figurec

# Comparative analysis of error mitigation techniques for variational quantum
eigensolver implementations on IBM quantum system

Shaobo Zhang<EMAIL_ADDRESS>School of Computing and
Information Systems, The University of Melbourne, Parkville, 3010, Australia
School of Physics, The University of Melbourne, Parkville, 3010, Australia
Charles D. Hill<EMAIL_ADDRESS>School of Physics, The University of
Melbourne, Parkville, 3010, Australia  School of Mathematics and Statistics,
The University of Melbourne, Parkville, 3010, Australia  Muhammad Usman
<EMAIL_ADDRESS>School of Computing and Information Systems, The
University of Melbourne, Parkville, 3010, Australia  School of Physics, The
University of Melbourne, Parkville, 3010, Australia  Data61, CSIRO, 3168,
Clayton Australia

###### Abstract

Quantum computers are anticipated to transcend classical supercomputers for
computationally intensive tasks by exploiting the principles of quantum
mechanics. However, the capabilities of the current generation of quantum
devices are limited due to noise or errors, and therefore implementation of
error mitigation and/or correction techniques is pivotal to reliably process
quantum algorithms. In this work, we have performed a comparative analysis of
the error mitigation capability of the [[4,2,2]] quantum error-detecting code
(QEC method), duplicate circuit technique, and the Bayesian read-out error
mitigation (BREM) approach in the context of proof-of-concept implementations
of variational quantum eigensolver (VQE) algorithm for determining the ground
state energy of H2 molecule. Based on experiments on IBM quantum device, our
results show that the duplicate circuit approach performs superior than the
QEC method in the presence of the hardware noise. A significant impact of
cross-talk noise was observed when multiple mappings of the duplicate circuit
and the QEC method were implemented simultaneously – again the duplicate
circuit approach overall performed better than the QEC method. To gain further
insights into the performance of the studied error mitigation techniques, we
also performed quantum simulations on IBM system with varying strengths of
depolarising circuit noise and read-out errors which further supported the
main finding of our work that the duplicate circuit offer superior performance
towards mitigating of errors when compared to the QEC method. Our work reports
a first assessment of the duplicate circuit approach for a quantum algorithm
implementation and the documented evidence will pave the way for future
scalable implementations of the duplicated circuit techniques for the error-
mitigated practical applications of near-term quantum computers.

##  I Introduction

Computationally expensive problems such as breaking modern encryption
algorithms or simulating quantum systems require exponentially large computing
resources and quickly become intractable on classical computers when the
problem size is increased. Contrarily, quantum computing, first envisioned by
Fyenman Feynman (1982), is an emerging paradigm of computing, which is
anticipated to be exponentially faster than a classical computer for certain
hard problems which are classically intractable. For instance, Shor’s
factoring algorithm is a quantum solution designed to factor large integers
which scales super-polynomially on classical computers Shor (1994). Recently,
two research groups have demonstrated the supremacy of quantum computers over
classical supercomputers, first based on a superconducting quantum processor
developed by Google Arute  _et al._ (2019) and subsequently by using a
photonic quantum computer developed by a Chinese research team Zhong  _et al._
(2020). The first experiment demonstrated by Google was to sample from the
output of a random quantum circuit a million times, which would take a
classical supercomputer 10,000 years, whereas, for a quantum computer with
only 53 qubits, it required merely around 200 seconds to complete the task.
The second experiment was to yield an output state space of the Gaussian boson
sampling with around $10^{30}$ dimension within 200 seconds and its sample
rate was around $10^{14}$ times faster than that of the state-of-the-art
classical supercomputers. While these two demonstrations clearly establish the
promise of immense computational capabilities of a quantum computer, a clear
quantum advantage for practical applications is yet to be demonstrated.

It is universally recognised that the primary limiting factor for today’s
quantum devices is noise or errors, which significantly reduce the accuracy of
quantum algorithms. Even with error rates tracking below 1%, their cumulative
impact on large quantum circuits relevant for practical applications is
detrimental to their capability to solve any real-world problems. Therefore,
the development and testing of efficient error mitigation or correction
strategies is crucial to fully unlock the promise of quantum computing. In
this work, we focus on three quantum error mitigation strategies and compare
their performance for experiments and quantum simulations implemented on IBM
quantum system ibm . The first approach is based on encoding a noisy quantum
state in a quantum error-detecting code to create error-mitigated logical
qubits Urbanek  _et al._ (2020); the second approach uses entanglement between
two (replicated) quantum states to calculate the error mitigated expectation
value of the copied state Huggins  _et al._ (2021); the third approach
iteratively optimizes the probability of measurement results of quantum
circuits based on the Bayes’ Theorem Richardson (1972); Lucy (1974). We tested
these three methods on a noisy quantum computer and a quantum circuit
simulator, respectively, using IBM quantum platform and performed a
comprehensive comparative analysis based on the results.

Figure 1: A flowchart diagram which describes the procedure of our
experiments. The fermionic Hamiltonian of H2 molecule is mapped to two-qubit
VQE circuit. In contrast to traditional approach where two-qubit circuit is
mapped on a quantum processor, our work also implements three error mitigation
schemes:  1 the [[4,2,2]] error-detecting code (QEC method),  2 the duplicate
circuit and  3 the Bayesian read-out error mitigation (BREM). Whilst  1 and  2
mitigates the effect of circuit noise,  3 is implemented to tame the impact of
measurement errors.

