Optimisation of Quantum Hamiltonian Evolution: From Two Projection Operators to Local Hamiltonians

TL;DR

This paper presents an optimized quantum Hamiltonian simulation method that scales efficiently by using large steps, reflections, and Chebyshev expansions, improving error control and computational complexity over traditional approaches.

Contribution

It extends Hamiltonian simulation techniques from two projection operators to general local Hamiltonians using large step evolution and advanced error management strategies.

Findings

Scales linearly in time and logarithmically in error bound

Exponential improvement over Lie-Trotter based schemes

Efficient control of total error using Chebyshev expansions

Abstract

Given a quantum Hamiltonian and its evolution time, the corresponding unitary evolution operator can be constructed in many different ways, corresponding to different trajectories between the desired end-points and different series expansions. A choice among these possibilities can then be made to obtain the best computational complexity and control over errors. It is shown how a construction based on Grover's algorithm scales linearly in time and logarithmically in the error bound, and is exponentially superior in error complexity to the scheme based on the straightforward application of the Lie-Trotter formula. The strategy is then extended first to simulation of any Hamiltonian that is a linear combination of two projection operators, and then to any local efficiently computable Hamiltonian. The key feature is to construct an evolution in terms of the largest possible steps instead of taking small time steps. Reflection operations and Chebyshev expansions are used to efficiently control the total error on the overall evolution, without worrying about discretisation errors for individual steps. We also use a digital implementation of quantum states that makes linear algebra operations rather simple to perform.