New Results on Quantum Property Testing

الأصل · EN

We present several new examples of speed-ups obtainable by quantum algorithms in the context of property testing. First, motivated by sampling algorithms, we consider probability distributions given in the form of an oracle f:[n]→[m]. Here the probability (j) of an outcome j∈[m] is the fraction of its domain that f maps to j. We give quantum algorithms for testing whether two such distributions are identical or ε-far in L₁-norm. Recently, Bravyi, Hassidim, and Harrow BHH10 showed that if and are both unknown (i.e., given by oracles f and g), then this testing can be done in roughly √m quantum queries to the functions. We consider the case where the second distribution is known, and show that testing can be done with roughly m¹/³ quantum queries, which we prove to be essentially optimal. In contrast, it is known that classical testing algorithms need about m²/³ queries in the unknown-unknown case and about √m queries in the known-unknown case. Based on this result, we also reduce the query complexity of graph isomorphism testers with quantum oracle access. While those examples provide polynomial quantum speed-ups, our third example gives a much larger improvement (constant quantum queries vs polynomial classical queries) for the problem of testing periodicity, based on Shor's algorithm and a modification of a classical lower bound by Lachish and Newman lachish&newman:periodicity. This provides an alternative to a recent constant-vs-polynomial speed-up due to Aaronson aaronson:bqpph.