Abstract
The variational method is a versatile tool for classical simulation of a variety of quantum systems. Great efforts have recently been devoted to its extension to quantum computing for efficiently solving static many-body problems and simulating real and imaginary time dynamics. In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel variational principle, the McLachlan's variational principle, and the time-dependent variational principle, for simulating real time dynamics. We focus on the simulation of dynamics and discuss the connections of the three variational principles. Previous works mainly focus on the unitary evolution of pure states. In this work, we introduce variational quantum simulation of mixed states under general stochastic evolution. We show how the results can be reduced to the pure state case with a correction term that takes accounts of global phase alignment. For variational simulation of imaginary time evolution, we also extend it to the mixed state scenario and discuss variational Gibbs state preparation. We further elaborate on the design of ansatz that is compatible with post-selection measurement and the implementation of the generalised variational algorithms with quantum circuits. Our work completes the theory of variational quantum simulation of general real and imaginary time evolution and it is applicable to near-term quantum hardware.
Highlights
Variational simulation is a widely used technique in many-body physics [1,2,3,4,5,6,7] and chemistry [8,9,10]
There exist many problems that may not be solved classically even with the classical variational method [17,18,19]. This is because there exist highly entangled many-body states that may not be efficiently represented via any classical method. This problem motivates the development of quantum simulation with universal quantum computers [20], because quantum systems can be efficiently encoded or represented by another quantum system and real time evolution of the Schrodinger equation can be efficiently realised via a unitary quantum circuit [21, 22]
After applying all the three variational principles to the general mixed state case, we find that McLachlan’s variational principle is the most appropriate variational principle that leads to a consistent theory of variational quantum simulation (VQS)
Summary
Variational simulation is a widely used technique in many-body physics [1,2,3,4,5,6,7] and chemistry [8,9,10]. This is because there exist highly entangled many-body states that may not be efficiently represented via any classical method This problem motivates the development of quantum simulation with universal quantum computers [20], because quantum systems can be efficiently encoded or represented by another quantum system and real time evolution of the Schrodinger equation can be efficiently realised via a unitary quantum circuit [21, 22]. The variational methods have been generalised to simulating the real [41, 54] and imaginary [55, 56] time evolution of pure quantum systems, and been experimentally implemented with superconducting qubits [57].
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