Theory of variational quantum simulation

Xiao Yuan1, Suguru Endo1, Qi Zhao2, Ying Li3, and Simon C. Benjamin1

1Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom
2Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing 100084, China
3Graduate School of China Academy of Engineering Physics, Beijing 100193, China

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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.

Universal quantum computers will eventually solve various classically intractable problems, but the exciting challenge is to demonstrate the first real quantum advantage as soon as possible -- with NISQ (for Noisy Intermediate Scaled Quantum) devices. In this regime, we may only be able to manipulate hundreds or thousands of qubits and the operations will be imperfect (or 'noisy'). With such a limited noisy quantum computer, it is unclear how to demonstrate any quantum advantage in any practical task.

This work solves this problem by exploring hybrid algorithms that only solve the core challenging problem with the quantum hardware and the higher level problem with a classical computer. This can be called the quantum coprocessor model: the quantum device handles only the bits that the conventional computer cannot. By considering different variational principles, we show how to simulate real and imaginary time dynamics of closed and open systems. Our work can thus be applied for solving static problems or simulating the dynamics of chemistry and general many-body physics with near-term quantum computers. These are tasks that, until recently, would have been thought to need a full scale fault-tolerant quantum computer in the more distant future.

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