A Probabilistic Imaginary Time Evolution Algorithm Based on Non-unitary Quantum Circuit Hao-Nan Xie1Shi-Jie Wei2yFan Yang1 Zheng-An Wang2 Chi-Tong Chen34 Heng Fan32 and Gui-Lu Long12z

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A Probabilistic Imaginary Time Evolution Algorithm Based on Non-unitary Quantum

Circuit

Hao-Nan Xie1,∗Shi-Jie Wei2,†Fan Yang1, Zheng-An Wang2, Chi-Tong Chen3,4, Heng Fan3,2, and Gui-Lu Long1,2‡

1Department of Physics, Tsinghua University, Beijing 100084, China

2Beijing Academy of Quantum Information Sciences, Beijing 100193, China

3Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China and

4School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

(Dated: October 12, 2022)

Imaginary time evolution is a powerful tool applied in quantum physics, while existing classical

algorithms for simulating imaginary time evolution suﬀer high computational complexity as the

quantum systems become larger and more complex. In this work, we propose a probabilistic al-

gorithm for implementing imaginary time evolution based on non-unitary quantum circuit. We

demonstrate the feasibility of this method by solving the ground state energy of several quantum

many-body systems, including H2, LiH molecules and the quantum Ising chain. Moreover, we

perform experiments on superconducting and trapped ion cloud platforms respectively to ﬁnd the

ground state energy of H2and its most stable molecular structure. We also analyze the successful

probability of the algorithm, which is a polynomial of the output error and introduce an approach

to increase the success probability by rearranging the terms of Hamiltonian.

I. INTRODUCTION

Imaginary time evolution (ITE), as a mathematical

tool, has impacted many problems in quantum physics,

such as solving the ground state of a Hamiltonian [1–

3], studying ﬁnite temperature properties [4–6] and the

quantum simulation of non-Hermitian systems [7, 8].

The concept of ITE can be understood by deﬁning the

imaginary time β=−it and substituting it into the

Schr¨odinger’s equation, i∂t|Φi=H|Φi, where His a

Hermitian Hamiltonian, which gives us the imaginary-

time Schr¨odinger’s equation:

−∂β|Φβi=H|Φβi.(1)

Given the initial state |Φ0i, the solution of Eq. 1 is

|Φβi=Ae−βH|Φ0i, where the corresponding evolution

operator e−βHis non-unitary, and Ais the normaliza-

tion constant. In classical simulations, one can directly

calculate e−βHand apply it to the initial state vector

|Φ0i, or employ some classical techniques such as quan-

tum Monte Carlo [9] and tensor networks [10]. However,

the dimension of the Hilbert space grows exponentially

with the size of the quantum system, making the tasks

intractable for classical computers [11].

Quantum computer is one of the promising tools for

eﬃciently simulating quantum systems [11–19]. For the

real time simulation, the evolution operator e−itHcan

be realized directly or be simply decomposed into a se-

quence of unitary quantum gates; but it is not the case

for imaginary time simulation, where the evolution oper-

ator e−βHis non-unitary. Therefore alternative methods

are required. Recently, some hybrid quantum-classical

∗xiehn19@mails.tsinghua.edu.cn

†weisj@baqis.ac.cn

‡gllong@mail.tsinghua.edu.cn

algorithms for simulating ITE have been proposed. For

example, variational quantum simulation methods [20–

22] utilize variational ansatz and simulate the evolution

of quantum states with classical optimization of parame-

ters; quantum imaginary time evolution (QITE) [23, 24]

ﬁnds a unitary operator to approach the ideal ITE in each

evolution step. The main drawbacks of these methods

include systematic error due to ﬁxed parametrization,

complexity of classical optimization [25, 26], limitation

of correlation length [3, 24], etc.

In 2004, Terashima and Ueda [27] proposed a method

to implement non-unitary quantum circuit by quantum

measurement. By applying unitary operations in the ex-

tended Hilbert space, one can obtain the desired ﬁnal

state in a certain subspace of the auxiliary qubits. Non-

unitary quantum circuit has been widely used in sim-

ulating non-Hermitian dynamics [8, 28], linear combina-

tion of unitary operators (LCU) [29], full quantum eigen-

solver (FQE) [30] and other algorithms. Based on non-

unitary circuit, Ref. [31] proposes a form of non-unitary

gate which applies to two-qubit ITE process; Refs. [3, 8]

show the form of the unitary operator acting on the total

Hilbert space to implement ITE, and give the examples

of circuits for two-qubit cases.

In this work, we propose a probabilistic imaginary time

evolution (PITE) algorithm which utilizes non-unitary

quantum circuit with one auxiliary qubit. In contrast to

previous works, we explicitly illustrates the construction

of the required quantum circuits using single- and double-

qubit gates, which applies to any number of qubits and

more generic Hamiltonians. We numerically apply the

PITE algorithm to calculate the ground-state energy of

several physical systems, and perform experiments on su-

perconducting and trapped ion cloud platforms. We also

give a detailed analysis about the computational com-

plexity of this method.