The benchmarking of the investigated error mitigation strategies is performed
based on quantum chemistry simulations which is considered as one of the most
promising applications of a quantum computer in near future Bharti  _et al._
(2022). Even for a simple molecule consisting of just a few atoms, the
complexity of finding its ground state energy by solving multi-particle
Hamiltonian would require tremendous computational power, surpassing the
limits of classical computational techniques Babbush  _et al._ (2014); Aspuru-
Guzik and Walther (2012). Variational quantum algorithms (VQA) such as
variational quantum eigensolver (VQE) Peruzzo  _et al._ (2014) and quantum
approximate optimization algorithm (QAOA) Farhi  _et al._ (2014) have been
demonstrated as the leading methods to find the ground state of molecules with
limited quantum resources on today’s quantum devices, solving problems in the
field of quantum chemistry and quantum physics Arute  _et al._ (2020);
Izmaylov  _et al._ (2019); Kandala  _et al._ (2017); Yung  _et al._ (2014);
McClean  _et al._ (2016); Li and Benjamin (2017); McArdle  _et al._ (2019);
Barkoutsos  _et al._ (2018); Dallaire-Demers  _et al._ (2019); Romero  _et
al._ (2018); Grimsley  _et al._ (2019). In this work, we have implemented H2
molecule VQE problem which allows us to perform a direct comparison of the
three error mitigation schemes. Figure 1 schematically illustrates the
flowchart of our approach, highlighting three error mitigation techniques
(labelled as  1 to  3) tested in this work. While the traditional H2 molecule
VQE is solved on a quantum processor by optimisation of a two-qubit circuit,
our work also implements 4-qubit [[4,2,2]] error-detecting code (QEC method)
encoded VQE and the duplicate circuit VQE solutions. We also implemented
Bayesian read-out error mitigation (BREM) to suppress the impact of
measurement errors. Our results show that the duplicate circuit approach
performs extremely well (and superior than the QEC method) against both the
hardware noise as well as the simulated depolarisation noise. We also test
multiple simultaneous implementations of the QEC and the duplicate circuit
techniques which indicate significant impact of the cross-talk noise. Based on
our results, we infer that a scalable implementation of duplicate circuit
approach could lead to large-scale quantum chemistry simulations on NISQ
devices, demonstrating the future direction of duplicate circuit.

The remainder of the paper is divided in the following sections: Section II
describes methods and the background literature. Section III reports our
results obtained from both the experiments on the IBM quantum device, as well
as from the quantum simulations with depolarisation noise. In Section IV, we
discuss future directions and Section V provides the overall conclusions of
our work.

##  II Methods and Background Literature

###  II.1 Variational Quantum Eigensolver

The estimation of the ground state energy of a molecular system is an multi-
particle electronic structure problem, which is one of the central problems in
quantum chemistry. Due to limited capabilities of the current generation of
quantum computers, the focus is on simulation of small-scale molecular systems
which serve as a benchmark for quantum algorithms. A number of studies in the
past have investigated the computation of the ground-state energy eigenvalue
of H2 molecule on a quantum computer or a quantum simulator using the well-
known Variational Quantum Eigensolver (VQE) algorithm Peruzzo  _et al._
(2014); Urbanek  _et al._ (2020); Kandala  _et al._ (2017); Colless  _et al._
(2018); Kandala  _et al._ (2019); O’Malley  _et al._ (2016). The overview of
VQA and VQE have been discussed in multiple review articles and interested
readers are referred to the following references for details: Cerezo  _et al._
(2021); Motta and Rice (2021); McArdle  _et al._ (2020); Cao  _et al._ (2019).

Here, we briefly summarise the VQE quantum circuit construction in the context
of H2 molecule. The VQE approach is a hybrid quantum-classical algorithm,
which has been designed to calculate and optimize the ground state energy (the
lowest eigenstate of a Hamiltonian) of a molecule efficiently Kandala  _et
al._ (2017); Kokail  _et al._ (2019); McClean  _et al._ (2016); O’Malley  _et
al._ (2016); Peruzzo  _et al._ (2014). In this paper, we tested the
performance of error mitigation techniques by implementing the VQE algorithm
to find the ground-state energy of H2 molecule.

We obtain the qubit Hamiltonian of H2 molecule by following the approach in
O’Malley  _et al._ (2016). The H2 Hamiltonian can be reduced from four qubits
to two qubits. First, we apply the Born-Oppenhemier approximation Born and
Oppenheimer (1927) to the second quantized form of the Hamiltonian of H2 to
assume the coordinates of two nuclei are fixed, choosing a minimal basis (we
use STO-3G basis Hehre  _et al._ (1972) in this paper), we can obtain a
Hamiltonian in the form of:

$\displaystyle H$
$\displaystyle=\sum_{i,j}f_{i,j}a^{\dagger}_{i}a_{j}+\frac{1}{2}\sum_{i,j,r,s}f_{i,j,r,s}a^{\dagger}_{i}a^{\dagger}_{j}a_{r}a_{s},$
(1)

where

$\displaystyle f_{ij}$ $\displaystyle=\int
dx\phi_{i}^{*}(x)(-\frac{\nabla^{2}}{2}-\sum_{I}\frac{Z_{I}}{\absolutevalue{r-R_{I}}})\phi_{j}(x),$
(2) $\displaystyle f_{ijrs}$ $\displaystyle=\int
dx_{1}dx_{2}\frac{\phi_{i}^{*}(x_{1})\phi_{j}^{*}(x_{2})\phi_{r}(x_{2})\phi_{s}(x_{1})}{\absolutevalue{r_{1}-r_{2}}}.$

In this formalism, the one-body operator
$\sum_{i,j}f_{i,j}a^{\dagger}_{i}a_{j}$ refers to moving a fermion from $jth$
spin orbital to $ith$ spin orbital, $f_{i,j}$ is the one-body integral, which
illustrates the Coulomb interaction between electrons and nuclei, and the
kinetic energies of the electrons; the two-body operator
$\frac{1}{2}\sum_{i,j,r,s}f_{i,j,r,s}a^{\dagger}_{i}a^{\dagger}_{j}a_{r}a_{s}$
indicates that the operator moves two fermions from $rth$ and $sth$ spin
orbitals to $ith$ and $jth$ spin orbitals, $f_{i,j,r,s}$ is the two-body
integral, which represents the repulsion energy between the electrons McArdle
_et al._ (2020). The Hamiltonian has an equal number of annihilation and
creation operators, since the number of electrons in a molecular system is
conserved.

