How to Sample From The Limiting Distribution of a Continuous-Time Quantum Walk

TL;DR

This paper introduces $ ext{ extepsilon}$-projectors to enable efficient exact sampling from the limiting distribution of continuous-time quantum walks, significantly improving over traditional methods in certain settings.

Contribution

The authors develop $ ext{ extepsilon}$-projectors that allow exact sampling from quantum walk distributions, reducing runtime from exponential to polynomial in specific scenarios.

Findings

Exact sampling achieved using $ ext{ extepsilon}$-projectors

Algorithm runs in time proportional to inverse eigenvalue gap in black-box setting

Exponential speedup demonstrated for certain graph classes

Abstract

We introduce $\varepsilon$-projectors, using which we can sample from limiting distributions of continuous-time quantum walks. The standard algorithm for sampling from a distribution that is close to the limiting distribution of a given quantum walk is to run the quantum walk for a time chosen uniformly at random from a large interval, and measure the resulting quantum state. This approach usually results in an exponential running time. We show that, using $\varepsilon$-projectors, we can sample exactly from the limiting distribution. In the black-box setting, where we only have query access to the adjacency matrix of the graph, our sampling algorithm runs in time proportional to $\Delta^{-1}$, where $\Delta$ is the minimum spacing between the distinct eigenvalues of the graph. In the non-black-box setting, we give examples of graphs for which our algorithm runs exponentially faster than the standard sampling algorithm.