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

The Efficiency of Quantum Identity Testing of Multiple States