Next, the Hamiltonian is mapped from fermions to qubits. This procedure can be
achieved by using the Jordan-Wigner (JW) transformation Jordan and Wigner
(1993) or the Bravyi-Kitaev (BK) transformation Bravyi and Kitaev (2002). By
applying the BK transformation, we can obtain a four-qubit Hamiltonian of the
hydrogen molecule,

$\displaystyle H=$ $\displaystyle f_{0}+f_{1}(Z_{0}+Z_{0}Z_{1})+f_{2}Z_{1}$
(3)
$\displaystyle+f_{3}(Z_{2}+Z_{1}Z_{2}Z_{3})+f_{4}(Z_{0}Z_{2}+Z_{0}Z_{2}Z_{3})+f_{5}Z_{1}Z_{3}$
$\displaystyle+f_{6}(X_{0}Z_{1}X_{2}+Y_{0}Z_{1}Y_{2}+X_{0}Z_{1}X_{2}Z_{3}+Y_{0}Z_{1}Y_{2}Z_{3})$
$\displaystyle+f_{7}(Z_{0}Z_{1}Z_{2}+Z_{0}Z_{1}Z_{2}Z_{3}),$

where $\\{f_{i}\\}$ are real coefficients which depend on the bond length of
the H2 molecule and $\\{X_{i},Y_{i},Z_{i}\\}$ denote the corresponding Pauli
operators acting on qubit $i$. Notice that qubit 1 and qubit 3 have only Pauli
Z operators in the Hamiltonian, and our initial state has been set in a
Hartree-Fock state McArdle  _et al._ (2020). This means that these two qubits
remain in the same state throughout the simulation, we can use this property
to efficiently reduce the number of qubits required in our experiment.
Finally, we further reduce our four-qubit Hamiltonian to two-qubit Hamiltonian
by removing qubit 1 and qubit 3. The reduced qubit Hamiltonian of the hydrogen
molecule acts on two qubits. These qubits and their corresponding coefficients
can be re-indexed to form the following reduced Hamiltonian:

$\displaystyle
H=g_{0}+g_{1}Z_{1}+g_{2}Z_{2}+g_{3}Z_{1}Z_{2}+g_{4}Y_{1}Y_{2}+g_{5}X_{1}X_{2},$
(4)

where $\\{g_{i}\\}$ are the coefficients calculated classically, which vary
based on different bond lengths of the H2 molecule. In this paper, we use
values of $g_{i}$ published in Ref. [Colless  _et al._ , 2018]. The
expectation value $\left\langle{H}\right\rangle$ of the Hamiltonian is
calculated as

$\displaystyle\left\langle{H}\right\rangle$
$\displaystyle=\bra{\psi(\theta)}H\ket{\psi(\theta)}$ (5)
$\displaystyle=\sum_{i}g_{i}\bra{\psi(\theta)}\prod_{j}\sigma^{j}_{i}\ket{\psi(\theta)}$
$\displaystyle=g_{1}+g_{2}\left\langle{Z_{1}}\right\rangle+g_{3}\left\langle{Z_{2}}\right\rangle$
$\displaystyle+g_{4}\left\langle{Z_{1}Z_{2}}\right\rangle+g_{5}\left\langle{Y_{1}Y_{2}}\right\rangle+g_{6}\left\langle{X_{1}X_{2}}\right\rangle.$

The variation principle of quantum mechanics implies that,
$\left\langle{H}\right\rangle\geq\varepsilon_{0}$ where $\varepsilon_{0}$ is
the exact ground state energy of the H2 molecule.

We prepared the corresponding parametrized quantum circuits based on the
simplified two-qubit Hamiltonian and to form the Ansatz state,

$\displaystyle\ket{\Psi(\theta)},$ (6)

where $\theta$ is the parameter that will be used in VQE circuit. Then we
repeatedly measured each term in the Hamiltonian on a quantum processor to
estimate the ground state energy of H2. The trial wavefunction
$\ket{\Psi(\theta)}$ in $\bra{\Psi(\theta)}H\ket{\Psi(\theta)}$ that we used
is the UCC ansatz Bartlett  _et al._ (1989), which has been calculated in Ref.
[Urbanek  _et al._ , 2020] and given by

$\displaystyle\ket{\Psi(\theta)}=e^{i(-\theta)Y_{1}X_{2}}\ket{\Phi},$ (7)

where in this paper, we use $\ket{\Phi}=\ket{00}$ as the reference state,
which is obtained from the Hartree-Fock method.

Figure 2: Quantum circuits are plotted for the [[4,2,2]] quantum error-
detecting code (QEC), duplicate circuit and Bayesian read-out error mitigation
scheme. BS gates denote basis transformations based on the corresponding
measured term in qubit Hamiltonian and M gates measure the outputs of qubits.
(a) The diagram of a two-qubit circuit based on trial wavefunction defined by
Eq. 9. The circuit in section I prepares the Hartree-Fock initial state
$\ket{00}$ as the initial state. The circuit in section II performs the UCC
exponential in the trial wavefunction. Measurements are applied in section
III. (b) The diagram of the QEC method based on the [[4,2,2]] quantum error-
detecting code in Eq. 9 and trial wavefunction in Eq. 7. In section I we
prepare the logical initial state $\ket{00}_{L}$, ancilla qubit $a_{1}$ is
used to detect if errors occur during the state preparation. In section II,
ancilla qubit $a_{2}$ is used to perform the UCC exponential. In section III
we apply the corresponding basis transformations and measure the outputs. (c)
The diagram of the duplicate circuit. In section I, we implement two two-qubit
circuits on the first two qubits and the last two qubits respectively,
$q_{i_{j}}$ refers to the $jth$ qubit in $ith$ subsystem. Basis
transformations have been applied in this section as well. The circuit in
section II is a single layer of B gates with matrix form defined by Eq. 15,
$B_{i}$ denotes B gate acts on $ith$ qubit in each subsystem. The circuit in
section III measures the outputs. (d) A realization of B gate on $ith$ qubit
of two subsystems.

