Abstract
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.
Highlights
Introduction.—The variational method is a powerful classical tool for simulating many-body quantum systems [1,2,3,4,5]
As quantum circuits can efficiently prepare states that may not be efficiently represented classically, the variational method has been recently generalized to the quantum regime with trial states efficiently prepared by a quantum circuit and information extracted from a coherent measurement of the state [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]
The trial state in variational quantum algorithms can be prepared with shallow quantum circuits [27,28,29,30], which is robust to a certain amount of device noise and is compatible with nearterm noisy intermediate scale quantum (NISQ) hardware [31]
Summary
Suguru Endo ,1,4,* Jinzhao Sun, Ying Li, Simon C. 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. We show its application in solving linear systems of equations and matrix-vector multiplications by converting these algebraic problems into generalized time evolution. Quantum circuits are unitary operations, the variational algorithm is not limited to energy minimization and unitary processes and it can be used to simulate dissipative imaginary time evolution that cannot be straightforwardly mapped to unitary gates [20,34]. Simulating the evolution of general open quantum systems is of great importance for understanding any
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