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Journal of Physics A Mathematical and General

Product formula methods for time-dependent Schrodinger problems

J. Huyghebaert, Hans A. De Raedt

Published Dec 21, 1990• 76 citations
Publisher

Abstract

The authors introduce a family of explicit and unconditionally stable algorithms for solving linear differential equations which contain a time-dependent Hermitian operator. Rigorous upper bounds are derived for two different 'time-ordered' approximation schemes and for errors resulting from approximating a time-ordered exponential by an ordinary exponential operator. The properties and the usefulness of the product formula algorithms are examined by applying them to the problem of Zener tunnelling. The most efficient algorithm is employed to solve the time-dependent Schrodinger equation for a wavepacket incident on a time-modulated rectangular barrier.

DOI
10.1088/0305-4470/23/24/019

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The authors introduce a family of explicit and unconditionally stable algorithms for solving linear differential equations which contain a time-dependent Hermitian operator. Rigorous upper bounds are derived for two different 'time-ordered' approximation schemes and for errors resulting from approximating a time-ordered exponential by an ordinary exponential operator. The properties and the usefulness of the product formula algorithms are examined by applying them to the problem of Zener tunnelling. The most efficient algorithm is employed to solve the time-dependent Schrodinger equation for a wavepacket incident on a time-modulated rectangular barrier.
We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrödinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time-displacement operator in every order of approximation. We study the usefulness of the first- and second-order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two-state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time-dependent force constant.
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 Trotter product formulas constitute a cornerstone quantum Hamiltonian simulation technique. However, the efficient implementation of Hamiltonian evolution of nested commutators remains an under explored area.In this work, we construct optimized product formulas of orders 3 to 6 approximating the exponential of a commutator of two arbitrary operators in terms of the exponentials of the operators involved. The new schemes require a reduced number of exponentials and thus provide more efficient approximations than other previouslypublished alternatives. They can also be used as basic methods in recursive procedures to increasethe order of approximation.We expect this research will improve the efficiency of quantum control protocols, as well as quantum algorithms such as the Zassenhaus-based product formula, Magnus operator-based time-dependent simulation, and product formula schemes with modified potentials.
We provide a recursive method for systematically constructing product formula approximations to exponentials of commutators, giving approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate unitary exponentials of (possibly nested) commutators using exponentials of the elementary operators, and we upper bound the number of elementary exponentials needed to implement the desired operation within a given error tolerance. By presenting an algorithm for quantum search using evolution according to a commutator, we show that the scaling of the number of exponentials in our product formulas with the evolution time is nearly optimal. Finally, we discuss applications of our product formulas to quantum control and to implementing anticommutators, providing new methods for simulating many-body interaction Hamiltonians.
This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs). Product formulas are used to simulate Hamiltonians which can be expressed as a sum of terms which can each be simulated individually. By simulating each of these terms in sequence, the net effect approximately simulates the total Hamiltonian. We find that the error of product formulas can be improved by randomizing over the order in which the Hamiltonian terms are simulated. We prove that this approach is asymptotically better than ordinary product formulas and present numerical comparisons for small numbers of qubits. The ODE algorithm applies to the initial value problem for time-independent first order linear ODEs. We approximate the propagator of the ODE by a truncated Taylor series, and we encode the initial value problem in a large linear system. We solve this linear system with a quantum linear system algorithm (QLSA) whose output we perform a post-selective measurement on. The resulting state encodes the solution to the initial value problem. We prove that our algorithm is asymptotically optimal with respect to several system parameters. The PDE algorithms apply the finite difference method (FDM) to Poisson's equation, the wave equation, and the Klein-Gordon equation. We use high order FDM approximations of the Laplacian operator to develop linear systems for Poisson's equation in cubic volumes under periodic, Neumann, and Dirichlet boundary conditions. Using QLSAs, we output states encoding solutions to Poisson's equation. We prove that our algorithm is exponentially faster with respect to the spatial dimension than analogous classical algorithms. We also consider how high order Laplacian approximations can be used for simulating the wave and Klein-Gordon equations. We consider under what conditions it suffices to use Hamiltonian simulation for time evolution, and we propose an algorithm for these cases that uses QLSAs for state preparation and post-processing.