This paper is organized as follows. In section II we

give a description of the PITE method. Section III shows

arXiv:2210.05293v1 [quant-ph] 11 Oct 2022

some experimental and numerical simulation results, and

the analysis about the error and the successful probabil-

ity. In Section IV, we discuss the generalization of the

PITE algorithm to the cases where Hamiltonian is not

composed of Pauli terms, and introduce a method to in-

crease the success probability.

II. METHOD

An n-qubit Hamiltonian, H=Pm

k=1 ckhk, is composed

of mPauli product terms, in which ckis a real coeﬃcient

and hk=⊗n

j=1σj

αj, where σj

αjis a Pauli matrix or the

identity acting on the j-th qubit, with αj∈ {0, x, y, z}

(here we use the notation σ0=I). We assume that

the Hamiltonian does not contain the identity term I⊗n

because it merely shift the spectrum of Hamiltonian.

Our goal is to implement the non-unitary operator

e−βHin quantum circuits. We ﬁrst apply the Trotter

decomposition [32, 33]

e−βH=e−c1h1∆t. . . e−cmhm∆tβ/∆t+O(∆t).(2)

For a single Trotter step, we wish to obtain |Φ0i=

e−ckhk∆t|Φi. Deﬁne e

Tk=e−ckhk∆t. Due to the fact

that ckhkonly has two diﬀerent eigenvalues ±|ck|, each

with the degeneracy of 2n−1, there exists a unitary Uk

satisfying

UkckhkU†

k=−|ck|σlk

z, lk∈ {1, . . . , n}(3)

which is a single qubit operator. Thus we have

Tk=U†

kexp −|ck|σlk

z∆tUk.(4)

In Fig. 1, we show how to implement e

Tkin a quantum

circuit. The construction of Ukrequires O(n) CNOT

gates and single qubit gates at most (see Appendix B for

details). After the action of Uk, we write the work-qubit

the projection of Uk|Φion the subspace where the lk-th

qubit is |0iand |1i, respectively. In Appendix B we will

show that a0, a1are also the amplitude of the projection

of |Φion the ground-state subspace and excited-state

subspace of ckhk, respectively.

To realize e−|ck|σz∆ton the lk-th work qubit, we add an

ancillary qubit |0i, and apply the controlled-Ryoperation

|0ih0|⊗ I+|1ih1|⊗ Ry(θk) on the lk-th work qubit and

the ancilla qubit, where

Ry(θ) = e−iθσy/2=cos θ/2−sin θ/2

sin θ/2 cos θ/2,

θk= 2 cos−1(e−2|ck|∆t),

(5)

which gives the state

a0|ψ0i|0ianc +a1e−2|ck|∆t|ψ1i|0ianc

+a1p1−e−4|ck|∆t|ψ1i|1ianc .(6)

Then we measure the ancilla qubit, and if the result is 0,

we obtain

|a0|2+|a1|2e−4|ck|∆ta0|ψ0i+a1e−2|ck|∆t|ψ1i(7)

which is equivalent to the result of e−|ck|σz∆tacting on

the lk-th work qubit up to normalization.

The probability of obtaining 0 in the measurement of

ancilla is |a0|2+|a1|2e−4|ck|∆t. If we success, then the

last step is to apply U†

kon the work qubits. The output

state will be exactly e

Tk|Φiup to normalization.

In summary, the non-unitary operator e

Tkacting on a

quantum state |Φican be written as

Tk|Φi=U†

k[h0|anc Rk·(Uk|Φi⊗|0ianc)] (8)

up to a constant coeﬃcient, where Uktransforms hkinto

a Pauli matrix acting on the lk-th work qubit, and Rk

represents the controlled-Rygate acting on the lk-th work

qubit and the ancilla qubit, with the rotation angle θk=

2 cos−1e−2|ck|∆t.



n work

qubits

|Φ〉



1-st

lk-th

n-th



auxiliary qubit 

exp(-ckσz

lkΔt)

UkUk

†

Ry(θk)

FIG. 1. Quantum circuit for implementing

Tk.

III. RESULTS

A. Calculation of H2, LiH and quantum Ising

model

To illustrate the performance of the PITE algorithm,

we apply the algorithm in the calculation of the ground-

state energy of three physical systems: H2molecules, LiH

molecules, and a quantum Ising spin chain with both

transverse and longitudinal ﬁelds. The calculations of

H2are carried out on the Quafu’s 10-qubit superconduct-

ing quantum processor and IonQ’s 10-qubit trapped ion

QPU, and the calculations of LiH and the Ising model

are carried out on a numerical simulator to study the

inﬂuence of noises and the success probability of the al-

gorithm.

