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arXiv:2104.07823 · PRX Quantum

Digital Quantum Simulation of Open Quantum Systems Using Quantum Imaginary–Time Evolution

Hirsh Kamakari, Shi-Ning Sun, Mário Motta, Austin J. Minnich

Published Feb 4, 2022• 141 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/prxquantum.3.010320

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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.
Digital quantum computers have the potential to simulate complex quantum systems. The spin-boson model is one of such systems, used in disparate physical domains. Importantly, in a number of setups, the spin-boson model is open, i.e., the system is in contact with an external environment which can, for instance, cause the decay of the spin state. Here, we study how to simulate such open quantum dynamics in a digital quantum computer, for which we use an IBM hardware. We consider in particular how accurate different implementations of the evolution result as a function of the level of noise in the hardware and of the parameters of the open dynamics. For the regimes studied, we show that the key aspect is to simulate the unitary portion of the dynamics, while the dissipative part can lead to a more noise-resistant simulation. We consider both a single spin coupled to a harmonic oscillator, and also two spins coupled to the oscillator. In the latter case, we show that it is possible to simulate the emergence of correlations between the spins via the oscillator.
Accurate simulation of the time evolution of a quantum system under the influence of an environment is critical to making accurate predictions in chemistry, condensed-matter physics, and materials sciences. Whereas there has been a recent surge in interest in quantum algorithms for the prediction of nonunitary time evolution in quantum systems, few studies offer a direct quantum analog to the Lindblad equation. Here, we present a quantum algorithm---utilizing a decomposition of nonunitary operators approach---that models dynamic processes via the unraveled Lindblad equation. This algorithm is employed to probe both a two-level system in an amplitude damping channel as well as the transverse field Ising model in a variety of parameter regimes; the resulting population dynamics demonstrate excellent agreement with classical simulation, showing the promise of predicting population dynamics utilizing quantum devices for a variety of important systems in molecular energy transport, quantum optics, and other open quantum systems.
Quantum simulation of dynamics in open quantum systems is crucial but poses a significant challenge due to the non-Hermitian nature leading to nonunitary evolution and the limited quantum resources on current quantum computers. Here we introduce a variational hybrid quantum-classical algorithm designed for simulating the time evolution governed by the Lindblad master equation. Our approach involves on a stochastic unveiling of the density matrix, transforming the Lindblad equation into a wave function-based quantum state diffusion (QSD) method with the aim of reducing qubit requirements. We then apply variational quantum simulation (VQS) to efficiently capture the nonunitary evolution in QSD. We demonstrate our QSD-VQS algorithm by investigating the quantum dynamics in a two-level system subjected to an amplitude damping channel and a four-level transverse field Ising model within a dissipative environment including time-independent and periodic Hamiltonian cases. The results reveal its promising utility with upcoming hardware in the near future.
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.
Designing quantum algorithms for simulating quantum systems has seen enormous progress, yet few studies have been done to develop quantum algorithms for open quantum dynamics despite its importance in modeling the system-environment interaction found in most realistic physical models. In this work we propose and demonstrate a general quantum algorithm to evolve open quantum dynamics on quantum computing devices. The Kraus operators governing the time evolution can be converted into unitary matrices with minimal dilation guaranteed by the Sz.-Nagy theorem. This allows the evolution of the initial state through unitary quantum gates, while using significantly less resource than required by the conventional Stinespring dilation. We demonstrate the algorithm on an amplitude damping channel using the IBM Qiskit quantum simulator and the IBM Q 5 Tenerife quantum device. The proposed algorithm does not require particular models of dynamics or decomposition of the quantum channel, and thus can be easily generalized to other open quantum dynamical models.
The imaginary-time evolution of quantum states is integral to various fields, ranging from natural sciences to classical optimization or machine learning. Since simulating quantum imaginary-time evolution generally requires storing an exponentially large wave function, quantum computers are emerging as a promising platform for this task. However, variational approaches, suitable for near-term quantum computers, struggle with a prohibitive number of measurements and impractical runtimes for relevant system sizes. Here, we suggest a stochastic approach to variational quantum imaginary-time evolution, which allows a significant reduction in runtimes. Our approach allows trading off invested resources and accuracy, which makes it also suitable for ground state preparation, where simulating the exact dynamics is not required. We demonstrate the efficiency of our algorithm in simulations and show a hardware experiment performing the imaginary-time evolution of the transverse field Ising model on 27 qubits.