Hamiltonian simulation for hyperbolic partial differential equations by scalable quantum circuits

Abstract

Solving partial differential equations for extremely large-scale systems within a feasible computation time serves in accelerating engineering developments. Quantum computing algorithms, particularly the Hamiltonian simulations, present a potential and promising approach to achieve this purpose. Actually, there are several oracle-based Hamiltonian simulations with potential quantum speedup, but their detailed implementations and accordingly the detailed computational complexities are all unclear. This paper presents a method that enables us to explicitly implement the quantum circuit for Hamiltonian simulation; the key technique is the explicit gate construction of differential operators contained in the target partial differential equation discretized by the finite difference method. Moreover, we show that the space and time complexities of the constructed circuit are exponentially smaller than those of conventional classical algorithms. We also provide numerical experiments and an experiment on a real device for the wave equation to demonstrate the validity of our proposed method.

25 pages, 6 figures

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• Quantum circuits for partial differential equations in Fourier space
• Quantum Calculation for Two-Stream Instability and Advection Test of Vlasov-Maxwell Equations: Numerical Evaluation of Hamiltonian Simulation
• Variational Quantum Simulation of the Fokker-Planck Equation applied to Quantum Radiation Reaction
• Nonlinear path-following via the asymptotic numerical method on a quantum processor
• Gate Efficient Composition of Hamiltonian Simulation and Block-Encoding with its Application on HUBO, Chemistry and Finite Difference Method
• Quantum Framework for Simulating Linear PDEs with Robin Boundary Conditions