Theory of variational quantum simulation

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.

arXiv comment41 pages, accepted by Quantum

References in corpus (41)

• Quantum Computing in the NISQ era and beyond
• Theory of Bose-Einstein condensation in trapped gases
• A variational eigenvalue solver on a quantum processor
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• Quantum Simulation
• QuTiP 2: A Python framework for the dynamics of open quantum systems
• The theory of variational hybrid quantum-classical algorithms
• QuTiP: An open-source Python framework for the dynamics of open quantum systems
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Error mitigation for short-depth quantum circuits
• Characterizing Quantum Supremacy in Near-Term Devices
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Extending the computational reach of a noisy superconducting quantum processor
• Time-dependent variational principle for quantum lattices
• Quantum Computational Supremacy
• Elucidating Reaction Mechanisms on Quantum Computers
• Simulation of Many-Body Fermi Systems on a Universal Quantum Computer
• Self-Verifying Variational Quantum Simulation of the Lattice Schwinger Model
• Roads towards fault-tolerant universal quantum computation
• Variational ansatz-based quantum simulation of imaginary time evolution
• Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States
• A blueprint for demonstrating quantum supremacy with superconducting qubits
• Robust determination of molecular spectra on a quantum processor
• Direct estimations of linear and non-linear functionals of a quantum state
• Universal Quantum Estimator
• Quantum Implementation of Unitary Coupled Cluster for Simulating Molecular Electronic Structure
• Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
• Variational quantum algorithms for discovering Hamiltonian spectra
• Error-mitigated digital quantum simulation
• Low-cost error mitigation by symmetry verification
• Molecular Dynamics for Fermions
• Post-Matrix Product State Methods: To tangent space and beyond
• Witnessing eigenstates for quantum simulation of Hamiltonian spectra
• Experimental Bayesian Quantum Phase Estimation on a Silicon Photonic Chip
• Review of Decoherence Free Subspaces, Noiseless Subsystems, and Dynamical Decoupling
• Quantum computation with realistic magic state factories
• Quantum Simulation of Helium Hydride in a Solid-State Spin Register
• Methodology for replacing indirect measurements with direct measurements
• Recovering noise-free quantum observables
• Generalized Hartree-Fock Theory for Interacting Fermions in Lattices: Numerical Methods
• Variational principle for quantum impurity systems in and out of equilibrium: application to Kondo problems

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