Quantum spatial search with multiple excitations

TL;DR

This paper extends quantum spatial search to multiple marked vertices, demonstrating that a continuous-time quantum walk can efficiently identify multiple marked sites with high fidelity, even as the state space grows significantly.

Contribution

It introduces a quantum walk approach for multiple marked vertices, showing efficient detection in the $k$-excitation subspace with practical implementation considerations.

Findings

Achieves $O(\sqrt{n})$ search time for multiple marked vertices.

Demonstrates implementation feasibility with $1/r^\alpha$ interactions.

Maintains high fidelity despite exponential growth of state space.

Abstract

Spatial search is the problem of finding a marked vertex in a graph. A continuous-time quantum walk in the single-excitation subspace of an $n$ spin system solves the problem of spatial search by finding the marked vertex in $O(\sqrt{n})$ time. Here, we investigate a natural extension of the spatial search problem, marking multiple vertices of a graph, which are still marked with local fields. We prove that a continuous-time quantum walk in the $k$-excitation subspace of $n$ spins can determine the binary string of $k$ marked vertices with an asymptotic fidelity in time $O(\sqrt{n})$, despite the size of the state space growing as $O(n^k)$. Numerically, we show that this algorithm can be implemented with interactions that decay as $1/r^\alpha$, where $r$ is the distance between spins, and an $\alpha$ that is readily available in current ion trap systems.