Eigenpath traversal by Poisson-distributed phase randomisation

TL;DR

This paper introduces a quantum computation framework based on the quantum Zeno effect and Poisson-distributed phase randomisation, providing optimal bounds for algorithm complexity and applications to Grover's search and linear systems.

Contribution

It develops a novel quantum computation approach using randomised dephasing and eigenstate filtering, achieving optimal complexity bounds with minimal problem-specific insights.

Findings

Achieves $O(1/\Delta_m)$ time complexity bounds for eigenspace tracking.

Provides optimal scaling results for Grover's algorithm and quantum linear systems.

Derives a differential equation for fidelity to analyze algorithm performance.

Abstract

We present a framework for quantum computation, similar to Adiabatic Quantum Computation (AQC), that is based on the quantum Zeno effect. By performing randomised dephasing operations at intervals determined by a Poisson process, we are able to track the eigenspace associated to a particular eigenvalue. We derive a simple differential equation for the fidelity, leading to general theorems bounding the time complexity of a whole class of algorithms. We also use eigenstate filtering to optimise the scaling of the complexity in the error tolerance $\epsilon$. In many cases the bounds given by our general theorems are optimal, giving a time complexity of $O(1/\Delta_m)$ with $\Delta_m$ the minimum of the gap. This allows us to prove optimal results using very general features of problems, minimising the problem-specific insight necessary. As two applications of our framework, we obtain optimal scaling for the Grover problem (i.e.\ $O(\sqrt{N})$ where $N$ is the database size) and the Quantum Linear System Problem (i.e.\ $O(\kappa\log(1/\epsilon))$ where $\kappa$ is the condition number and $\epsilon$ the error tolerance) by direct applications of our theorems.