The Efficiency of Quantum Identity Testing of Multiple States

TL;DR

This paper analyzes the efficiency and optimality of quantum identity tests, specifically the Permutation and Circle Tests, extending the Swap Test, and introduces semi-classical protocols for approximation.

Contribution

It establishes the optimality of the Permutation and Circle Tests and proposes semi-classical methods to efficiently approximate these tests for multiple quantum states.

Findings

Permutation Test is optimal for any input size n.

Circle Test is optimal for three input states.

A semi-classical protocol approximates the Circle Test efficiently.

Abstract

We examine two quantum operations, the Permutation Test and the Circle Test, which test the identity of n quantum states. These operations naturally extend the well-studied Swap Test on two quantum states. We first show the optimality of the Permutation Test for any input size n as well as the optimality of the Circle Test for three input states. In particular, when n=3, we present a semi-classical protocol, incorporated with the Swap Test, which approximates the Circle Test efficiently. Furthermore, we show that, with help of classical preprocessing, a single use of the Circle Test can approximate the Permutation Test efficiently for an arbitrary input size n.