Quantum algorithms for Simon's problem over general groups

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Daniel Simon's 1994 discovery of an efficient quantum algorithm for solving the hidden subgroup problem (HSP) over Zn2 provided one of the first algebraic problems for which quantum computers are exponentially faster than their classical counterparts. In this paper, we study the generalization of Simon's problem to arbitrary groups. Fixing a finite group G, this is the problem of recovering an involution m = (m1,...,mn) e Gn from an oracle f with the property that f(x) = f(x · y) ⇔ y e {1, m}. In the current parlance, this is the hidden subgroup problem (HSP) over groups of the form Gn, where G is a nonabelian group of constant size, and where the hidden subgroup is either trivial or has order two.Although groups of the form Gn have a simple product structure, they share important representation-theoretic properties with the symmetric groups Sn, where a solution to the HSP would yield a quantum algorithm for Graph Isomorphism. In particular, solving their HSP with the so-called standard method requires highly entangled measurements on the tensor product of many coset states.Here we give quantum algorithms with time complexity 2O(√n log n) that recover hidden involutions m = (m1,..., mn) e Gn where, as in Simon's problem, each mi is either the identity or the conjugate of a known element m and there is a character X of G for which X(m) = - X(1). Our approach combines the general idea behind Kuperberg's sieve for dihedral groups with the missing harmonic approach of Moore and Russell. These are the first nontrivial hidden subgroup algorithms for group families that require highly entangled multiregister Fourier sampling.