likelihood function Herbert  _et al._ (2021), and the second is to introduce
more decay parameters into the likelihood function Tanaka  _et al._ (2022). In
summary, one either measures the nuisance parameters in a simplified noise
model or adds extra nuisance parameters in the likelihood function.

Our approach is to tailor the device noise into stochastic noise, thus
producing an effective noise model closer to the exponential decay. While Fig.
1 highlights three different methods to make quantum amplitude estimation
feasible for near-term applications, we emphasize that they are not mutually
exclusive and some combined procedure might eventually be the best. Using
noisy simulations, we show that the measurement data from noise-tailored
amplitude amplification circuits come closer to the exponential decay model
for likelihood function. Furthermore, we assess the performance of our
technique at the task of estimating expectation values using current quantum
processors.

Figure 1: Three techniques for dealing with device noise in quantum amplitude
estimation: (a) construct accurate noise models Herbert  _et al._ (2021), (b)
add more parameters to the existing noise model Tanaka  _et al._ (2022), (c)
tailor device noise into existing noise model [this work].

Our paper is organized as follows. We begin in §II by briefly reviewing the
robust amplitude estimation algorithm, and then inspect the impact of both
incoherent and coherent errors on the likelihood function. In §III we
summarize randomized compiling, an efficient noise-tailoring technique for
quantum devices, and study its effect on this likelihood function. Finally we
present our results in §IV on the success and limitations of this noise-
tailoring technique for robust amplitude estimation using IBM devices, both
simulators and real hardware, and conclude our work in §V.

##  II Noise in Robust Amplitude Estimation

Robust amplitude estimation (RAE) is a quantum-enhanced algorithm for
estimating the expectation value of a Hermitian operator with higher accuracy
and precision as compared to the standard sampling technique of direct
averaging Katabarwa  _et al._ (2021). In this section, we first briefly
explain the RAE algorithm, then we introduce mathematical models describing
major sources of incoherent and coherent errors in superconducting qubits.
Finally, we discuss the impact of these errors on the RAE algorithm.

$\ket{0}^{\otimes n}$$A$$G$$G$$G$ (a) An enhanced sampling circuit with three
Grover iterates ($G$).

$P$$A^{\dagger}$$R_{0}$$A$ (b) Circuit for $G$.

Figure 2: The structure and components of a n-qubit quantum circuit used in
estimating $\langle A|P|A\rangle$ by enhanced sampling. $A$ is a parametrized
quantum circuit, $P$ is the operator to be measured and $R_{0}$ creates
reflection about $\ket{0}^{\otimes n}$.

### RAE algorithm

Given a $n$-qubit Hermitian operator $P$ with eigenvalues $\pm 1$ and a
quantum state $\ket{A}=A\ket{0}^{\otimes n}$, the goal is to estimate the
expectation value $\Pi:=\langle A|P|A\rangle$. To do this, the RAE algorithm
uses measurement data from enhanced sampling circuits Wang  _et al._ (2021),
and then performs the classical postprocessing of maximum likelihood
estimation (MLE) Katabarwa  _et al._ (2021). For a noisy enhanced sampling
circuit with $L$ layers, each layer resembles a Grover iterate for quantum
amplification and has a fidelity $f\in[0,1]$; see Fig. 2. Furthermore, the
likelihood of obtaining a measurement bit-string with parity $d\in\\{0,1\\}$
is

$\mathcal{L}(d|\Pi)=\frac{1}{2}\left(1+(-1)^{d}f^{(L+\nicefrac{{1}}{{2}})}\operatorname{cos}((2L+1)\operatorname{cos}^{-1}(\Pi))\right).$
(2)

The layer fidelity parameter is in-turn related to the nuisance parameter
$\lambda$ of the exponential decay noise model as $f=e^{-\lambda}$, thus
incorporating the impact of noise into the above likelihood function.
Combining likelihood functions for enhanced sampling circuits with different
$L$s and subsequently using MLE technique, the RAE algorithm yields an
estimate of $\Pi$.

It was empirically shown that RAE affords not only a way to improve the
precision of one’s estimate but also a way to mitigate the bias in one’s
estimate at least for a two-qubit system Katabarwa  _et al._ (2021). The
crucial insight behind this observation is that as long as the noise model
describing the quantum hardware is not far from the assumed noise model for
RAE, i.e. the exponential decay model, or has the same effect on the algorithm
as this assumed noise model, the effect of the noise in MLE is simply to slow
the rate of information gain as opposed to biasing the estimates. In this
work, we push this insight further and endeavour to tailor the real and
complicated noise on a quantum hardware to produce an effective noisy circuit
so that $f$ in likelihood function (2) becomes a better approximation to the
effect of noise in the estimation process.