###  II.2 Depolarizing Error Model

The depolarizing error, which is also known as a depolarizing channel,
implements noise on single-qubit and is defined as

$\displaystyle E(\rho)$ $\displaystyle=p\rho_{cms}+(1-p)\rho,$ (8)
$\displaystyle\rho_{cms}$
$\displaystyle=\frac{I}{2^{n}}=\frac{1}{2^{n}}\sum_{i=1}^{2^{n}}\ket{\psi_{i}}\bra{\psi_{i}},$

where for a single qubit, with probability $p$, that qubit is depolarized to a
completely mixed state $\rho_{cms}=I/2$, and with probability $1-p$ the qubit
stays untouched Nielsen and Chuang (2002). More generally, in Eq. 8, $n$ is
the number of qubits the depolarizing channel acts on.

###  II.3 Quantum Error-Detecting Code

With the increasing size of quantum computers, mitigating and correcting the
effect of noise is becoming more and more vital, which means quantum circuits
should be more noise resilient, to allow Noisy Intermediate-Scale Quantum
(NISQ) devices Preskill (2018) to achieve greater performance. The noise can
come from a variety of sources, most fundamental of which is decoherence
induced by the interaction between qubits and the environment and leads to
errors occurring on the quantum computer. To mitigate the effect of noise or
errors, researchers have developed quantum error correction and detection
codes to try to correct induced errors Babbush  _et al._ (2014); Fowler  _et
al._ (2009); Vaidman  _et al._ (1996); Gottesman (2010). Here we explore the
performance of a QEC method Grassl  _et al._ (1997); Gottesman (2016); Urbanek
_et al._ (2020); Linke  _et al._ (2017) to encode four physical qubits to two
logical qubits. The QEC has the stabilizers $Z_{4}Z_{3}Z_{2}Z_{1}$ and
$X_{4}X_{3}X_{2}X_{1}$, which can detect any single-qubit error in the encoded
state. The distance between any two logical states is two. The logical code
words are:

$\displaystyle\ket{00}_{L}$
$\displaystyle\to\frac{1}{\sqrt{2}}(\ket{0000}+\ket{1111}),$ (9)
$\displaystyle\ket{01}_{L}$
$\displaystyle\to\frac{1}{\sqrt{2}}(\ket{0011}+\ket{1100}),$
$\displaystyle\ket{10}_{L}$
$\displaystyle\to\frac{1}{\sqrt{2}}(\ket{0101}+\ket{1010}),$
$\displaystyle\ket{11}_{L}$
$\displaystyle\to\frac{1}{\sqrt{2}}(\ket{0110}+\ket{1001}),$

where the first qubit refers to the rightmost qubit in both logical and
physical states.

There is no need to use ancilla qubits to perform logical operations on a
single logical states. Some operators on single logical qubit are the product
of single physical qubit Pauli operators. For instance, the logical $X$
operator acting on the first logical qubit is $I_{4}I_{3}X_{2}X_{1}$ and the
logical $X$ operator acting on the second logical qubit is
$I_{4}X_{3}I_{2}X_{1}$. $I_{4}Z_{3}I_{2}Z_{1}$ is the logical $Z$ operator
applied to the first logical qubit and $I_{4}I_{3}Z_{2}Z_{1}$ is the logical
$Z$ operator applied to the second logical qubit. The logical $H$ operation
acting on both logical qubits is $H_{4}H_{3}H_{2}H_{1}$. For logical $CNOT$
gate, which performs $CNOT_{L,12}=SWAP_{13}$, where $CNOT_{L,12}$ denotes
control qubit on the first logical qubit and target qubit on the second
logical qubit, similarly, $CNOT_{L,21}=SWAP_{12}$. The implementations of the
two-qubit circuit to prepare the trial state, and the equivalent logical
circuit for the QEC are shown in (a) and (b) in Fig. 2 respectively.

###  II.4 Duplicate Circuit

The duplicate circuit method is a second technique to mitigate the effect of
errors in a quantum circuit. First proposed by Cotler et al Cotler  _et al._
(2019), it makes use of multiple copies of a circuit being performed on a
quantum computing device and makes correlated measurements between these in an
attempt to reduce the resulting error. We also note that another study by
Huggins et al Huggins  _et al._ (2021) translated a similar approach from
bosonic systems to qubit systems. We apply this approach to our work by
connecting two identical two-qubit circuits (hence named ”duplicate circuit”
in this paper) with a layer of two-qubit gates and measure the qubits in
computational basis.

We calculate the expectation value of duplicate circuit according to

$\displaystyle
E(O)=\frac{\Tr(O\rho^{N})}{\Tr(\rho^{N})}=\frac{\Tr(O^{(N)}S^{(N)}\rho^{\otimes
N})}{\Tr(S^{(N)}\rho^{\otimes N})},$ (10)

where $\rho$ denotes a density matrix of the quantum system, in our case,
$\rho$ is a $4\times 4$ matrix generated from the two-qubit circuit, which can
be diagonalised to $\rho=\sum p_{i}\ket{\phi_{i}}\bra{\phi_{i}}$. Raising
$\rho$ to the $N$ power gives,

$\displaystyle\rho^{N}=\sum_{i}p^{N}_{i}\ket{\phi_{i}}\bra{\phi_{i}},$ (11)

where $\ket{\phi_{i}}$ are the four eigenstates of the original density
matrix, $\rho$. Under exponentiation, the dominant eigenvalue is amplified
exponentially in $N$, whereas, in contrast, the non-dominant eigenvalues are
suppressed exponentially in $N$. $O^{(N)}$ represents the observable operator
$O$ that acts on $N$ quantum systems,

$\displaystyle O^{(N)}$ $\displaystyle=\frac{1}{N}\sum_{i=1}^{N}O^{i},$ (12)
$\displaystyle O^{i}$ $\displaystyle=I^{\otimes M}_{1}\otimes I^{\otimes
M}_{2}\otimes\dots\otimes O^{\otimes M}_{i}\otimes\dots\otimes I^{\otimes
M}_{N},$

where each system has $M$ qubits. In our case, $O$ is the Pauli $Z$ operator
acting on two copied systems, namely

$\displaystyle Z^{(2)}=\frac{1}{2}(Z^{1}+Z^{2}).$ (13)

$S^{(N)}$ defined as the cyclic shift operator acts on $N$ quantum systems,

$\displaystyle
S^{(N)}\ket{\psi_{1}\psi_{2}\dots\psi_{N}}=\ket{\psi_{2}\psi_{3}\dots\psi_{1}}.$
(14)