To calculate the ground state of H2and LiH on a

quantum computer, we ﬁrst need to encode the molec-

ular Hamiltonian onto qubits. Here we choose the STO-

3G basis [34] and use the Jordan-Wigner transforma-

tion (JWT) (see details in Appendix C). We eventu-

ally obtain Hamiltonians composed of Pauli matrices

H(R) = Pkck(R)σ1

α1. . . σn

αn, which can be acted on n

qubits, and the coeﬃcients ckvary with the interatomic

distance R. In this way the Hamiltonian for H2and LiH

can be encoded onto 4 and 6 qubits, respectively. Fur-

ther mapping is applied on H2to compactly encode the

H2Hamiltonian onto 2 qubits (see details in Ref. [35]),

which gives the H2Hamiltonian in the form of

HH2=c0(R) + c1(R)σ1

z+c1(R)σ2

+c2(R)σ1

zσ2

z+c3(R)σ1

xσ2

x,(9)

where the coeﬃcients at diﬀerent Rare available in Ap-

pendix C. The Hamiltonian for LiH at its lowest-energy

interatomic distance (bond distance) is given explicitly

also in Appendix C.

In our experiments, we use 2 qubits as the work qubits

to represent the H2molecule, and 1 qubit as the ancilla

qubit. The Hatree-Fock state of H2is |ΦHFi=|00iin

the qubit representation, which is chosen as the initial

state of the work qubits. Following the PITE method,

we ﬁrst do the calculation at a ﬁxed interatomic dis-

tance R= 0.75˚

A, and the experiments are carried out on

Quafu’s superconducting QPU P-10 and IonQ’s trapped

ion QPU (see information about Quafu in Appendix A).

After each Trotter step, the quantum state of work qubits

is tomographed, with 2000 shots on Quafu and 1000 shots

on IonQ, and then we use the state to calculate the en-

ergy value, and set it as the initial state for the next step.

The quantum circuits and related details are shown in

Appendix D. The results of the energy expectation value

as a function of the imaginary time βare shown in Fig.

2(a), compared with the theoretical PITE results. As β

increases, the energy rapidly converges to the exact solu-

tion in 5 evolution steps, within an error of ∼10−4a.u,

which is in the chemical precision.

To obtain the most stable molecular structure, we vary

the interatomic distance and plot the potential-energy

surface for H2molecule, as shown in Fig. 2(b). The

series of experiments is only carried out on Quafu’s P-

10, and the results (β= 0.5 and β= 1) are compared

with the Hatree-Fock state energies (β= 0) and the exact

ground-state energies obtained by diagonalization. The

lowest energy in the potential-energy surface corresponds

to the bound distance of H2molecule, which is around

0.75˚

In our numerical simulations, we calculate the ground-

state energy of LiH and quantum Ising chain to study the

success probability and the inﬂuence of noises. For LiH,

we use 6 work qubits and 1 ancilla qubit. The Hatree-

Fock state is |ΦHFi=|110000iin the qubit represen-

tation. Here |ΦHFiis very close to the exact ground

state, so it takes few steps for the state to converge.

Therefore, to show more about the convergence process,

FIG. 2. Experimental results of PITE on Quafu and IonQ

cloud platform. (a) H2energy expectation value as a func-

tion of β, at a ﬁxed interatomic distance R= 0.75˚

A. The

identity term in the Hamiltonian is considered when calculat-

ing the energy value, but not considered when executing the

algorithm. (b) H2energy as a function of βand the inter-

atomic distance R. The black line is the exact ground state

energy obtained by diagonalization.

we use a superposition of |ΦHFiand an excited state,

|Φ0i=√0.99 |ΦHFi+ 0.1|000011i, as the initial state.

Fig. 3(a) and Fig. 3 (b) show the convergence of the

energy E(β) and the ﬁdelity Fas a function of β, re-

spectively, where Fis the ﬁdelity between |Φβiand the

exact ground state |ψGi. The inﬂuence of quantum noises

are studied by applying quantum channels on all qubits

before each measurement of the ancilla qubit. The noise

is described by

E(ρ) =

ν=1

Eνρˆ

E†

ν(10)

with Kraus operators

E1=1 0

0√1−r−d,

E2=0√d

0 0 ,ˆ

E3=0 0

0√r(11)

where rand dare relaxation parameter and dephasing

parameter, respectively (see details in Appendix G). The

simulation results are also shown in Fig. 3(a) and Fig.

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AProbabilisticImaginaryTimeEvolutionAlgorithmBasedonNon-unitaryQuantumCircuitHao-NanXie1,Shi-JieWei2,yFanYang1,Zheng-AnWang2,Chi-TongChen3;4,HengFan3;2,andGui-LuLong1;2z1DepartmentofPhysics,TsinghuaUniversity,Beijing100084,China2BeijingAcademyofQuantumInformationSciences,Beijing100193,China3Institu...

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A Probabilistic Imaginary Time Evolution Algorithm Based on Non-unitary Quantum Circuit Hao-Nan Xie1Shi-Jie Wei2yFan Yang1 Zheng-An Wang2 Chi-Tong Chen34 Heng Fan32 and Gui-Lu Long12z

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