Quantum phase discrimination with applications to quantum search on graphs

We study the phase discrimination problem, in which we want to decide whether the eigenphase $\theta\in(-\pi,\pi]$ of a given eigenstate $|\psi\rangle$ with eigenvalue $e^{i\theta}$ is zero or not, using applications of the unitary $U$ provided as a black box oracle.We propose a quantum algorithm named {\it quantum phase discrimination(QPD)} for this task, with optimal query complexity $\Theta(\frac{1}{\lambda}\log\frac{1}{\delta})$ to the oracle $U$, where $\lambda$ is the gap between zero and non-zero eigenphases and $\delta$ the allowed one-sided error. The quantum circuit is simple, consisting of only one ancillary qubit and a sequence of controlled-$U$ interleaved with single qubit $Y$ rotations, whose angles are given by a simple analytical formula. Quantum phase discrimination could become a fundamental subroutine in other quantum algorithms, as we present two applications to quantum search on graphs: i) Spatial search on graphs. Inspired by the structure of QPD, we propose a new quantum walk model, and based on them we tackle the spatial search problem, obtaining a novel quantum search algorithm. For any graph with any number of marked vertices, the quantum algorithm that can find a marked vertex with probability $\Omega(1)$ in total evolution time $O(\frac{1}{\lambda \sqrt{\varepsilon}})$ and query complexity $O(\frac{1}{\sqrt{\varepsilon}})$, where $\lambda$ is the gap between the zero and non-zero eigenvalues of the graph Laplacian and $\varepsilon$ is a lower bound on the proportion of marked vertices. ii) Path-finding on graphs.} By using QPD, we reduce the query complexity of a path-finding algorithm proposed by Li and Zur [arxiv: 2311.07372] from $\tilde{O}(n^{11})$ to $\tilde{O}(n^8)$, in a welded-tree circuit graph with $\Theta(n2^n)$ vertices. Besides these two applications, we argue that more quantum algorithms might benefit from QPD.