### Relevant noise models

For our simulations, we use the model of a noisy superconducting quantum
processor and include incoherent errors from amplitude damping and dephasing
and coherent errors from residual $ZZ$ interactions between qubits, which are
the major sources of error for these processors. Although amplitude and phase
damping errors are bad, coherent error is the major performance-limiting
factor for quantum algorithms because these errors can accumulate
adversarially with increasing circuit depth. Ref. Karamlou  _et al._ (2021)
demonstrates that, for superconducting quantum processors, the residual $ZZ$
errors exert a detrimental impact on hybrid algorithms using parameterized
quantum circuits (PQC).

The combined effect of phase and amplitude damping errors is described by a
stochastic error channel. For a single-qubit, with decay rate $T_{1}$,
dephasing rate $T_{2}$, and a time-step parameter $t_{\text{step}}$, this
channel is defined using the three Kraus matrices

$\displaystyle E_{1}$ $\displaystyle=\begin{pmatrix}1&0\\\
0&\sqrt{1-\alpha-\beta}\end{pmatrix},$ $\displaystyle E_{2}$
$\displaystyle=\begin{pmatrix}0&\sqrt{\alpha}\\\ 0&0\end{pmatrix},$
$\displaystyle E_{3}$ $\displaystyle=\begin{pmatrix}0&0\\\
0&\sqrt{\beta}\end{pmatrix},$ (3)

where $\alpha=1-e^{-\frac{t_{\text{step}}}{T_{1}}}$ and
$\beta=1-e^{-\frac{t_{\text{step}}}{T_{2}}}$. Extending this formulation to
$n$ qubits, the incoherent error channel transforms a $n$-qubit density matrix
$\rho$ into

$\Lambda(\rho)=\sum_{i_{1}=1}^{i_{1}=3}\cdots\sum^{i_{n}=3}_{i_{n}=1}E_{i_{1}}\otimes\dots\otimes
E_{i_{n}}\rho E_{i_{1}}^{\dagger}\otimes\dots\otimes E_{i_{n}}^{\dagger}.$ (4)

For our simulations, we use the implementation of this error channel provided
by IBM’s qiskit T1T (2021).

The coherent error arising from undesired $ZZ$ interactions between transmon
qubits is modelled as a modification to the ideal CNOT gate Karamlou  _et al._
(2021). These $ZZ$ interactions are produced by small anharmonicities in the
qubits. The overall effect of these interactions on a pair of qubits, namely
control $c$ and target $t$, is expressed as

$\displaystyle\Xi=\xi_{c,t}(ZZ)_{c,t}+\sum_{i}^{|\text{spec}(c)|}\xi_{c,i}(ZZ)_{c,i}+\sum_{j}^{|\text{spec}(t)|}\xi_{t,j}(ZZ)_{t,j},$
(5)

where $\text{spec}(c)$ and $\text{spec}(t)$ are the sets of qubits coupled to
$c$ and $t$, respectively, with $\xi_{c,i}$ and $\xi_{t,j}$ being the
corresponding $ZZ$ interactions. In the presence of these coherent errors, the
expression for CNOT over gate time $t_{\text{gate}}$, in terms of the native
cross-resonance gate Rigetti and Devoret (2010); Chow  _et al._ (2011), is
modified as

CNOT $\displaystyle:=e^{-\frac{i\pi}{4}}e^{i\frac{\pi}{4}Z\otimes
I}e^{-i\frac{\pi}{4}Z\otimes X}e^{i\frac{\pi}{4}I\otimes X}$
$\displaystyle\longmapsto e^{-\frac{i\pi}{4}}e^{i\frac{\pi}{4}Z\otimes
I}e^{i(\Gamma Z\otimes X+\Xi)t_{\text{gate}}}e^{i\frac{\pi}{4}I\otimes X},$
(6)

where $\Gamma=\frac{\pi}{4t_{\text{gate}}}$ is the coupling between the
control and target qubits.

### Impact of noise on the likelihood function

$\ket{\reg_{\idx}}$$X$$X$$R_{y}(\theta_{0})$ Figure 3: A 4-qubit parameterized
circuit for Hydrogen molecule in a minimal basis.

