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The Journal of Chemical Physics

Variational quantum simulation of time-local quantum master equations via quantum jump

Zhihao Lan, Jie Liu, Zhenyu Li, WanZhen Liang

Published Oct 9, 2025• 2 citations
Publisher

Abstract

Strong coupling and environmental memory render many open quantum systems intractable to classical computation. To overcome this barrier, we present a variational quantum algorithm capable of solving generalized form time-local quantum master equations directly on noisy intermediate-scale quantum (NISQ) processors. Our approach combines four key innovations: (1) a pair-vector stochastic Schrödinger equation to unravel time-local quantum master equations, (2) deterministic evolution projected onto parameterized circuits via McLachlan's principle and stochastic jump implementation via singular-value decomposition, (3) using Hadamard tests circuits to measure the jump rates and the element of reduced density matrices, and (4) using no-evolution sampling to sample trajectories. The resulting protocol remains resilient under realistic hardware noise. We validate it on both classical simulators and a 66-qubit superconducting processor for solving the Redfield master equation and fourth-order time-local non-Markovian master equation and find that it can successfully capture key features of complex open quantum dynamics-including non-Markovian oscillations and strong-coupling effects. Our approach establishes a practical pathway toward scalable simulations of open quantum systems in the NISQ era.

DOI
10.1063/5.0291793

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Strong coupling and environmental memory render many open quantum systems intractable to classical computation. To overcome this barrier, we present a variational quantum algorithm capable of solving generalized form time-local quantum master equations directly on noisy intermediate-scale quantum (NISQ) processors. Our approach combines four key innovations: (1) a pair-vector stochastic Schrödinger equation to unravel time-local quantum master equations, (2) deterministic evolution projected onto parameterized circuits via McLachlan's principle and stochastic jump implementation via singular-value decomposition, (3) using Hadamard tests circuits to measure the jump rates and the element of reduced density matrices, and (4) using no-evolution sampling to sample trajectories. The resulting protocol remains resilient under realistic hardware noise. We validate it on both classical simulators and a 66-qubit superconducting processor for solving the Redfield master equation and fourth-order time-local non-Markovian master equation and find that it can successfully capture key features of complex open quantum dynamics-including non-Markovian oscillations and strong-coupling effects. Our approach establishes a practical pathway toward scalable simulations of open quantum systems in the NISQ era.
We present a quantum algorithm based on the generalized quantum master equation (GQME) approach to simulate open quantum system dynamics on noisy intermediate-scale quantum (NISQ) computers. This approach overcomes the limitations of the Lindblad equation, which assumes weak system-bath coupling and Markovity, by providing a rigorous derivation of the equations of motion for any subset of elements of the reduced density matrix. The memory kernel resulting from the effect of the remaining degrees of freedom is used as input to calculate the corresponding non-unitary propagator. We demonstrate how the Sz.-Nagy dilation theorem can be employed to transform the non-unitary propagator into a unitary one in a higher-dimensional Hilbert space, which can then be implemented on quantum circuits of NISQ computers. We validate our quantum algorithm as applied to the spin-boson benchmark model by analyzing the impact of the quantum circuit depth on the accuracy of the results when the subset is limited to the diagonal elements of the reduced density matrix. Our findings demonstrate that our approach yields reliable results on NISQ IBM computers.
Variational quantum algorithms have been proposed to solve static and dynamic problems of closed many-body quantum systems. Here we investigate variational quantum simulation of three general types of tasks-generalized time evolution with a non-Hermitian Hamiltonian, linear algebra problems, and open quantum system dynamics. The algorithm for generalized time evolution provides a unified framework for variational quantum simulation. In particular, we show its application in solving linear systems of equations and matrix-vector multiplications by converting these algebraic problems into generalized time evolution. Meanwhile, assuming a tensor product structure of the matrices, we also propose another variational approach for these two tasks by combining variational real and imaginary time evolution. Finally, we introduce variational quantum simulation for open system dynamics. We variationally implement the stochastic Schrödinger equation, which consists of dissipative evolution and stochastic jump processes. We numerically test the algorithm with a 6-qubit 2D transverse field Ising model under dissipation.
