Variational quantum algorithms for scanning the complex spectrum of non-Hermitian systems

doi:10.1007/s11467-023-1382-3

Front. Phys.

2024, Vol. 19

Issue (4) : 41202 https://doi.org/10.1007/s11467-023-1382-3

Variational quantum algorithms for scanning the complex spectrum of non-Hermitian systems

Xu-Dan Xie¹, Zheng-Yuan Xue^1,²(

), Dan-Bo Zhang^1,²(

)

¹. Key Laboratory of Atomic and Subatomic Structure and Quantum Control (Ministry of Education), Guangdong Basic Research Center of Excellence for Structure and Fundamental Interactions of Matter, and School of Physics, South China Normal University, Guangzhou 510006, China
². Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, Guangdong−Hong Kong Joint Laboratory of Quantum Matter, and Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China

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Abstract

Solving non-Hermitian quantum many-body systems on a quantum computer by minimizing the variational energy is challenging as the energy can be complex. Here, we propose a variational quantum algorithm for solving the non-Hermitian Hamiltonian by minimizing a type of energy variance, where zero variance can naturally determine the eigenvalues and the associated left and right eigenstates. Moreover, the energy is set as a parameter in the cost function and can be tuned to scan the whole spectrum efficiently by using a two-step optimization scheme. Through numerical simulations, we demonstrate the algorithm for preparing the left and right eigenstates, verifying the biorthogonal relations, as well as evaluating the observables. We also investigate the impact of quantum noise on our algorithm and show that its performance can be largely improved using error mitigation techniques. Therefore, our work suggests an avenue for solving non-Hermitian quantum many-body systems with variational quantum algorithms on near-term noisy quantum computers.

Keywords quantum algorithm non-Hermitian physics quantum many-body systems

Corresponding Author(s): Zheng-Yuan Xue,Dan-Bo Zhang

Issue Date: 05 February 2024

Cite this article:

Xu-Dan Xie,Zheng-Yuan Xue,Dan-Bo Zhang. Variational quantum algorithms for scanning the complex spectrum of non-Hermitian systems[J]. Front. Phys. , 2024, 19(4): 41202.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1382-3
https://academic.hep.com.cn/fop/EN/Y2024/V19/I4/41202

Fig.1 Illustration of the variational quantum algorithm, which can prepare the eigenstates and compute the corresponding eigenenergies. The quantum circuit

U (θ)

is parameterized by

θ

and should be performed on a quantum computer. The variable parameters

(E, θ)

are updated and optimized with classical computing in order to minimize the cost function.

Fig.2 (a) The quantum circuit diagram for the real part of

? 0 | U ? (θ m l) O^U (θ n r) | 0 ?

by the Hadamard test, in which H denotes the Hadamard gate. (b) The quantum circuit diagram for the imaginary part of

? 0 | U ? (θ m l) O^U (θ n r) | 0 ?

by the Hadamard test, in which S denotes the

π 2

phase gate.

Fig.3 The logarithm of the cost function

L (E, θ)

as a function of the circuit depth

P

, for various system sizes

L

. Here,

λ = 1, κ = 0.8

Fig.4 The variability of iterations in proportion to system size. As the system size expands, there is a corresponding increase in the requisite number of iterations for the loss function to converge to its minimum value. For various system sizes

L

, we set the depth of the quantum circuit

P

equal to

L

in order to ensure the convergence of the loss function to zero. Here,

λ = 1, κ = 0.8

Fig.5 The energy levels of

H λ, κ

as a function of

κ

, in the case

L = 4, λ = 1

. (a) Illustrates the real component of the energy levels, while (b) depicts the imaginary part of the energy levels. Those lines denoted by

E 0

E 1

E 2

and

E 3

are obtained by exact diagonalization.

	$n = 0$	$n = 1$	$n = 2$	$n = 3$	$n = 4$	$n = 5$	$n = 6$	$n = 7$

VQA ( $κ = 0.4$ )	0.7988	0.7988	0.9165	0.7719	0.7719	0.9165	0.7978	0.7994
Exact	0.7988	0.7988	0.9165	0.7719	0.7719	0.9165	0.7979	0.7994
VQA ( $κ = 0.2$ )	0.1680	0.1680	0.9798	0.5822	0.5821	0.9798	0.9537	0.9541
Exact	0.1681	0.1681	0.9798	0.5821	0.5821	0.9798	0.9537	0.9541

Tab.1 Fidelity between the right eigenstates

| ψ n r ?

and the left eigenstates

| ψ n l ?

, with

κ = 0.4

and

κ = 0.2

, respectively. Here

L = 3, P = 3, λ = 1

Fig.6 The fidelity between

| ψ n r ?

and

| ψ n l ?

for

κ = 0.4

in (a) and

κ = 0.2

in (b). Here

L = 3, λ = 1

Fig.7 The expected value of the Hamiltonian operator

H λ, κ

according to Eq. (14) for

κ = 0.4

in (a) and

κ = 0.2

in (b). Here

L = 3, λ = 1

Fig.8 Illustration of effects of quantum noises on the landscapes for the real and imaginary part of the energy parameter. The left and right plots depict the landscape without noise and under the depolarization noise, respectively. The red star in the figure is the minimum value of the loss function, which corresponds to the optimal energy parameter. A comparison shows that noise will lead to a shift of the optimized energy.

Fig.9 Comparison of optimization processes for ideal, depolarizing noisy and mitigated variational quantum algorithm. The figure illustrates the impact of depolarization noise on quantum algorithms and the effectiveness of noise mitigation techniques. (a) The cost function as a function of the number of iterations. (b) The fidelity respect to target ground state as a function of the number of iterations. (c) The real component of the energy

E

as a function of the number of iterations. (d) The imaginary component of the energy

E

as a function of the number of iterations. In all cases

L = 4, P = 4, λ = 1, κ = 0.4

Fig.A1 Bit-flip noise and its error mitigation. In (a?d) the dependence of the cost function, the fidelity, the real and the imaginary components of the energy with the iteration count are shown respectively. In all cases

L = 4,

P = 4, λ = 1, κ = 0.4

Fig.A2 Phase-flip noise and its error mitigation. All other setups are the same as .

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