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Abstract

We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the so-called pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp threshold as a function of the density ν = k/ log 2 N , where k is the number of copies of the hidden subgroup state and 2N is the order of the dihedral group. In particular, for ν < 1 the optimal measurement (and hence any measurement) identifies the hidden subgroup with a probability that is exponentially small in log N , while for ν > 1 the optimal measurement identifies the hidden subgroup with a probability of order unity. Thus the dihedral group provides an example of a group G for which Ω(log |G|) hidden subgroup states are necessary to solve the hidden subgroup problem. We also consider the optimal measurement for determining a single bit of the answer, and show that it exhibits the same threshold. Finally, we consider implementing the optimal measurement by a quantum circuit, and thereby establish further connections between the dihedral hidden subgroup problem and average case subset sum problems. In particular, we show that an efficient quantum algorithm for a restricted version of the optimal measurement would imply an efficient quantum algorithm for the subset sum problem, and conversely, that the ability to quantum sample from subset sum solutions allows one to implement the optimal measurement.

How a Clebsch-Gordan transform helps to solve the Heisenberg hidden subgroup problem

“…Further Ip showed that for the dihedral hidden subgroup problem, the optimal measurement was not to perform a quantum Fourier transform over the dihedral group followed by a projective measurement. Continuing on in this line of inquiry, Bacon, Childs, and van Dam derived an exact expression for the optimal measurement for the dihedral hidden subgroup problem [19]. This measurement turned out to be the so-called pretty good measurement [43].…”

“…Thus, even though measurement on a single register containing the hidden subgroup state is enough to information theoretically reconstruct the hidden subgroup state, a measurement across many registers containing the hidden subgroup is optimal for solving this problem. Unfortunately, it is not known how to efficiently implement the optimal measurement described in [19]. However, building upon the optimal measurement approach, Bacon, Childs, and van Dam then applied the apparatus of optimal measurements for the HSP to the HSP for certain semidirect product groups of the for Z r p ⋊ Z, for a fixed r and prime p [14].…”

“…Towards this end, a series of efficient quantum algorithms for certain non-Abelian hidden subgroups have been developed [8,9,10,11,12,13,14,15]. At the same time a series of negative results towards the standard approach to solving the hidden subgroup problem on a quantum computer have also appeared [8,9,16,17,18,19,20]. Viewed pessimistically, these results cast doubt on whether quantum computers can be used to efficiently solve non-Abelian hidden subgroup problems.…”

“…One such group which admits an efficient quantum algorithm is the Heisenberg group, Z 2 p ⋊Z p . The algorithm of Bacon, Childs, and van Dam [14] was discovered using the framework of pretty-good measurements [14,15,19,21,22]. This yields a particular algorithm for solving the Heisenberg hidden subgroup problem which is optimal and which is related to the solution of certain algebraic equations.…”

Minimal scheme for certifying three-outcome qubit measurements in the prepare-and-measure scenario

“…The set of measurements described in this way contains instances which are neither projective nor can they be obtained by combining projective measurements. This class of genuinely nonprojective measurements can be utilized, for example, in quantum computing [1,2], quantum cryptography [3,4], randomness certification [5], and quantum tomography [6]. But since genuinely nonprojective measurements cannot be combined from projective measurements, their experimental implementation is difficult and typically requires control over additional degrees of freedoms [7].…”