Doubling the Success of Quantum Walk Search Using Internal-State Measurements

TL;DR

Measuring the internal state of a discrete-time quantum walk significantly enhances the success probability of quantum search algorithms, enabling certain problems like Grover's search to be solved with certainty and doubling their efficiency.

Contribution

This work demonstrates that internal-state measurements in quantum walks can double success probabilities and provides conditions and methods for applying this to spatial search problems.

Findings

Measuring internal states doubles success probability in quantum walks.

Grover's search can be achieved with certainty using internal-state measurements.

Conditions identified for applying this doubling technique on regular graphs.

Abstract

In typical discrete-time quantum walk algorithms, one measures the position of the walker while ignoring its internal spin/coin state. Rather than neglecting the information in this internal state, we show that additionally measuring it doubles the success probability of many quantum spatial search algorithms. For example, this allows Grover's unstructured search problem to be solved with certainty, rather than with probability 1/2 if only the walker's position is measured, so the additional measurement yields a search algorithm that is twice as fast as without it, on average. Thus the internal state of discrete-time quantum walks holds valuable information that can be utilized to improve algorithms. Furthermore, we determine conditions for which spatial search problems on regular graphs are amenable to this doubling of the success probability, and this involves diagrammatically analyzing search using degenerate perturbation theory and deriving a useful formula for how the quantum walk acts in its reduced subspace.