In our case, we have two copied systems in the duplicate circuit, hence
$S^{(2)}$ works the same as a quantum SWAP gate. In order to simultaneously
diagonalize $S^{(2)}$ and $Z^{(2)}$ which act on the pairs of qubits
independently as the factorized tensor product of operators, Huggins et al
Huggins  _et al._ (2021) defined a two-qubit unitary gate called beam-splitter
unitary gate ($B$ gate),

$\displaystyle B^{(2)}$ $\displaystyle=B^{(2)}_{1}\otimes
B^{(2)}_{2}\otimes\dots\otimes B^{(2)}_{M},$ (15) $\displaystyle B_{i}$
$\displaystyle=\begin{bmatrix}1&0&0&0\\\
0&\frac{\sqrt{2}}{2}&-\frac{\sqrt{2}}{2}&0\\\
0&\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}&0\\\ 0&0&0&1\end{bmatrix},$

where $M$ is the total number of qubits in each system and the index $i$,
refers to the $ith$ duplicate system. An implementation of the duplicate
circuit using two two-qubit logical circuits is shown in Fig. 2(c) and a
realization of $B$ gate is shown in Fig. 2(d).

Figure 3: Chip geometry of $ibmq\\_toronto$. Numbers represent the index of
qubits. Blue boxes circle the qubits that are mapped by two-qubit circuit,
orange boxes circle the qubits that are mapped by the QEC method and green
boxes circled the qubits that are mapped by duplicate circuit.

###  II.5 Bayesian Read-Out Error Mitigation

In this work, we implemented Bayesian read-out error mitigation (BREM) (also
known as Bayesian unfolding) to mitigate the impact of read-out error Urbanek
_et al._ (2020); Nachman  _et al._ (2020). read-out error is the error that
occurs in measurements, where the state of a qubit is incorrectly determined.
That is, a qubit in state $\ket{0}$ is measured as $\ket{1}$ and vice versa.
The BREM approach Richardson (1972); Lucy (1974) is an iterative technique
based on Bayes’ Theorem to correct read-out errors. The recursive equation
used is:

$\displaystyle t_{i}^{r+1}=\sum_{j}\frac{Pr(\mathrm{measure}\ j|\mathrm{truth\
is}\ i)t_{i}^{r}}{\sum_{s}Pr(\mathrm{measure}\ j|\mathrm{truth\ is}\
s)t_{s}^{r}}Pr(\mathrm{truth\ is}\ j),$ (16)

where $r$ is the iteration number, $t_{i}^{0}$ is called a prior truth
spectrum, we set it as a uniform distribution. $Pr(\mathrm{measure}\
j|\mathrm{truth\ is}\ i)$ is a response matrix, which is the probability of
measuring state $j$ when given the state $i$. The response matrix can be
determined experimentally by constructing $2^{n}$ calibration circuits, where
$n$ is the number of qubits, each circuit is constructed with only Pauli $X$
gates and measured on $Z$ basis. $Pr(truth\ is\ j)$ is the measured spectrum,
which is the probability of measuring state $j$ when given the state $j$. $i$,
$j$ and $s$ are the number of states, which is $2^{n}$.

##  III Implementation on IBM System

We summarise our VQE experiments as follows. We calculate the expectation
value of four terms: $\left\langle{Z_{1}}\right\rangle$,
$\left\langle{Z_{2}}\right\rangle$, $\left\langle{Z_{1}Z_{2}}\right\rangle$,
and $\left\langle{X_{1}X_{2}}\right\rangle$ independently in our qubit
Hamiltonian on an IBM quantum device by using three circuits shown in Figure
2. According to the Pauli operators in these four terms, we apply the
corresponding basis transformation to measure them on the correct basis. To be
specific, for $\left\langle{Z_{1}}\right\rangle$,
$\left\langle{Z_{2}}\right\rangle$, and
$\left\langle{Z_{1}Z_{2}}\right\rangle$ terms, no basis transformation needs
to be applied; for $\left\langle{X_{1}X_{2}}\right\rangle$ term, Hadamard
gates will be used as the basis transformation gates to transform the basis
from Z basis to X basis. For the values of the variational parameter $\theta$,
we choose 257 values of $\theta\in[-\pi,\pi]$. Since we have only one
parameter, all of the 257 values will be implemented for all three circuits.
We sample each expectation value with 8192 shots for each value of the
variational parameter. For inter-atomic distance between electrons in H2, we
choose 78 values of $r\in[0.1,3.95]\ \mathring{\mathrm{A}}$, ground state
energy for each separation is estimated by determining the minimum energy
$\left\langle{E}\right\rangle_{\theta}$ among each of the 257 angles. We will
compare each of the methods described above to determine their effectiveness
at mitigating errors present in the devices for this VQE experiment.

###  III.1 Quantum Hardware Experiments

We executed three circuits on $ibmq\\_toronto$, which is one of the 27-qubit
IBM Quantum Falcon Processors ibm  (Geometry shown in Fig. 3), to test the
performance for each mitigation technique applied to the VQE algorithm. Since
the noise level of each qubit varies over time, we arbitrarily choose the
qubits to implement these circuits. For each circuit, we first used the
measured outputs of the circuit to calculate the raw ground state energy of H2
molecule. Then we calculated the optimized ground state energy by using the
BREM technique described earlier in section II E, which can be obtained by
applying the BREM approach to the raw measurement results to mitigate the
effect of read-out error. For the QEC method, we then discard the states for
which an error was detected and renormalise the remaining states.

