Limitations of quantum coset states for graph isomorphism
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
An Efficient Quantum Algorithm for the Hidden Subgroup Problem over Weyl-Heisenberg Groups
Mathematical Methods in Computer Science
Efficient Quantum Tensor Product Expanders and k-Designs
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Quantum measurements for hidden subgroup problems with optimal sample complexity
Quantum Information & Computation
Limitations on quantum dimensionality reduction
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Machine learning in a quantum world
AI'06 Proceedings of the 19th international conference on Advances in Artificial Intelligence: Canadian Society for Computational Studies of Intelligence
Hi-index | 0.00 |
We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finite ensemble of quantum states there is a single POVM that distinguishes between every pair of states from the ensemble by at least a constant times their Frobenius distance; in fact, with high probability a random POVM, under a suitable definition of randomness, suffices. There are examples of ensembles with constant pairwise trace distance where a single POVM cannot distinguish pairs of states by much better than their Frobenius distance, including the important ensemble of coset states of hidden subgroups of the symmetric group [MRS05]. We next consider the random Fourier method for the hidden subgroup problem (HSP) which consists of Fourier sampling the coset state of the hidden subgroup using random orthonormal bases for the group representations. In cases where every representation of the group has polynomially bounded rank when averaged over the hidden subgroup, the random Fourier method gives a POVM for the HSP operating on one coset state at a time and using totally a polynomial number of coset states. In particular, we get such POVMs whenever the group and the hidden subgroup form a Gel'fand pair, e. g., abelian, dihedral and Heisenberg groups. This gives a positive counterpart to earlier negative results about random Fourier sampling when the above rank is exponentially large [GSVV04], which happens for example in the HSP in the symmetric group. The drawback of random POVMs is that they are not efficient to implement, since measuring in a random basis takes exponential time as can be seen by a counting argument. This leads us to the open question of efficiently implementable pseudo-random measurement bases.