To understand the effect of incoherent and coherent errors on the RAE
algorithm, we investigate how these errors impact the likelihood function (2).
To do this, we use the likelihood of getting even parities as

$P(0|\Pi,L)=\frac{1}{2}\left(1+f^{(L+\nicefrac{{1}}{{2}})}\operatorname{cos}((2L+1)\operatorname{cos}^{-1}\Pi)\right),$
(7)

where $L=0$ is the standard sampling case. We perform noisy simulations of
enhanced sampling circuits with $A$ being the PQC describing a 4-qubit
hydrogen molecule (Fig. 3), with $\theta_{0}=-6.057$, and $P$ being the
4-qubit Pauli operator $XXXX$. In Fig. 4(a), we present results from
simulations incorporating only amplitude and phase damping, with typical
$T_{1}$ and $T_{2}$ values for IBM devices, for a fixed $\Pi$ and increasing
$L$. We observe that the likelihood values obtained from these simulations
yield a good fit to the desired functional form (7), as evident from the near-
unity $R^{2}$ value. This implies that despite the noise we should be able to
get a good estimate of $f$ and the estimation process could be largely
salvaged.

(a) Fitting simulation results to $P(0|\Pi=0.2238,L)$ in the presence of only
incoherent noise: amplitude damping with $T_{1}=84~{}\upmu$s and phase damping
with $T_{2}=110~{}\upmu$s.

(b) Fitting simulation results to $P(0|\Pi=0.2238,L)$ in the presence of both
incoherent and coherent noise: amplitude damping with $T_{1}=84~{}\upmu$s,
phase damping with $T_{2}=110~{}\upmu$s and residual $ZZ$ interactions with
$\nicefrac{{\xi}}{{2\pi}}=45$ kHz.

Figure 4: Impact of incoherent and coherent noise on the likelihood function
$P(0|\Pi,L)$ (7) obtained from noisy simulations of enhanced sampling
circuits, with $A$ being the 4-qubit PQC (Fig. 3) and $\Pi=\langle
XXXX\rangle=0.2238$.

On the other hand, adding moderate level of coherent errors from $ZZ$
interactions drives the simulation results far away from our assumed
functional form with little hope of getting a good estimate of $f$; see Fig.
4(b). This observation serves as a strong evidence of how coherent errors
impact likelihood values, and can consequently deteriorate the performance of
RAE. From these results we infer that with the assumed noise model behind the
derivation of the likelihood function (2), RAE will not always offer the
sought after sampling power.

##  III Randomized Compiling for RAE

Motivated by our findings on how detrimental coherent errors are for
likelihood values, we aim to engineer an effective noise profile whose impact
on RAE is similar to that of an exponential decay model. To this end, we
employ randomized compiling (RC) that efficiently tailors coherent errors into
stochastic errors Wallman and Emerson (2016). In this section, we first
briefly state the background of RC, followed by an explanation of the basic
procedure for RC. Finally, we study the impact of this technique on likelihood
values in the presence of different magnitudes of coherent error.

The idea of RC was originally proposed in the context of quantum error
correction and fault tolerant quantum computing, where thresholds for
different error-correcting codes are estimated with an assumption of
stochastic Pauli noise Aliferis  _et al._ (2006, 2008). These estimates can be
way off if coherent errors are present Darmawan and Poulin (2017) since they
pile up systematically. In this regard, RC was introduced to construct an
effective quantum channel, which is described by only stochastic incoherent
noise. Recently, RC has been used for improving algorithmic performances of
NISQ devices Ville  _et al._ (2021); Hashim  _et al._ (2021); Urbanek  _et
al._ (2021); Ferracin  _et al._ (2022); Gu  _et al._ (2022). Thus, besides
making RAE feasible for NISQ devices, our work adds on to the growing field of
research in noise mitigation and noise tailoring within the NISQ era.

### RC procedure

Given a $n$-qubit “bare” circuit $\mathcal{C}_{\text{bare}}$, RC produces a
set of $N$ random circuits
$\\{\tilde{\mathcal{C}_{\ell}}\\}~{}\forall\ell\in\\{1,2,\cdots,N\\}$ that are
all logically equivalent to $\mathcal{C}_{\text{bare}}$. Logical equivalence
means that all these circuits produce exactly the same wave-function in the
noiseless limit. Each $\tilde{\mathcal{C}_{\ell}}$, which we refer to as a
“random duplicate”, is constructed by the following procedure:

  1. 1.

Partition $\mathcal{C}_{\text{bare}}$ into “easy” and “hard” cycles, where we
define an easy cycle to be a layer of single-qubit gates and a hard cycle to
be a layer of two-qubit entangling gates,

  2. 2.

Uniformly sample from the $n$-qubit Pauli group $\mathcal{P}_{n}$ to get an
element,

  3. 3.

Insert the element after an easy cycle,

  4. 4.

