Random measurement bases, quantum state distinction and applications to the hidden subgroup problem
Pranab Sen
TL;DR
The paper develops an information-theoretic framework showing that random POVMs can distinguish between quantum states with total variation distance proportional to their Frobenius distance, under plausible rank constraints. It then leverages this to derive general bounds for quantum state identification and to derive polynomial-copy algorithms for the hidden subgroup problem via random Fourier sampling, especially when Fourier blocks have polynomially bounded rank (notably in Gel'fand-pair cases). The results provide the first information-theoretic upper bounds on the number of copies needed for general state identification and establish single-register algorithms for several HSP instances, while highlighting open questions about efficiently implementable pseudo-random measurement bases. Overall, the work connects state-distinction, state-identification, and HSP through a unified random-measurement approach with concrete quantitative guarantees and clear limitations.
Abstract
We show that measuring any two quantum states by a random POVM, under a suitable definition of randomness, gives probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states, with high probability. This result gives us the first sufficient condition and an information-theoretic solution for the following quantum state distinction problem: given an a priori known ensemble of quantum states, is there a single POVM that gives reasonably large total variation distance between every pair of states from the ensemble? Our random POVM method also gives us the first information-theoretic upper bound on the number of copies required to solve the quantum state identification problem for general ensembles, i.e., given some number of independent copies of a quantum state from an a priori known ensemble, identify the state. The standard quantum approach to solving the hidden subgroup problem (HSP) is a special case of the state identification problem where the ensemble consists of so-called coset states of candidate hidden subgroups. Combining Fourier sampling with our random POVM result gives us single register algorithms using polynomially many copies of the coset state that identify hidden subgroups having polynomially bounded rank in every representation of the ambient group. These HSP algorithms complement earlier results about the powerlessness of random Fourier sampling when the ranks are exponentially large, which happens for example in the HSP over the symmetric group. The drawback of random Fourier sampling based algorithms is that they are not efficient because measuring in a random basis is not. This leads us to the open question of efficiently implementable pseudo-random measurement bases.