Quantum Algorithms for Estimating Gauss Sums and Calculating Discrete Logarithms

An efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings is presented. Also a quantum algorithm for the discrete logarithm problem is described, which is different from Shor’s algorithm. A reduction from the discrete logarithm problem to Gauss sum estimation gives evidence that the latter is classically a hard problem. A crucial ingredient of the algorithms is a quantum state that needs to be constructed before we can perform the computation. After one copy of this state is created, the algorithm can be executed arbitrarily many times.