Insert gates after a hard cycle to ensure the logical equivalence to
$\mathcal{C}_{\text{bare}}$.

Additionally, we combine all adjacent single-qubit gates to preserve the
circuit depth. Using a convenient tensor product notation

$\vec{\mathcal{E}_{i}}=\mathcal{E}_{i1}\otimes\mathcal{E}_{i2}\otimes\cdots\otimes\mathcal{E}_{in},$
(8)

where $\mathcal{E}_{ij}$ is the single-qubit gate acting on qubit $j$ in an
easy cycle $i$, Fig. 5 shows the construction and structure of a random
duplicate for a 2-qubit circuit.

$\ket{\reg_{\idx}}$${\mathcal{E}_{11}}$${\mathcal{E}_{12}}$$\mathcal{H}_{1}$${\mathcal{E}_{21}}$${\mathcal{E}_{22}}$$\mathcal{H}_{2}$${\mathcal{E}_{31}}$${\mathcal{E}_{32}}$
(a) A 2-qubit $\mathcal{C}_{\text{bare}}$ with alternating layers of 3 easy
cycles ($\vec{\mathcal{E}_{1}}$, $\vec{\mathcal{E}_{2}}$ and
$\vec{\mathcal{E}_{3}}$) and 2 hard cycles ($\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$).

$\ket{\reg_{\idx}}$$\mathcal{E}_{11}$$\mathcal{E}_{12}$$P_{11}$$P_{12}$$\mathcal{H}_{1}$$P_{11}^{c}$$P_{12}^{c}$$\mathcal{E}_{21}$$\mathcal{E}_{22}$$P_{21}$$P_{22}$$\mathcal{H}_{2}$$P_{21}^{c}$$P_{22}^{c}$$\mathcal{E}_{31}$$\mathcal{E}_{32}$
(b) A random duplicate with $\vec{P}_{i}$ sampled from $\mathcal{P}_{2}$ and
$\vec{P}_{i}^{c}=\mathcal{H}_{i}\vec{P}_{i}^{\dagger}\mathcal{H}_{i}^{\dagger}$.
We assume that $\mathcal{P}_{0}^{c}=\mathbb{I}^{\otimes 2}$.

$\ket{\reg_{\idx}}$$\widetilde{\mathcal{E}_{11}}$$\widetilde{\mathcal{E}_{12}}$$\mathcal{H}_{1}$$\widetilde{\mathcal{E}_{21}}$$\widetilde{\mathcal{E}_{22}}$$\mathcal{H}_{2}$$\widetilde{\mathcal{E}_{31}}$$\widetilde{\mathcal{E}_{32}}$
(c) The circuit is re-compiled to preserve the depth of
$\mathcal{C}_{\text{bare}}$. The new easy cycles
$\vec{\widetilde{\mathcal{E}_{i}}}=\vec{P}_{i}\vec{\mathcal{E}_{i}}\vec{P}_{i-1}^{c}$

Figure 5: Circuit representations for logically-equivalent circuits before,
during and after RC.

In our implementation, the total number of measurements obtained by executing
the RC procedure is kept equal to the number of measurements allocated to
$\mathcal{C}_{\text{bare}}$. By defining a tuple $(\mathcal{C},M)$ for a
circuit $\mathcal{C}$ with $M$ measurements, we can thus formulate RC as the
map

$f_{\text{RC}}:(\mathcal{C}_{\text{bare}},M)\longmapsto(\mathcal{C}_{\text{RC}},M)$
(9)

where $(\mathcal{C}_{\text{RC}},M)$ represents a virtual circuit with
measurements being the union of measurement outcomes from all random
duplicates. For the set $\\{(\tilde{\mathcal{C}_{\ell}},M_{\ell})\\}$ and the
identity $\sum_{\ell=1}^{N}M_{\ell}=M$, we choose

$\displaystyle M_{\ell}$
$\displaystyle=\lfloor\nicefrac{{M}}{{N}}\rfloor,\;\forall\ell\in\\{1,2,\cdots,N-1\\},$
$\displaystyle M_{N}$ $\displaystyle=M-\sum_{\ell=1}^{N-1}M_{\ell}.$ (10)

All these random duplicates in the noisy setting are only distinguished by how
the noise affects them since they are all equivalent in the noiseless setting.
Then the intuition is that upon running them all on hardware and collecting
the results we get bit-strings that behave as though they came from a circuit
$\mathcal{C}_{\text{RC}}$ with no coherent noise but simply stochastic noise.

