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arXiv:2001.02552 · Chinese Physics Letters

Variational Quantum Algorithms for the Steady States of Open Quantum Systems

Huan-Yu Liu, Tai-Ping Sun, Yu-Chun Wu, Guo-Ping Guo

Published Sep 1, 2021• 14 citations
arXivHTMLPDF

Abstract

The solutions of the problems related to open quantum systems have attracted considerable interest. We propose a variational quantum algorithm to find the steady state of open quantum systems. In this algorithm, we employ parameterized quantum circuits to prepare the purification of the steady state and define the cost function based on the Lindblad master equation, which can be efficiently evaluated with quantum circuits. We then optimize the parameters of the quantum circuit to find the steady state. Numerical simulations are performed on the one-dimensional transverse field Ising model with dissipative channels. The result shows that the fidelity between the optimal mixed state and the true steady state is over 99%. This algorithm is derived from the natural idea of expressing mixed states with purification and it provides a reference for the study of open quantum systems.

DOI
10.1088/0256-307x/38/8/080301

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The solutions of the problems related to open quantum systems have attracted considerable interest. We propose a variational quantum algorithm to find the steady state of open quantum systems. In this algorithm, we employ parameterized quantum circuits to prepare the purification of the steady state and define the cost function based on the Lindblad master equation, which can be efficiently evaluated with quantum circuits. We then optimize the parameters of the quantum circuit to find the steady state. Numerical simulations are performed on the one-dimensional transverse field Ising model with dissipative channels. The result shows that the fidelity between the optimal mixed state and the true steady state is over 99%. This algorithm is derived from the natural idea of expressing mixed states with purification and it provides a reference for the study of open quantum systems.
Non-equilibrium steady states are a focal point of research in the study of open quantum systems. Previous variational algorithms for searching these steady states have suffered from resource-intensive implementations due to vectorization or purification of the system density matrix, requiring large qubit resources and long-range coupling. In this work, we present a novel variational quantum algorithm that efficiently searches for non-equilibrium steady states by simulating the operator-sum form of the Lindblad equation. By introducing the technique of random measurement, we are able to estimate the nonlinear cost function while reducing the required qubit resources by half compared to previous methods. Additionally, we prove the existence of the parameter shift rule in our variational algorithm, enabling efficient updates of circuit parameters using gradient-based classical algorithms. To demonstrate the performance of our algorithm, we conduct simulations for dissipative quantum transverse Ising and Heisenberg models, achieving highly accurate results. Our approach offers a promising solution for effectively addressing non-equilibrium steady state problems while overcoming computational limitations and implementation challenges.
We propose a quantum-classical hybrid algorithm to simulate the nonequilibrium steady state of an open quantum many-body system, named the dissipative-system variational quantum eigensolver (dVQE).To employ the variational optimization technique for a unitary quantum circuit, we map a mixed state into a pure state with a doubled number of qubits and design the unitary quantum circuit to fulfill the requirements for a density matrix.This allows us to define a cost function that consists of the time evolution generator of the quantum master equation.Evaluation of physical observables is, in turn, carried out by a quantum circuit with the original number of qubits.We demonstrate our dVQE scheme by both numerical simulation on a classical computer and actual quantum simulation that makes use of the device provided in Rigetti quantum cloud service.
We propose a variational quantum algorithm named variational open quantum eigensolver (VOQE) for solving steady states of open quantum systems described by either Lindblad master equations or non-Hermitian Hamiltonians. In VOQE, density matrices of mixed states are represented by pure states in doubled Hilbert space. We give a framework for building a circuit ansatz which we call the Hermitian-preserving ansatz to restrict the searching space. We also give a method to efficiently measure the operators' expectation values by postselection measurements. We show the workflow of VOQE on solving steady states of the Lindblad master equations of the driven XXZ model and implement VOQE to solve the spectrum of the non-Hermitian Hamiltonians of the Ising spin chain in an imaginary field.
The simulation of many-body open quantum systems is key to solving numerous outstanding problems in physics, chemistry, material science, and in the development of quantum technologies. Near-term quantum computers may bring considerable advantage for the efficient simulation of their static and dynamical properties, thanks to hybrid quantum-classical variational algorithms to approximate the dynamics of the density matrix describing the quantum state in terms of an ensemble average. Here, a variational quantum algorithm is developed to simulate the real-time evolution of the density matrix governed by the Lindblad master equation, under the assumption that the quantum state has a bounded entropy along the dynamics, entailing a low-rank representation of its density matrix. The algorithm encodes each pure state of the statistical mixture as a parametrized quantum circuit, and the associated probabilities as additional variational parameters stored classically, thereby requiring a significantly lower number of qubits than algorithms where the full density matrix is encoded in the quantum memory. Two variational ansatze are proposed, and their effectiveness is assessed in the simulation of the dynamics of a 2D dissipative transverse field Ising model. The results underscore the algorithm's efficiency in simulating the dynamics of open quantum systems in the low-rank regime with limited quantum resources on a near-term quantum device.
We investigate dynamical quantum phase transitions (DQPTs) in Markovian open quantum systems using a variational quantum simulation (VQS) algorithm based on quantum state diffusion (QSD). This approach reformulates the Lindblad master equation as an ensemble of pure-state trajectories, enabling efficient simulation of dissipative quantum dynamics with effectively reduced quantum resources. Focusing on the one-dimensional transverse-field Ising model (TFIM), we simulate quench dynamics under both local and global Lindblad dissipation. The QSD-VQS algorithm accurately captures the nonanalytic cusps in the Loschmidt rate function, and reveals their modulation by dissipation strength and system size. Notably, DQPTs are gradually suppressed under strong local dissipation, while they persist under strong global dissipation due to collective environmental effects. Benchmarking against exact Lindblad solutions confirms the high accuracy and scalability of our method.
Variational quantum algorithms have emerged as a promising approach for simulating open quantum systems, yet their expressive power and trainability remain poorly understood. We propose a variational framework for solving Lindblad master equations by minimizing a residual cost function using a compositional ansatz obtained by combining a classical neural network with a parameterized quantum circuit. The density matrix is represented through a purification on an enlarged Hilbert space, ensuring positivity, Hermiticity, and unit trace by construction, while all gradients are evaluated exactly via the parameter-shift rule. Beyond the algorithmic formulation, we derive an upper bound for the covering number of the proposed compositional model and compare it with that of purely quantum neural networks. Our analysis shows that the classical parametrization enlarges the effective hypothesis class while providing finer control over its complexity, thereby establishing a theoretical connection between expressivity, covering numbers, effective dimension, and the barren plateau phenomenon. Numerical experiments validate both the accuracy of the proposed method for representative Lindblad master equations and the theoretical predictions regarding gradient concentration. These results provide a unified variational and theoretical framework for the simulation of dissipative quantum dynamics using compositional quantum machine learning models.