Quadratic speedup for spatial search by continuous-time quantum walk

TL;DR

This paper introduces a new continuous-time quantum walk algorithm that achieves a quadratic speedup for spatial search in any graph with any number of marked nodes, surpassing previous limitations.

Contribution

The authors present a simple, general quantum walk algorithm that quadratically accelerates spatial search for any graph and number of marked nodes, with a novel analog procedure for Hamiltonian operations.

Findings

Achieves quadratic speedup for spatial search in any graph.

Develops an analog method to perform $e^{-tH^2}$ operations efficiently.

Enables potential applications in quantum ground state preparation and optimization.

Abstract

Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this article, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analog procedure to perform operations on a state of the form $e^{-tH^2}|\psi\rangle$, for a given Hamiltonian $H$: it only requires evolving $H$ for time scaling as $\sqrt{t}$. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our work thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.