### Impact of RC on the likelihood function

(a) Fitting simulation results (with and without RC) to $P(0|\Pi,L=1)$, for
1000 different parameterization of $A$, in the presence of both incoherent and
coherent noise: amplitude damping with $T_{1}=84~{}\upmu$s, phase damping with
$T_{2}=110~{}\upmu$s and residual $ZZ$ interactions with
$\nicefrac{{\xi}}{{2\pi}}=75$kHz.

(b) Fitting simulation results (with and without RC) to $P(0|\Pi=0.2238,L)$ in
the presence of both incoherent and coherent noise: amplitude damping with
$T_{1}=84~{}\upmu$s, phase damping with $T_{2}=110~{}\upmu$s and residual $ZZ$
interactions with $\nicefrac{{\xi}}{{2\pi}}=10$kHz (circles) and
$\nicefrac{{\xi}}{{2\pi}}=75$ kHz (crosses).

Figure 6: Impact of the RC procedure on the likelihood function $P(0|\Pi,L)$
(7) obtained from noisy simulations of enhanced sampling circuits, with $A$
being the 4-qubit PQC (Fig. 3) and $P=XXXX$.

We study how RC impacts the likelihood of getting even parities (7) from noisy
simulations of enhanced sampling circuits (Fig. 2), with the same $A$ and $P$
as in the previous section. As this likelihood function depends on the exact
expectation value $\Pi$ and number of layers, i.e. Grover iterates, $L$, our
analysis is performed in the following two ways:

  1. 1.

Fix $L=1$ and vary $\Pi\in[-1,1]$ by choosing 1000 different values for
$\theta_{0}$ in Fig. 3,

  2. 2.

Fix $\Pi=0.2238$ and increase $L$ from 0 to 10.

Furthermore, we generate $N=50$ random duplicates for each enhanced sampling
circuit.

In Fig. 6(a), we observe that even in the presence of moderate $ZZ$
interactions with strength $\nicefrac{{\xi}}{{2\pi}}=75$ kHz, RC successfully
tailors this coherent error into incoherent error for circuits with one Grover
iterate, and consequently brings the likelihood function close to the desired
form (7). The tailored circuits fit the function $P(0|\Pi,L=1)$ closely and
yields a layer fidelity of 0.34, signifying that the resultant depolarizing
channel is very noisy. As we increase circuit depth beyond one Grover iterate,
coherent noise piles up producing no signal, as seen in Fig. 6(b). The
resultant likelihood function is essentially flat and uninformative, with
negligible enhanced sampling power; on the other hand for weak coherent noise,
i.e. with $\nicefrac{{\xi}}{{2\pi}}=10$ kHz, RC is able to consistently tailor
the likelihood values to the desired values for increasing $L$ and the
resultant function $P(0|\Pi=0.2238,L)$ is not very destructive. From these
observations we can hypothesize that RC can improve RAE in the presence of
coherent noise as long as the noise is not very strong.

##  IV RC-assisted RAE in practice

In this section, we present results on estimating Pauli expectation values
using our technique of RC-assisted RAE. This technique involves calculating
maximum likelihood estimates using measurement data from randomly-compiled
enhanced sampling circuits, as opposed to bare enhanced sampling circuits used
in the RAE algorithm Katabarwa  _et al._ (2021). As RC tailors coherent error
into stochastic error, consequently yielding a likelihood function resembling
the desired functional form (2), we expect to obtain more precise and accurate
estimates using the RC-assisted RAE algorithm. We compare performances between
three estimation techniques, namely RAE, RC-assisted RAE and direct averaging,
for two different parameterized circuits. In addition to the previously-used
4-qubit Hydrogen molecule (Fig. 3), we test these techniques for a 2-qubit
low-depth circuit ansatz (LDCA) Dallaire-Demers  _et al._ (2019) with 20 CNOTs
(Fig. 7). Our tests are performed on both IBM simulator and a real device.

The classical postprocessing for MLE uses measurement data from multiple
enhanced sampling circuits (Fig. 2) with increasing number of layers. For a
given maximum layer number $L_{\text{max}}$, we calculate an estimate by
collecting equal number of shots from circuits with
$L\in\\{0,1,\dots,L_{\text{max}}\\}$ and then employing the MLE technique on
this data. By fixing the total number of shots to be $M$, we take
$\lfloor\nicefrac{{M}}{{L_{\text{max}}+1}}\rfloor$ measurements from each
circuit. This is in contrast to techniques using $M$ measurements of each
circuit Giurgica-Tiron  _et al._ (2022b), i.e. a total of
$M(L_{\text{max}}+1)$ shots for each $L_{\text{max}}$, or optimizing number of
shots for each circuit based on nuisance parameter Herbert  _et al._ (2021).

