The Efficiency of Quantum Identity Testing of Multiple States

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.