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Computer Physics Communications

Numerical errors resulting from finite-difference approximation in computations of a one-dimensional Schrödinger equation with the Trotter formula

Norio Inui

Published Jul 12, 2007• 1 citation
Publisher
DOI
10.1016/j.cpc.2007.06.014

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We study algorithm to calculate the time dependent Schrödinger equation numerically and to visualize the results. We extend the product formula by the Trotter-Suzuki decomposition to cases of Hamiltonians with vector potentials. This method is unconditionally stable and it can be used for a time dependent Hamiltonian so that one can apply it to various problems without any difficulty. After examining numerical errors for time evolutions of one dimensional wave packets, we present the errors under a constant magnetic field. Also we discuss a cyclotron motion in quantum mechanics, where an advantage for the unconditional stability is demonstrated. In order to visualize the results we make animations of the time evolutions, where the phase of the wave function is represented by colors and its absolute value is shown by brightness. Here we suggest a new way to represent a complex number by colors.
Abstract. Known as the no fast-forwarding theorem in quantum computing (see, e.g., Theorem 3 in [D. W. Berry et al., Comm. Math. Phys., 270 (2007), pp. 359–371], Theorem 5 in [A. M. Childs, Comm. Math. Phys., 294 (2010), pp. 581–603], and [R. Kothari, Efficient Simulation of Hamiltonians, M.S. thesis, University of Waterloo, 2010]), the simulation time for the Hamiltonian evolution is typically [Formula: see text], which essentially states that one cannot go across the multiple scales as the simulation time for the Hamiltonian evolution needs to be strictly greater than the physical time. We demonstrated in the context of the semiclassical Schrödinger equation that the computational cost for a class of observables can be much lower than the state-of-the-art bounds. In the semiclassical regime (the effective Planck constant [Formula: see text]), the operator norm of the Hamiltonian is [Formula: see text]. We show that the number of Trotter steps used for the observable evolution can be [Formula: see text], that is, to simulate some observables of the Schrödinger equation on a quantum scale only takes the simulation time comparable to the classical scale. In terms of error analysis, we improve the additive observable error bounds in [C. Lasser and C. Lubich, Acta Numer., 29 (2020), pp. 229–401] to uniform-in-[Formula: see text] observable error bounds. This is, to our knowledge, the first uniform observable error bound for the semiclassical Schrödinger equation without sacrificing the convergence order of the numerical method. Based on semiclassical calculus and discrete microlocal analysis, our result showcases the potential improvements taking advantage of multiscale properties, such as the smallness of the effective Planck constant, and of the underlying dynamics, and sheds light on going across the scale for quantum dynamics simulation.
Abstract The extent to which quantum computers can simulate physical phenomena and solve the partial differential equations (PDEs) that govern them remains a central open question. In this work, one of the most fundamental PDEs is addressed: the multidimensional transport equation with space- and time-dependent coefficients. We present a quantum numerical scheme based on three steps: quantum state preparation, evolution, and measurement of relevant observables. The evolution step combines a high-order centered finite difference with a time-splitting scheme based on product formula approximations, also known as Trotterization. We introduce a novel vector-norm analysis and prove that the number of time-steps can be reduced by a factor exponential in the number of qubits, compared with previously established operator-norm analysis. This new scaling significantly reduces the projected computational resources, independently of the circuit implementation of the trotterized evolution operator. We also present efficient quantum circuits based on (sparse) Walsh approximations along with numerical simulations that confirm the predicted vector-norm scaling. We report results on real quantum hardware for the one-dimensional convection equation, and solve a non-linear ordinary differential equation via its associated Liouville equation, a particular case of transport equations. This work provides a practical framework for efficiently simulating transport phenomena on quantum computers, with potential applications in plasma physics, molecular gas dynamics and non-linear dynamical systems, including chaotic systems.