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arXiv:2305.06915 · Quantum

Adaptive variational simulation for open quantum systems

Huo Chen, Niladri Gomes, Siyuan Niu, Wibe Albert de Jong

Published Feb 13, 2024• 24 citations
arXivHTMLPDF

Abstract

Emerging quantum hardware provides new possibilities for quantum simulation. While much of the research has focused on simulating closed quantum systems, the real-world quantum systems are mostly open. Therefore, it is essential to develop quantum algorithms that can effectively simulate open quantum systems. Here we present an adaptive variational quantum algorithm for simulating open quantum system dynamics described by the Lindblad equation. The algorithm is designed to build resource-efficient ansatze through the dynamical addition of operators by maintaining the simulation accuracy. We validate the effectiveness of our algorithm on both noiseless simulators and IBM quantum processors and observe good quantitative and qualitative agreement with the exact solution. We also investigate the scaling of the required resources with system size and accuracy and find polynomial behavior. Our results demonstrate that near-future quantum processors are capable of simulating open quantum systems.

DOI
10.22331/q-2024-02-13-1252

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Emerging quantum hardware provides new possibilities for quantum simulation. While much of the research has focused on simulating closed quantum systems, the real-world quantum systems are mostly open. Therefore, it is essential to develop quantum algorithms that can effectively simulate open quantum systems. Here we present an adaptive variational quantum algorithm for simulating open quantum system dynamics described by the Lindblad equation. The algorithm is designed to build resource-efficient ansatze through the dynamical addition of operators by maintaining the simulation accuracy. We validate the effectiveness of our algorithm on both noiseless simulators and IBM quantum processors and observe good quantitative and qualitative agreement with the exact solution. We also investigate the scaling of the required resources with system size and accuracy and find polynomial behavior. Our results demonstrate that near-future quantum processors are capable of simulating open quantum systems.
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.
Abstract We introduce variational hybrid classical-quantum algorithms to simulate the Lindblad master equation and its adjoint for time-evolving Markovian open quantum systems and quantum observables. Our methods are based on a direct representation of density matrices and quantum observables as quantum superstates. We design and optimize low-depth variational quantum circuits that efficiently capture the unitary and non-unitary dynamics of the solutions. We benchmark and test the algorithms on different models and system sizes, showing their potential for utility with near-future hardware.
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.
Simulating open quantum systems remains one of the most demanding problems in quantum dynamics, as environmental interactions lead to non-unitary evolution that challenges standard quantum simulation techniques. Variational quantum algorithms (VQAs) offer a practical way forward by combining classical optimization with quantum hardware, making them well suited for near-term devices. Here, in Part V of the QFlux implementation series, we introduce adaptive variational-ansatz methods for solving the Lindblad master equation. This tutorial presents the Python-based qmad module of the QFlux toolkit, which is designed to help students and researchers easily build and explore adaptive VQA strategies. The tutorial walks through each step of the workflow and applies the methods to two illustrative examples. The first is the amplitude-damping channel, which models spontaneous emission in a two-level system. The second is the Fenna-Matthews-Olson (FMO) complex, a prominent model for excitonic energy transfer in photosynthesis. The two workflows utilizing adaptive variational approaches implemented here are the unrestricted adaptive variational quantum dynamics (UAVQD) scheme and the stochastic Schrödinger equation (SSE) based trajectory method. By working through these examples, readers gain both intuition and hands-on experience with variational techniques for simulating dissipative quantum dynamics.
Simulating open quantum systems remains one of the most demanding problems in quantum dynamics, as environmental interactions lead to non-unitary evolution that challenges standard quantum simulation techniques. Variational quantum algorithms (VQAs) offer a practical way forward by combining classical optimization with quantum hardware, making them well suited for near-term devices. Here, in Part V of the QFlux implementation series, we introduce adaptive variational-ansatz methods for solving the Lindblad master equation. This tutorial presents the Python-based qmad module of the QFlux toolkit, which is designed to help students and researchers easily build and explore adaptive VQA strategies. The tutorial walks through each step of the workflow and applies the methods to two illustrative examples. The first is the amplitude-damping channel, which models spontaneous emission in a two-level system. The second is the Fenna-Matthews-Olson (FMO) complex, a prominent model for excitonic energy transfer in photosynthesis. The two workflows utilizing adaptive variational approaches implemented here are the unrestricted adaptive variational quantum dynamics (UAVQD) scheme and the stochastic Schrödinger equation (SSE) based trajectory method. By working through these examples, readers gain both intuition and hands-on experience with variational techniques for simulating dissipative quantum dynamics.
Quantum simulation on emerging quantum hardware is a topic of intense interest.While many studies focus on computing ground-state properties or simulating unitary dynamics of closed systems, open quantum systems are an interesting target of study owing to their ubiquity and rich physical behavior.However, their nonunitary dynamics are also not natural to simulate on digital quantum devices.Here, we report algorithms for the digital quantum simulation of the dynamics of open quantum systems governed by a Lindblad equation using adaptations of the quantum imaginary-time evolution algorithm.We demonstrate the algorithms on IBM Quantum's hardware with simulations of the spontaneous emission of a two-level system and the dissipative transverse field Ising model.Our work advances efforts to simulate the dynamics of open quantum systems on quantum hardware.