Abstract

Many of the most successful quantum algorithms to date have exploited the ability of the Fourier transform to capture subgroup structure. However, the ability of the Fourier transform to capture shift structure has not been explored. We present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. By reducing the problem of breaking algebraically homomorphic cryptosystems to one of these problems, we show that such cryptosystems can be broken on a quantum computer. These shift problems and the hidden subgroup problem can be cast in the framework of the hidden coset problem. This framework gives us a unified way of viewing the quantum Fourier transform's ability to capture subgroup and shift structure.

Citations