####  III.1.1 Two-Qubit Circuit

Figure 4: The comparison of experimental results of two-qubit circuit, the
[[4,2,2]] quantum error-detecting code (QEC) circuit and duplicate circuit,
where Raw in legends for all figures denotes the values calculated from raw
data and BREM in legends denotes the application of the Bayesian read-out
error mitigation technique, which shows the optimized energies. (a) and (b)
illustrate the results from two-qubit experiments, Mapping i,j in legends
denotes the indices of qubits that are circled by blue boxes in Fig. 3; (c)
and (d) show the results of the QEC method, Mappings A and B in legends denote
qubits that are circled by orange boxes with the corresponding colored labels
in Fig. 3, respectively; (e) and (f) represent the results of duplicate
circuit, Mappings C, D and E in legends denote qubits that are circled by
green boxes with the corresponding colored labels in Fig. 3, respectively. (a)
Raw ground state energy of four mappings for two-qubit circuit. (b) Optimized
ground state energy of four mappings for two-qubit circuit. (c) Raw ground
state energy of two mappings for the QEC method. (d) Optimized ground state
energy of two mappings for the QEC method. (e) Raw ground state energy of
three mappings for the duplicate circuit. (f) Optimized ground state energy of
three mappings for the duplicate circuit.  Figure 5: The comparison of
experimental results of the [[4,2,2]] quantum error-detecting code (QEC)
circuit and duplicate circuit for parallel (simultaneous) execution of two
mappings and three mappings, respectively. Mapping A and Mapping B in (a) and
(b) denote qubits that are circled by orange boxes in Fig. 3 with the
corresponding colored labels, respectively; Mapping C, Mapping D and Mapping E
in (c) and (d) denote qubits that are circled by green boxes in Fig. 3 with
the corresponding coloured labels, respectively. (a) The raw experimental
results of two mappings of the QEC method with parallel execution. (b) The
experimental results of two mappings of the QEC method with parallel execution
after the application of BREM scheme. (c) The raw experimental results of
three mappings of duplicate circuit with parallel execution. (d) The
experimental results of three mappings of duplicate circuit with parallel
execution after the application of BREM scheme.

We implemented the two-qubit circuit with four mappings onto $ibmq\\_toronto$
(see Figure 3) and the VQE experimental results are shown in Fig. 4(a) and
Fig. 4(b). The BREM improves the fidelity of the measured VQE results in each
mapping as expected. The results for each of the four mappings investigated
vary due to the different noise level on the mapped qubits, both raw ground
state energy curve and optimized ground state energy curve with mapping. The
mapping utilizing qubits three and five had have the best performance among
all raw energy curves and all optimized energy curves, which indicate qubit 3
and qubit 5 have the lowest noise level among all mapped qubits during the
execution. Contrarily, the mapping utilizing qubits two and three had exhibits
least agreement with the exact energy calculation and in the next two
sections, we will explore if better results can be achieved by using error-
mitigation strategies.

####  III.1.2 [[4,2,2]] Error-Detecting Circuit

We tested two mappings (A and B, see Figure 3) of the QEC method under two
different scenarios on $ibmq\\_toronto$. In the first scenario, we tested the
performance of two mappings by executing them independently (or serially),
i.e., we first implemented VQE with mapping A, and then we implemented VQE
with mapping B (refer to Fig. 3). For the second scenario, we tested the
performance of the two mappings by executing them in parallel, which is, we
apply mapping A and mapping B simultaneously, measuring the outputs of the two
mappings at the same time at the stage of obtaining the raw data in VQE,
remaining other procedures of VQE for each mapping the same.

The VQE results of the first scenario are shown in Fig. 4(c) and Fig. 4(d).
These results show that mapping B performs better than mapping A in both raw
ground state energy and optimized ground state energy. The curve for raw
ground state energy of mapping B is closer to the exact energy curve than the
curve for optimized ground state energy when the inter-atomic distance is
larger than around 1.25 $\mathring{\mathrm{A}}$, which indicates the choice of
mapping qubits is vital which can further improves the fidelity of VQE
results.

The experimental results of the second scenario are represented in Fig. 5(a)
and Fig. 5(b). In this case, both mapping A and mapping B generate highly
noisy results, since there are twelve qubits used during the experiment,
whereas an individual mapping requires only six qubits. More qubits induce a
larger error on a noisy quantum computer. One possible explanation for this is
that cross-talk error is present. Cross-talk error is generally referred to
the unwanted coupling generated between signal paths Mazda (2014), which
commonly occurs between qubits on noisy quantum devices. In scenario two, the
effect of the cross-talk errors is stronger than in scenario one since the
qubits in two mappings are close to each other according to the geometry of
the quantum chip as shown in Fig. 3. This causes enhanced inaccuracy in the
experimental results for both mappings.

Comparing the experimental results from the two scenarios for the QEC method
to the experimental results of the two-qubit circuit, our results show that
both of the scenarios perform worse than the two-qubit circuit in the VQE
experiments. The QEC method consists of six qubits, each of the qubits
increases the probability of errors occurring during the execution, so it is
naturally noisier than the two-qubit circuit. In addition, the two-qubit
circuit can be directly mapped on the corresponding qubits, whereas larger
circuits like the QEC method requires auxiliary gates, such as SWAP gate, to
map the circuit on the corresponding qubits. This procedure also adds noise to
the circuit. Overall, the mapping structures of quantum circuits and the size
of quantum circuits both affect the experimental results. We note that our
results are in contrast to an earlier study using IBM Sydney machine Urbanek
_et al._ (2020), where QEC method was shown to perform better than two-qubit
mapping. We believe that this could be due to different noise in IBM Sydney
and IBM Toronto machines. We could not repeat our experiments on IBM Sydney
machine which is no longer available from IBM Quantum platform.

####  III.1.3 Duplicate Circuit

In this section, we analyse a second error-mitigation scheme (duplicate
circuit) which is described earlier in Section II D. For this case, we
considered three 4-qubit mappings (C, D, and E) of the duplicate circuit onto
the $ibmq\\_toronto$ processor as illustrated in Fig. 3. We again consider two
scenarios as discussed in the last section. For the first scenario, we tested
the performance of three mappings by executing them independently (one by
one). For the second scenario, we tested the performance of the three mappings
by executing all of them in parallel, which is, we apply mapping C, D and E
simultaneously, measuring the outputs of three mappings at the same time at
the stage of obtaining the raw data in VQE, remaining other procedures of the
VQE implementation for each mapping the same.