For RC-assisted RAE, we also fix the number $N$ of duplicates for increasing
layers. An estimate $\widetilde{\Pi}$ of the expectation value is taken as the
average of estimates from $B$ Bayesian runs or MLEs, where the value of $B$ is
judiciously chosen such that $\widetilde{\Pi}$ is converged. Using 50
different measurement datasets, we obtain 50 estimates and then report their
mean and root-mean-square error (RMSE), along with their corresponding
standard errors. The total time, in units of $A$, required for calculating an
estimate is the runtime

$\displaystyle R=$
$\displaystyle\frac{M}{L_{\text{max}}+1}\sum_{k=0}^{L_{\text{max}}}\left(\left(2k+1\right)+\frac{n_{O}}{n_{A}}k\right)$
$\displaystyle=M\left(L_{\text{max}}+1\right)+\frac{Mn_{O}}{2n_{A}}L_{\text{max}},$
(11)

where $n_{O}$ and $n_{A}$ are the number of two-qubit gates in $R_{0}$ and
$A$, respectively.

$\ket{\reg_{\idx}}$$X$$R_{z}(\theta_{0})$$R_{x}(\frac{\pi}{2})$$R_{z}(\theta_{1})$$H$$R_{z}(2\theta_{2})$$R_{x}(-\frac{\pi}{2})$$H$$H$$R_{x}(\frac{\pi}{2})$$R_{z}(-2\theta_{2})$$H$$R_{x}(-\frac{\pi}{2})$$R_{z}(2\theta_{3})$$R_{x}(\frac{\pi}{2})$$R_{x}(\frac{\pi}{2})$$R_{z}(2\theta_{4})$$R_{x}(-\frac{\pi}{2})$$H$$R_{x}(-\frac{\pi}{2})$$H$$R_{z}(2\theta_{4})$$H$$H$$R_{x}(\frac{\pi}{2})$$H$$R_{z}(2\theta_{5})$$R_{x}(-\frac{\pi}{2})$$H$$H$$R_{x}(\frac{\pi}{2})$$R_{z}(-2\theta_{5})$$H$$R_{x}(-\frac{\pi}{2})$$R_{z}(2\theta_{6})$$R_{x}(\frac{\pi}{2})$$R_{x}(\frac{\pi}{2})$$R_{z}(2\theta_{7})$$R_{x}(-\frac{\pi}{2})$$H$$R_{x}(-\frac{\pi}{2})$$H$$R_{z}(2\theta_{7})$$H$$H$

Figure 7: A 2-qubit LDCA with
$\bm{\theta}=[-1.491,1.838,1.977,2.305,-3.124,2.049,1.254,-1.791]$.

### Performance on a noisy simulator

Through simulations of noisy quantum circuits, we study how RC impacts the two
features of RAE, namely enhanced sampling for improving RMSE and error
mitigation for improving bias of estimates. In the presence of amplitude and
phase damping noise and a low magnitude of residual $ZZ$ coupling, RC-assisted
RAE achieves slightly lower variances and RMSEs as compared to RAE (Fig.
8(a)), although both techniques have equivalent scaling of RMSE with runtime
(Fig. 9(a)). In this noise regime, both techniques also have comparable biases
because our choice of $B$ can not resolve changes in biases at the third
decimal point.

(a) Estimation results in the presence of amplitude damping with
$T_{1}=84~{}\upmu$s, phase damping with $T_{2}=110~{}\upmu$s and residual $ZZ$
interactions with $\nicefrac{{\xi}}{{2\pi}}=10$ kHz. (b) Estimation results in
the presence of amplitude damping with $T_{1}=84~{}\upmu$s, phase damping with
$T_{2}=110~{}\upmu$s and residual $ZZ$ interactions with
$\nicefrac{{\xi}}{{2\pi}}=75$ kHz.

Figure 8: Error mitigation properties of RAE and RC-assisted RAE (RC-RAE) as
observed by simulating enhanced sampling circuits in the presence of both
incoherent and coherent noise. Each point and its error bar represent the mean
and standard error of 50 estimates of $\Pi=\langle A|XXXX|A\rangle=0.2238$,
respectively, with $A$ being the 4-qubit PQC (Fig. 3). Each estimate is
obtained from $B=500000$ runs and $M=20000$ shots, with RC using $N=50$. Blue
line indicates the estimate obtained by direct averaging over $M$ shots from
the SS circuit, and the blue band denotes the corresponding standard error.

