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Dynamics Simulation of Arbitrary Non-Hermitian Systems Based on Quantum Monte Carlo

Xiaogang Li, Kecheng Liu, Qiming Ding

TL;DR

This algorithm combines the advantages of both classical and quantum computation and exhibits good applicability and adaptability, making it promising for simulating arbitrary non-Hermitian systems such as PT-symmetric systems, non-physical processes, and open quantum systems.

Abstract

Non-Hermitian quantum systems exhibit unique properties and hold significant promise for diverse applications, yet their dynamical simulation poses a particular challenge due to intrinsic openness and non-unitary evolution. Here, we introduce a hybrid classical-quantum algorithm based on Quantum Monte Carlo (QMC) for simulating the dynamics of arbitrary time-dependent non-Hermitian systems. Notably, this approach constitutes a natural extension of the quantum imaginary-time evolution (QITE) algorithm. This algorithm combines the advantages of both classical and quantum computation and exhibits good applicability and adaptability, making it promising for simulating arbitrary non-Hermitian systems such as PT-symmetric systems, non-physical processes, and open quantum systems. To validate the algorithm, we applied it to the dynamic simulation of open quantum systems and achieved the desired results.

Dynamics Simulation of Arbitrary Non-Hermitian Systems Based on Quantum Monte Carlo

TL;DR

This algorithm combines the advantages of both classical and quantum computation and exhibits good applicability and adaptability, making it promising for simulating arbitrary non-Hermitian systems such as PT-symmetric systems, non-physical processes, and open quantum systems.

Abstract

Non-Hermitian quantum systems exhibit unique properties and hold significant promise for diverse applications, yet their dynamical simulation poses a particular challenge due to intrinsic openness and non-unitary evolution. Here, we introduce a hybrid classical-quantum algorithm based on Quantum Monte Carlo (QMC) for simulating the dynamics of arbitrary time-dependent non-Hermitian systems. Notably, this approach constitutes a natural extension of the quantum imaginary-time evolution (QITE) algorithm. This algorithm combines the advantages of both classical and quantum computation and exhibits good applicability and adaptability, making it promising for simulating arbitrary non-Hermitian systems such as PT-symmetric systems, non-physical processes, and open quantum systems. To validate the algorithm, we applied it to the dynamic simulation of open quantum systems and achieved the desired results.

Paper Structure

This paper contains 10 sections, 2 theorems, 49 equations, 3 figures, 1 algorithm.

Table of Contents

  1. The kernel functions
  2. Proof of Theorem \ref{['theorem_O_QMC']}
  3. Proof of Theorem \ref{['theorem_O_QMC_error']} (Error and complexity analysis)
  4. Truncation error, quadrature error and the query complexity
  5. Statistical error and the sampling complexity
  6. The subroutine for Hamiltonian simulation without time discretization error
  7. The measurements of $N(O)$ and $D$
  8. The isomorphic relation between open quantum systems and non-Hermitian systems with vectorization
  9. The variance bound
  10. Proof of $L_i\geqslant0$ for any normal jump operators

Key Result

Theorem 1

Given an initial state $|\psi\rangle$, the arbitrary finite-dimensional time-dependent non-Hermitian Hamiltonian $H(t)$, a new Hamiltonian $\tilde{H}(s)=H_r(s)-i(H_i(s)-{E_i}_0(s))$ with a manually chosen time-dependent quantity ${E_i}_0(s)$, the observable $O=\sum_n o_n O_n$, then the estimation $\ Here $\langle [\cdot](k',k)\rangle\equiv$$\langle\psi|\mathcal{\overline{T}} e^{i \int_0^T K_{s'}(k

Figures (3)

  • Figure 1: Hybrid classical-quantum algorithm for simulating arbitrary time-dependent non-Hermitian systems based on Quantum Monte Carlo. $X$ ($Y$) denotes the measurement basis, which is related to the real (imaginary) part of estimated value.
  • Figure 2: The 4-qubit transverse Ising model with periodic boundary conditions subjected to dissipation (amplitude damping noise). For convenience, the dissipation only acts on the first qubit.
  • Figure 3: Dynamics simulation of a 4-qubit transverse Ising model with periodic boundary conditions subjected to dissipation. The related parameters in the model are $h=2, J=1$, the initial state $|\psi\rangle=|1000\rangle$, the population observable is $O=|1000\rangle\langle1000|$, and the jump operator of amplitude damping noise $\Gamma=\sqrt{\gamma}|0\rangle_1\langle1|$, where $\gamma=1.5$. (a) The relation between population and time $t$. The simulation duration is $T=2$. The time step of both Trotter, qDrift subroutines is $\Delta t=0.05$, while for HSWDE, the minimum rotation angle (time step) is $\tau=0.05$, the compensation coefficient $c_p=0.3607$, the number of samples $N_s=10^5$ in the HSWDE subroutine. (b) The relation between the absolute error and time $t$. The vertical axis scale is logarithmic.The codes are available in xiaogang_2025_17309084.

Theorems & Definitions (7)

  • Theorem 1: Quantum-Classical Monte Carlo Estimator
  • proof
  • Theorem 2
  • proof
  • proof
  • proof
  • Definition 1