The comparison of experimental results for duplicate circuit implementations
under scenario one is shown in Fig. 4(e) and Fig. 4(f). We find that all three
mappings exhibit high fidelity results, performing better than the two-qubit
and QEC method results. The duplicate circuit consists of many more gates
where each gate adds additional noise. In theory, it is easier to obtain an
accurate expectation value of the duplicated system when the number of copied
systems $N$ in Eq. 10 is larger, but in practice, more gates introduce more
noise to the quantum system. Thus, there is a trade-off of the advantage
conferred by having more duplicate systems, and the additional noise that
increasingly larger depth $B$ gates add. As the results for duplicate circuit
indicate very close agreement with the exact energy, it is concluded that the
duplicate circuit provides significant advantage over the QEC method in
mitigating hardware errors.

The VQE results of scenario two are shown in Fig. 5(c) and Fig. 5(d), where
all three duplicate circuit mappings are executed concurrently. Mapping E
generated the highest fidelity results in both raw experimental data and after
the application of BREM approach. Overall our results show that while cross-
talk impacts duplicate circuit performance, generally the duplicate circuit
scheme was still observed to be quite effective in mitigating this kind of
complex noise mechanism.

Figure 6: The comparison of VQE results of three circuits on the simulator
with read-out error rates 0.5%, 2% and 4%.

###  III.2 Analysis of Noise Interplay based on Quantum Simulations

In the previous section, the results were reported from experiments on IBM
quantum devices which include hardware noise. However, in those experiments,
it was not possible to individually understand the impact of read-out error
and circuit noise. In this section, our aim is to test and compare the
performance of the BREM technique for the two-qubit circuit, the QEC method
and the duplicate circuit on the AerSimulator (a noisy quantum circuit
simulator in qiskit qis ) under three different noisy simulation scenarios.
The simulator in each scenario included varying types and degrees of noise. To
be specific, for the first scenario, we added only read-out error to the
simulator; for the second scenario, we added only depolarizing error to the
simulator; and for the third scenario, we added both read-out error and
depolarizing error to the simulator. This analysis helps us to understand the
relative impact of each type of noise on the VQE implementation and the
comparative performance of the investigated error mitigation techniques.

####  III.2.1 read-out errors

In first set of simulations, we added only read-out errors to the noisy
simulator, that is, only read-out errors can occur upon the measurements
during the simulations (all quantum gates were assumed to be noise-free). We
tested the performance of the three circuits (Figure 2) under three read-out
error levels, 0.5%, 2% and 4% respectively. The corresponding results are
plotted in Figure 6. The read-out error can be regarded as a conditional
probabilistic error since it can be summarized as
$P(\mathrm{measure}\ket{0}|\mathrm{truth\ is}\ket{0})$,
$P(\mathrm{measure}\ket{1}|\mathrm{truth\ is}\ket{0})$ given input state
$\ket{0}$ and $P(\mathrm{measure}\ket{0}|\mathrm{truth\ is}\ket{1})$,
$P(\mathrm{measure}\ket{1}|\mathrm{truth\ is}\ket{1})$ given input state
$\ket{1}$ to the measured qubit, matching the conditions of the BREM technique
to iteratively solve the problems based on Bayes’ Theorem. Therefore, read-out
errors can be largely mitigated by BREM approach on quantum circuits.

Figure 7: The comparison of VQE results of three circuits on the simulator
with depolarizing error rates 0.1%, 1% and 2%. Figure 8: Energy error of five
circuits with depolarizing error model in
$0.75\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ inter-atomic distance. The
[[4,2,2]] error-detecting circuit (QEC method) and duplicate circuit perform
better than two-qubit circuit when gate error rate is lower than around
$30\%$, the duplicate circuit with ideal B gate shows the best performance out
of the five circuits. Note that these simulations do not include read-out
error or BREM implementation.

The computed ground state energy plots are shown in Fig. 6. These results show
that for each read-out error level, the BREM approach can indeed significantly
improve the fidelity of the estimated ground state energy of the H2 molecule
in two-qubit circuits and the QEC method – the corresponding optimized curves
are very close to the exact energy. Our results also indicate that the
performance of the BREM technique is approximately similar for all three
circuit implementations, roughly independent of read-out noise level.
Therefore, we conclude that BREM could in general improve quantum circuit
fidelities, irrespective of the other error mitigation strategies implemented
to cancel the impact of circuit noise.

####  III.2.2 Depolarizing Noise

In these second set of simulations, we implemented three circuits (Figure 2)
on a noisy simulator deployed with only depolarizing error model (see Section
II-B for details), whereas the read-out error was assumed to be zero. Since we
only use single-qubit gates and two-qubit gates in three circuits, based on
the definition above, we added single-qubit depolarizing noise and two-qubit
depolarizing noise to the simulator at the same rate.

The simulation results are shown in Fig. 7, for a variety of different
depolarizing noise rates (0.5%, 1%, and 2%). In the absence of any explicit
read-out error, it is apparent that the BREM technique barely mitigates the
effect of depolarizing error for three circuits; all three optimized curves
are very close to the corresponding raw curves, since the depolarizing error
can occur multiple times on each of qubits. For instance, when implementing a
one-qubit circuit with multiple single-qubit gates, each gate has a
probability $p$ to occur a depolarizing error, according to the BREM approach,
$Pr(\mathrm{truth\ is}\ j)$ in Eq. 16 is hard to be the corresponding
probability to the likelihood $Pr(\mathrm{measure}\ j|\mathrm{truth\ is}\ i)$,
which has the different depolarizing error distribution on the gates. Whereas
for the circuit with only read-out errors, which occur only at the stage of
measurement, so $Pr(\mathrm{truth\ is}\ j)$ has the corresponding probability
distribution to $Pr(\mathrm{measure}\ j|\mathrm{truth\ is}\ i)$, thus it can
be used to optimize the measurement results.

Figure 9: The comparison of VQE results of three circuits on the mixed noisy
simulator with the combination of three depolarizing error rates (0.4%, 1.2%
and 2%) and fixed read-out error rate 1%.