(a) Estimation results in the presence of amplitude damping with
$T_{1}=84~{}\upmu$s, phase damping with $T_{2}=110~{}\upmu$s and residual $ZZ$
interactions with $\nicefrac{{\xi}}{{2\pi}}=10$ kHz.

(b) Estimation results in the presence of amplitude damping with
$T_{1}=84~{}\upmu$s, phase damping with $T_{2}=110~{}\upmu$s and residual $ZZ$
interactions with $\nicefrac{{\xi}}{{2\pi}}=75$ kHz.

Figure 9: Enhanced sampling properties of RAE and RC-assisted RAE (RC-RAE) as
observed by simulating enhanced sampling circuits in the presence of both
incoherent and coherent noise. Each point and its error bar represent the RMSE
of 50 estimates of $\Pi=0.2238$ (Fig. 8) and the corresponding standard error,
respectively.

By increasing the coherent noise to a moderate value, we observe that RC-
assisted RAE performs significantly better than RAE. In this regime, the
scaling of RMSE with runtime for RC-assisted RAE exceeds the scaling for RAE
up to $L_{\text{max}}=7$ (Fig. 9(b)); this signifies that RC helps RAE to
improve RMSE faster. As we have fixed $M$ for increasing $L_{\text{max}}$,
this perceived improvement in RMSE with increasing runtime is solely due to
using more layers, and not due to increasing shots. This feature is also
evident from Fig. 8(b) where error bars on the low-biased estimates from our
RC-assisted RAE decreases with increasing $L_{\text{max}}$. Moreover, due to
the inherent ability of RC to mitigate the impact of coherent errors on
quantum-algorithmic performance, our algorithm is able to significantly reduce
bias as compared to RAE in the presence of moderate coherent noise (Fig.
8(b)).

Similar to our previous experiments Katabarwa  _et al._ (2021), we notice that
the estimates oscillate with $L_{\text{max}}$ (Fig. 8). This might be due to
the impact of coherent error being accumulated in some systematic manner as we
include additional layers in the estimation process. These oscillations
persist in RC-assisted RAE possibly due to the fact that we do not optimize
the number of RC duplicates and there might exist some residual coherent
errors in the circuits. By allocating the same number of shots for SS and RAE,
we observe that RC-assisted RAE also beats SS in the presence of low to
moderate coherent error by achieving far less bias and standard deviation
(Fig. 8).

(a) Error mitigation property

(b) Enhanced sampling property

Figure 10: Estimation of expectation values using ibmq_belem for a 4-qubit
Hydrogen Ansatz (Fig. 3) with $\Pi=\langle XXXX\rangle=0.2238$. Each point and
its error bar are obtained by taking statistics over 50 independent estimates;
see captions of Figs. 8 and 9.

(a) Error mitigation property

(b) Enhanced sampling property

Figure 11: Estimation of expectation value for a 2-qubit LDCA (Fig. 7), with
$\Pi=\langle XX\rangle=0.39$, using ibmq_belem. Similar to Fig. 10, each point
and its error bar are obtained by taking statistics over 50 independent
estimates. In this experiment, each estimate is obtained from $B=200000$ runs
and $M=8000$ shots, with RC using $N=20$.

### Performance on a real hardware

Having established the importance and limitations of RC for the RAE algorithm
in simulations, we now test its efficacy on real devices. We showcase results
from one publicly-available IBM device: ibmq_belem. For the 4-qubit circuits,
RC-assisted RAE has only slight advantage over RAE in terms of reducing bias,
but no improvement in precision is observed (Fig. 10(a)). Consequently, RMSEs
from both techniques scale similarly with increasing runtime, where RC-
assisted RAE yields slightly less RMSEs than RAE (Fig. 10(b)), and this
reduction of RMSEs is solely due to the corresponding decrease in biases. As
these circuits experience huge coherent noise from various sources including
residual $ZZ$ couplings and other kinds of crosstalk between qubits, the
stochastic noise model achieved after Pauli twirling renders the likelihood
function uninformative. Our observation from Fig. 10(a) suggests a major
limitation of RC for RAE: when coherent noise is very high, the effective
noise model after RC doesn’t assist RAE and thus we are better off opting for
SS.

For the 2-qubit circuits, although we see a significant reduction in bias with
RC-assisted RAE, no noticeable advantage occurs for precision of estimates
(Fig. 11(a)). The impact of coherent error sources on these 2-qubit circuits
is lesser than that on 4-qubit circuits, thus enabling RC to mitigate noise
better. A possible reason for getting lesser biases, both with and without RC,
for 4-qubit circuits is that the readout error correction may be less
effective for more qubits. These readout errors can introduce additional bias
to the estimates, which cannot be tackled by RC. These two-qubit circuits
experience a noise regime where RC-assisted RAE beats SS by reducing bias of
estimates quite significantly. Consequently, RMSEs obtained from our technique
decrease up to $L_{\text{max}}=4$, and have lesser values than those achieved
by RAE, as evident from Fig. 11(b).

