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Theory of variational quantum simulation
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.Comment: 41 pages, accepted by Quantu
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