Optimisation of Quantum Evolution Algorithms

TL;DR

This paper explores optimizing quantum evolution algorithms by selecting trajectories that minimize complexity and errors, demonstrating significant improvements in error control and efficiency through different discretization methods.

Contribution

It introduces a framework for optimizing quantum evolution trajectories, comparing Lie-Trotter and reflection-based methods, and explains their impact on computational complexity and error management.

Findings

Power-law error dependence with Lie-Trotter discretization

Logarithmic error dependence with reflection operators

Overrelaxation algorithms outperform small step size algorithms

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. A choice among these trajectories can then be made to obtain the best computational complexity and control over errors. As an explicit example, Grover's quantum search algorithm is described as a Hamiltonian evolution problem. It is shown that the computational complexity has a power-law dependence on error when a straightforward Lie-Trotter discretisation formula is used, and it becomes logarithmic in error when reflection operators are used. The exponential change in error control is striking, and can be used to improve many importance sampling methods. The key concept is to make the evolution steps as large as possible while obeying the constraints of the problem. In particular, we can understand why overrelaxation algorithms are superior to small step size algorithms.