Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation

Abstract

We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.

Quantum simulation of real-space dynamics

“…Theorem 8). Consider an instance of the Schrödinger equation (1) for η particles in d dimensions, with a time-dependent potential f (x, t) that is bounded for any fixed t ≥ 0 and is L-Lipschitz continuous in t. Hamiltonian simulation with a rescaled Dyson series and interaction picture can produce an approximated wave function at time T on a set of grid nodes, with ℓ 2 error at most ǫ, with asymptotic gate complexity…”

“…Theorem 4 (kth-order product formula simulation of real-space dynamics). Consider an instance of the Schrödinger equation in (1) with a time-independent potential f (x) satisfying f (x) L ∞ ≤ f max and a given T > 0. Let g ′ = max t Φ (n/2) (•, t) L 1 as in (51).…”

Quantum simulation of real-space dynamics

“…Theorem 7). Consider an instance of the Schrödinger equation (1) for η particles in d dimensions, with a time-independent potential f (x) satisfying f (x) L ∞ ≤ f max . Hamiltonian simulation with a truncated Taylor series can produce an approximated wave function at time T on a set of grid nodes, with 2 error at most , with asymptotic gate complexity ηd(ηd + f max )T poly log ηdT g / , (4) where g defined in (51) upper bounds the high-order derivatives of the wave function.…”

“…Theorem 8). Consider an instance of the Schrödinger equation (1) for η particles in d dimensions, with a time-dependent potential f (x, t) that is bounded for any fixed t ≥ 0 and is L-Lipschitz continuous in t. Hamiltonian simulation with a rescaled Dyson series and interaction picture can produce an approximated wave function at time T on a set of grid nodes, with 2 error at most , with asymptotic gate complexity ηd f max,1 poly log L f max,1 g / , (5) where f max,1 := T 0 f (t) max dt measures the integrated strength of the potential f , and g defined in (51) upper bounds the high-order derivatives of the wave function.…”

“…Theorem 4 (kth-order product formula simulation of real-space dynamics). Consider an instance of the Schrödinger equation in (1) with a time-independent potential f (x) satisfying f (x) L ∞ ≤ f max and a given T > 0. Let g = max t Φ (n/2) (•, t) L 1 as in (51).…”