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QUANTUM ALGORITHMS FOR SOME HIDDEN SHIFT PROBLEMS

Wim van Dam, Sean Hallgren, Lawrence Ip

Published Nov 21, 2002• 52 citations
OpenAlex

Abstract

Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.

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Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.
Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.
Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structure of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of ``unknown shift'' problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.
The search for practical problems that can be solved exponentially faster on a quantum computer is one of the primary goals in research into algorithms for quantum computers. It has long been known that such speedups are only possible for highly structured problems. We consider two problems with algebraic structure: (1) the hidden subgroup problem, where we have an unknown subgroup, a function that is constant on cosets of the subgroup, and the task is to find the unknown subgroup; (2) the hidden shift problem, where we have an unknown element of a group, a function that has been shifted by that group element, and the task is to find the unknown group element. We show that the Fourier transform is intimately connected to both these problems. This is because the Fourier transform respects group symmetries. For the hidden subgroup problem, we introduce the idea of using information theoretic techniques to design quantum algorithms. This consists of first finding an optimal set of measurements for distinguishing between different subgroups, followed by attempts to implement this set of measurements. We show that the standard algorithm for the abelian hidden subgroup problem is optimal. For the nonabelian hidden subgroup problem, the optimal measurements respect the Fourier transform in that they can be implemented by first performing a Fourier transform on each coset state, measuring the name of the irreducible representation for each state, followed by a joint measurement over the resulting lower dimensional space. This shows that the “strong standard method” cannot be used to implement the optimal set of measurements. For some instances of the hidden shift problem, we give a quantum algorithms that are exponentially faster than the best known classical algorithm. The algorithms make use of the Fourier transform in a way that may be interpreted as solving a deconvolution problem. As evidence of hardness, we give a reduction from algebraically homomorphic cryptosystems to the shifted Legendre symbol problem. This is an example of a problem which admits an exponential quantum speedup yet is not an instance of the abelian hidden subgroup problem.
We give a quantum algorithm for solving a shifted multiplicative character problem over Z/nZ and finite fields. We show that the algorithm can be interpreted as a matrix factorization or as solving a deconvolution problem and give sufficient conditions for a shift problem to be solved efficiently by our algorithm. We also show that combining the shift problem with the hidden subgroup problem results in a hidden coset problem. This naturally captures the redundancy in the shift due to the periodic structure of multiplicative characters over Z/nZ.
The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor's celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitely-generated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewise-continuous function on R. Finally, we show that this problem of period-finding over R, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, period-finding over R lies outside the complexity class MA, a class which contains period-finding over the integers.