Deterministic Discrimination of Phase-Modified Permutation Oracles via Single Qubit Measurement

TL;DR

The promise is intrinsically quantum, since the two cases differ only in their relative-phase structure and therefore have no direct classical counterpart in the usual black-box model.

Abstract

I study a promise problem for an unknown unitary operator acting on an $n$-qubit system. The operator is promised to take one of two forms: either it implements a fixed permutation of computational basis states, or it implements the same permutation together with a conditional sign change determined by a designated input qubit. I show that these two cases can be distinguished with certainty using a single query to the unknown operator and a measurement of only one qubit. The procedure requires no ancilla qubits and uses only $n+1$ Hadamard gates in addition to the oracle call. The promise is intrinsically quantum, since the two cases differ only in their relative-phase structure and therefore have no direct classical counterpart in the usual black-box model.

Figures (1)

• Figure 1: An example circuit implementation of the algorithm with $n=3$ and $L=2$.