Furthermore, we observe that the raw result curve for duplicate circuit show
the best performance compare to that of the other two circuits in a large
error rate (2%). The positive effects of this approach outweigh the negative
effects of this approach, that is, the noise introduced by 30 gates does not
affects to calculate an accurate expectation value by amplifying the largest
eigenvalue exponentially, which verifies our assumption in Fig. 8; the
optimized curve for the QEC method shows improved accuracy compared to the
corresponding raw curve, since after the optimization, we discarded the states
with non-zero ancilla value and renormalized the probability of remaining
states. When the noise level is high and the BREM optimization does not work,
many incorrect states can be discarded during these two procedures, the
probability of correct states move closer to the corresponding exact value,
which can improve the fidelity of optimized curves in the QEC method.

Figure 10: The comparison of VQE results of three circuits on the mixed noisy
simulator with the combination of three depolarizing error rates (0.4%, 1.2%
and 2%) and fixed read-out error rate 2%. Figure 11: The comparison of VQE
results of three circuits on the mixed noisy simulator with the combination of
three depolarizing error rates (0.4%, 1.2% and 2%) and fixed read-out error
rate 3%.

####  III.2.3 Relative Performance of Error Mitigation Schemes

To further quantify the performance of the QEC method and duplicate circuits,
we increase depolarisation error rate from 0 to 12% and compute the ground
state energy at a selected inter-atomic distance of
$0.75\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. In these simulations, we do not
include any measurement or read-out errors. Figure 8 shows the error in
ground-state energy of H2 molecule as a function of quantum gate error rate
for each circuit. The results show that the duplicate circuit and the QEC
method both outperform the two-qubit circuit when the gate error rate is
increased from 0 to 12%, which is significantly larger than the actual gate
error rate on $ibmq\\_toronto$ (around 8% for CNOT error). Furthermore,
duplicate circuit out-performs the QEC method for the whole noise range. The
inset in Fig. 8 shows that the two-qubit circuit implementation performs
better than the duplicate circuit and QEC method only when error rate is
increased above 32%, which is unrealistic for practical devices.

Quantum devices generally exhibit much large error rate for two-qubit gates
compared to single qubit gates, therefore, we also simulated duplicate circuit
with only two-qubit gate noise (ideal single qubit gates). Our results in Fig.
8 indicate that such scenario results in only a small improvement in the
accuracy. Finally, we simulated duplicate circuit with ideal (zero noise) $B$
gates, whereas the actual VQE circuit consists of noisy single and two-qubit
gates. In this scenario, we find that the accuracy of energy calculation is
drastically improved. Based on these results, we conclude that the performance
of duplicate circuit can be drastically enhanced by mapping $B$ gates on least
noise qubits available on a quantum processor.

####  III.2.4 Depolarizing Noise and read-out errors

For the third set of simulations, we added both depolarizing gate noise and
read-out errors to a noisy simulator to test the performance of the BREM
optimization on three circuits shown in Fig. 2. The computed results are
summarised in Fig. 9, Fig. 10 and Fig. 11, where each figure compares the VQE
results from the three circuits on a simulator with the combination of a fixed
read-out error rate and three depolarizing error rates. Overall, we find that
the implementation of BREM scheme improves the fidelity of all three circuits.
However, interestingly, the improvement from BREM is much larger for two-qubit
circuit and QEC method when compared to the duplicate circuit. This is clearly
evident from the results at 3% read-out error results, where the duplicate
circuit wins based on the raw data, but is no longer the best option after the
BREM implementation. Another finding from our results is that the
implementation of BREM technique is less effective in the presence of both
depolarising and read-out noise when compared to both type of noise
individually present. We also conclude that the hardware noise is quite
different from simple depolarisation noise and read-out errors. Importantly,
the performance of duplicate circuit is extremely well when implemented on IBM
device, compared to noisy simulations, which is more relevant for practical
applications.

##  IV Conclusions and Future Directions

In this work, we have provided a comprehensive analysis of the performance of
the three error mitigation techniques in the context of proof-of-concept H2
molecule ground-state energy calculations. Our results provide key new
insights into the working of the studied error mitigation schemes based on
both quantum hardware experiments and noisy simulation environment with
varying strengths and types of noise models. When the three circuits are
executed on $ibmq\\_toronto$, the duplicate circuit gives significantly better
performance when compared to the two-qubit implementation and the QEC method.
We also find that the BREM scheme is very efficient in mitigating the impact
of read-out errors. When we implemented multiple mappings simultaneously on
quantum devices, our results indicate that strong cross-talk impacts both
duplicate circuit and the QEC experiments. To gain further insights of the
comparative working of the three error mitigation schemes under various noise
configurations, we performed simulations with varying degrees of depolarising
and read-out errors. Overall the simulated results support the findings from
the hardware experiments that the duplicate circuit approach provides the best
fidelity by mitigating the impact of depolarisation noise.

We also identify a few directions for future research. First, one of the
differences between $ibmq\\_toronto$ quantum processor and a noisy simulator
is that actual device has complex type of errors not fully captured by simple
depolarising error models. Future investigations can focus on the performance
of three circuits on complex error models including cross-talk and
spatially/temporally inhomogeneous and correlated noise. Secondly, based on
the experimental results of the duplicate circuit in this paper, we found that
the positive effects in calculating the expectation value of the subsystem
outweigh the negative effects of noise brought by bringing more quantum gates.
Future work can focus to expand this study by working in two directions: one
is to test this method in a quantum system with more subsystems by designing a
complicated cascaded $B$ gate; another one is to focus on designing the $B$
gate efficiently, to introduce less noise to the system. Although the
duplicate circuit in our work shows excellent performance when compared
against both hardware noise and simulated error models, its benchmarking for
larger molecular systems such as LiH or BeH2 will be an important topic of
future research.

In summary, our results provide key new insights into the working of the
duplicate circuit, the QEC method and BREM technique for noise mitigation in
quantum chemistry simulations which is anticipated as one of the killer
applications of quantum computing.

###### Acknowledgements.

We acknowledge the use of IBM Quantum services for this work. The views
expressed are those of the authors, and do not reflect the official policy or
position of IBM or the IBM Quantum team.

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