### Summary of results

In practice, the performance of our RC-assisted RAE algorithm greatly depends
on the impact of device noise on the enhanced sampling circuits, and
consequently, this algorithm does not always deliver better performance than
those from RAE or SS. The effective stochastic noise after RC can be very
detrimental when the bare circuit has a high degree of coherent error. This is
due to the fact that although RC successfully tailors the likelihood function
into the desired form (2), the advantage of RC-assisted RAE ultimately depends
on the value of $f$ for this tailored function. A low value for $f$,
corresponding to an effective stochastic noise of high strength, reduces the
signal used in the estimation process and thus deteriorates the algorithmic
performance.

From our experiments using simulators and real devices, we can thus infer the
following:

  1. 1.

RC can help in improving RAE’s performance if coherent errors are not high,
where the high, moderate and low coherent error regimes can be algorithm
dependant.

  2. 2.

Study of optimal number of RC duplicates is needed since this might impact the
bias of estimate.

##  V Conclusion

The excitement of near-term quantum computing is not one for merely academics
but also for practitioners in the industrial world where quantum advantage
could scalably be turned into a great boon for the world at large. One major
roadblock in achieving quantum advantage on NISQ devices is the high cost for
estimating expectation values of operators with desired accuracies. Although
RAE can potentially squeeze quantum coherence from deep circuits to improve
precision and bias of estimates, the presence of highly correlated noise and
decoherence restricts its applicability only to small circuits. In this work,
we show that by altering complicated device noise using RC we can make RAE
more feasible on NISQ devices.

By running RC-assisted RAE on IBM devices, both simulator and quantum
processor, we propose three broad regimes for coherent noise, which is found
to be the main culprit for deteriorating algorithmic performances. In the low-
noise regime, the performance of our algorithm is not conclusively better (we
see disagreements in the third decimal point) than RAE’s performance; this
might not be surprising as there is little coherent noise to be removed and we
suspect that more careful optimization of randomized compiling is needed. But
it is clear that both techniques provide significant advantage over SS. We
demonstrate these effects in simulations by modelling coherent errors as
residual $ZZ$ couplings between superconducting qubits.

Upon increasing this coupling strength to a moderate magnitude, RC yields an
effective stochastic noise channel, which is not very destructive.
Consequently, our RC-assisted RAE can be used to recover the sampling power of
RAE and to further enhance the noise mitigating property of RAE in this
regime. As coherent error increases, RC can still help to reduce the bias but
the tailored likelihood function has very little signal. Thus, the inferencing
procedure loses its sampling power and using Grover iterates does not
particularly improve the scaling of the precision and RMSE, as observed by
running 2-qubit circuits on today’s devices.

On the extreme end where the coherent noise gets very high, reduction in bias
after RC is insignificant; thus, the noise mitigation property of RAE vanishes
as well and using SS is enough. We find that 4-qubit circuits on NISQ devices
fall in this regime. Thus RC-assisted RAE performs differently in this high-
noise regime as compared to the moderate-noise regime, where this algorithm
enhances at least one of the two features of RAE.

Our work highlights the role of noise tailoring in improving performances of
NISQ algorithms that rely on quantum amplitude estimation. From our
experiments on practical NISQ computers, we realize that further quantum-
control techniques are essential for decreasing crosstalk errors Ni  _et al._
(2022); Tripathi  _et al._ (2021), and, consequently, for extending the
usefulness of our RC-assisted RAE on these devices. Nevertheless, our
technique can possibly make implementations of RAE on early fault-tolerant and
fault-tolerant quantum computers perform more closely to the existing
performance models Johnson  _et al._ (2022). Moreover, performance of our
algorithm can be improved by using optimal number of random duplicates for
each enhanced sampling circuit. A possible future direction is to combine the
existing techniques for dealing with device noise (Fig. 1) in order to recover
the ability of RAE to improve precision even in particularly noisy conditions.

## Acknowledgements

We acknowledge insightful discussions on RAE and RC with Peter Johnson and
Athena Caesura, respectively. We also thank Razieh Annabestani, Jerome
Gonthier and George Umbrarescu for reviewing the manuscript. Our experiments
are run using the Orquestra™ platform of Zapata Computing.

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