Classical simulations of real-space quantum dynamics are challenging due to the exponential scaling of computational cost with system dimensions. Quantum computers offer the potential to simulate quantum dynamics with polynomial complexity; however, existing quantum algorithms based on the split-operator techniques require large-scale fault-tolerant quantum computers that remain elusive in the near future. Here, we present variational simulations of real-space quantum dynamics suitable for implementation in noisy intermediate-scale quantum (NISQ) devices. The Hamiltonian is first encoded onto qubits using a discrete variable representation and binary encoding scheme. We show that direct application of a real-time variational quantum algorithm based on the McLachlan's principle is inefficient as the measurement cost grows exponentially with the qubit number for general potential energy, and an extremely small time-step size is required to achieve accurate results. Motivated by the insights that many chemical dynamics occur in the low-energy subspace, we propose a subspace expansion method by projecting the total Hamiltonian, including the time-dependent driving field, onto the system low-energy eigenstate subspace using quantum computers, and the exact quantum dynamics within the subspace can then be solved classically. We show that the measurement cost of the subspace approach grows polynomially with dimensionality for general potential energy. Our numerical examples demonstrate the capability of our approach, even under intense laser fields. Our work opens the possibility of simulating chemical dynamics with NISQ hardware.
Quantum computing holds great potential for simulating quantum systems, such as molecular systems, due to its inherent ability to represent and manipulate quantum states. However, simulating nonunitary dynamics of open quantum systems on a quantum computer is still challenging. Here, we present a scheme denoted as VQS-QJ-LTLME, which adopts the trajectory average of quantum jump Monte Carlo wave function variational evolution to evolve the system's density matrix and local thermalizing Lindblad operators to describe the system-environment interactions. This combination allows the quantum circuit to be initialized after a period of evolution, thereby eliminating the accumulation of errors. The VQS-QJ-LTLME algorithm requires only log_2 qubits for system size nnn and holds a time complexity of O(Tn^3), making it particularly suitable for operation on noisy intermediate-scale quantum devices. To accelerate the Monte Carlo sampling process in VQS-QJ-LTLME, we introduce an efficient sampling method, named No-evolution sampling (NES). The VQS-QJ-LTLME aided by NES requires simulating only (n^3+1) trajectories on quantum computers, and then makes 10^6 ~ 10^7 samplings on classical computers in a few seconds. To demonstrate the performance of the VQS-QJ-LTLME algorithm, we simulate the dynamics of a two-level spin-boson model and a four-level reduced Fenna-Matthews-Olson system with real superconducting quantum computers and classical simulators. It is shown that the simulations produced by VQS-QJ-LTLME algorithm align closely with those obtained from the purely classical methods.
Quantum computing offers promising new avenues for tackling the long-standing challenge of simulating the quantum dynamics of complex chemical systems, particularly open quantum systems coupled to external baths. However, simulating such nonunitary dynamics on quantum computers is challenging since quantum circuits are specifically designed to carry out unitary transformations. Furthermore, chemical systems are often strongly coupled to the surrounding environment, rendering the dynamics non-Markovian and beyond the scope of Markovian quantum master equations like Lindblad or Redfield. In this work, we introduce a quantum algorithm designed to simulate non-Markovian dynamics of open quantum systems. Our approach enables the implementation of arbitrary quantum master equations on noisy intermediate-scale quantum (NISQ) computers. We illustrate the method as applied in conjunction with the numerically exact hierarchical equations of motion (HEOM) method. The effectiveness of the resulting quantum HEOM algorithm is demonstrated as applied to simulations of the non-Lindbladian electronic energy and charge transfer dynamics in models of the carotenoid-porphyrin-C 60 molecular triad dissolved in tetrahydrofuran and the Fenna–Matthews–Olson complex.
Quantum simulation of open quantum systems in the noisy intermediate-scale quantum (NISQ) era is hindered by the non-unitary nature of dissipative dynamics and the limited quantum resources available on near-term quantum processors. In this work, we propose a resource-efficient algorithm for simulating Lindbladian dynamics on NISQ devices. For open quantum systems with Pauli dissipations, we first derive a compact and stable mixed-unitary adjoint channel that approximates the target dissipative dynamics and enables ancilla-free implementation through trajectory sampling. To further reduce the circuit depth required for implementing the sampled trajectories, we introduce an adaptive variational quantum trajectory compression framework. In this framework, a depth-adaptive parameterized quantum circuit is trained to approximate repeated Trotterized Hamiltonian simulation operators, which are then used to replace repeated unitary segments appearing in the sampled trajectories. Importantly, the training procedure can also be performed without auxiliary qubits. Numerical simulations of the dissipative quantum XYXYXY model demonstrate the accuracy and resource efficiency of the proposed algorithm. Our results provide a practical route toward ancilla-free and depth-reduced simulation of open quantum systems on near-term quantum hardware.