The efficiency of quantum identity testing of multiple states

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 the help of classical preprocessing, a single use of the circle test can approximate the permutation test efficiently for an arbitrary input size n.