Multi-state Swap Test Algorithm

TL;DR

This paper introduces a recursive quantum circuit that efficiently measures all pairwise overlaps among multiple quantum states simultaneously, reducing resource consumption and increasing precision compared to existing methods.

Contribution

A novel recursive quantum circuit design that measures multiple quantum state overlaps concurrently with fewer resources and higher accuracy.

Findings

Achieves $O(n ext{log} n)$ controlled-swap gates and $O( ext{log} n)$ ancillary qubits.

Provides higher precision than existing schemes.

Validated through simulations on IBM quantum cloud platform.

Abstract

Estimating the overlap between two states is an important task with several applications in quantum information. However, the typical swap test circuit can only measure a sole pair of quantum states at a time. In this study we designed a recursive quantum circuit to measure overlaps of multiple quantum states $|\phi_1...\phi_n\rangle$ concurrently with $O(n\log n)$ controlled-swap (CSWAP) gates and $O(\log n)$ ancillary qubits. This circuit enables us to get all pairwise overlaps among input quantum states $|\langle\phi_i|\phi_j\rangle|^2$. Compared with existing schemes for measuring the overlap of multiple quantum states, our scheme provides higher precision and less consumption of ancillary qubits. In addition, we performed simulation experiments on IBM quantum cloud platform to verify the superiority of the scheme.