Theory of variational quantum simulation

By Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, Simon C. Benjamin. Quantum, 2018

530 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/983e2dfe563857d8c16116c3e71b721941af1e78

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Citations in the corpus

Error mitigation with Clifford quantum-circuit data (2020, background): “[11] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin, “Theory of variational quantum simulation,” Quantum 3, 191 (2019).”
Variational Quantum Linear Solver (2019, background): “Other VHQCAs have been proposed for chemistry [20–23], simulation [24–28], data compression [29], state diagonalization [30–32], compiling [33, 34], quantum foundations [35], fidelity estimation [36], and metrology [37].”
Noise-induced barren plateaus in variational quantum algorithms (2020, background): “By keeping the quantum circuit depth relatively short, VQAs mitigate hardware noise and may enable near-term applications including electronic structure [5–8], dynamics [9– 12], optimization [13–16], linear systems [17, 18], metrology [19, 20], factoring [21], compiling [22–24], and others [25–30].”
The future of quantum computing with superconducting qubits (2022, methodology): “Heuristic quantum algorithms can be employed near-term to solve classical optimization 98 , machine learning 99 , and quantum simulation 100 problems.”
Variational quantum algorithms (2020, methodology): “U(θ)|ψ0〉 for some initial state |ψ0〉), are sometimes used in sophisticated optimization algorithms [71–73] (see Section IID) and variational quantum simulation [74–76]”
Connecting ansatz expressibility to gradient magnitudes and barren plateaus (2021, background): “[19] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin, “Theory of variational quantum simulation,” Quantum 3, 191 (2019).”
Quantum Natural Gradient (2019, background): “[27] appeared which considers general evolution of variational density matrices in both real and imaginary time, from a different perspective.”
Programmable quantum simulations of spin systems with trapped ions (2019, methodology): “There are several ways a time evolution or adiabatic ramp has been approximated by discrete operations in a DQS, Trotterized evolution (Suzuki, 1985; Trotter, 1959) or variational approximations (Yuan et al., 2019), in particular the Variational Quantum Eig solver (VQE) (Pe uzzo et al.”