On the Power of Random Bases in Fourier Sampling: Hidden Subgroup Problem in the Heisenberg Group

Abstract

The hidden subgroup problem (HSP) provides a unified framework to study problems of grouptheoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an efficient solution in the abelian case, not much is known for general non-abelian groups. Recently, some authors raised the question as to whether post-processing the Fourier spectrum by measuring in a random orthonormal basis helps for solving the HSP. Several negative results on the shortcomings of this random strong method are known. In this paper however, we show that the random strong method can be quite powerful under certain conditions on the group G. We define a parameter r(G) for a group G and show that O((log |G|/r(G))2 ) iterations of the random strong method give enough classical information to identify a hidden subgroup in G. We illustrate the power of the random strong method via a concrete example of the HSP over finite Heisenberg groups. We show that r(G) = Ω(1) for these groups; hence the HSP can be solved using polynomially many random strong Fourier samplings followed by a possibly exponential classical post-processing without further queries. The quantum part of our algorithm consists of a polynomial computation followed by measuring in a random orthonormal basis. This gives the first example of a group where random representation bases do help in solving the HSP and for which no explicit representation bases are known that solve the problem with (log G) O(1) Fourier samplings. As an interesting by-product of our work, we get an algorithm for solving the state identification problem for a set of nearly orthogonal pure quantum states.

Quantum t-designs: t-wise Independence in the Quantum World

“…Radhakrishnan et al [15] have shown Theorem 2 Let |ψ 1 , |ψ 2 be two orthogonal quantum states in C N . Then,…”

From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups

“…In other words, the running time of the algorithm is poly(log p). The algorithm is particularly simple in the case r = 2, where the only nontrivial semidirect product is known as the Heisenberg group (for which it was recently shown that there is an efficient quantum algorithm whose output information theoretically determines the solution of the hsp [25]). We present the algorithm for r = 2 in Section 6.1, and then proceed to the general case in Section 6.2.…”

Random Measurement Bases, Quantum State Distinction and Applications to the Hidden Subgroup Problem

“…If the ranks of the blocks of σ H 1 , σ H 2 are polynomially bounded, we show that r(H 1 , H 2 ) is at least polynomially large. Earlier, Radhakrishnan et al [23] had proposed a similar distance function r (H 1 , H 2 ), but it was difficult to estimate r (H 1 , H 2 ) except for very special cases. Also, the function r (H 1 , H 2 ) was not powerful enough to even show that polynomially many iterations of the random Fourier method suffice to identify a hidden subgroup in the dihedral group with high probability.…”

“…For many concrete examples of Gel'fand pair HSPs like dihedral and Heisenberg groups, the number of iterations of random Fourier sampling can be brought down to O(log |G|) by a more careful analysis. Gel'fand pairs have been studied extensively in group theory, and much recent work on the HSP has involved Gel'fand pairs, e.g., dihedral group [5,7], affine group [20], Heisenberg group [4,23], optimality of the pretty good measurement for identifying hidden subgroups from a known conjugacy class [19]. For suitable subgroups of dihedral and affine groups, it is possible to give explicit efficient measurement bases for the single register Fourier sampling procedure that identify the hidden subgroup with high probability using polynomially many copies.…”

“…Interestingly, for the Heisenberg group no such explicit basis for single register Fourier sampling is known, though an explicit efficient entangled basis for two-register Fourier sampling is known [4]. The only proof that polynomially many iterations of single register Fourier sampling suffice information-theoretically to identify hidden subgroups in the Heisenberg group is through random Fourier sampling, and was first observed in [23].…”