Quantum query complexity of symmetric oracle problems

Daniel Copeland; Jamie Pommersheim

doi:10.22331/q-2021-03-07-403

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arXiv · 1812.09428

Quantum query complexity of symmetric oracle problems

Daniel Copeland·Jamie Pommersheim

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Abstract

We study the query complexity of quantum learning problems in which the oracles form a group $G$ of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a $t$-query quantum algorithm in terms of group characters. As an application, we show that $\Omega(n)$ queries are required to identify a random permutation in $S_n$. More generally, suppose $H$ is a fixed subgroup of the group $G$ of oracles, and given access to an oracle sampled uniformly from $G$, we want to learn which coset of $H$ the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a $t$-query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on $n$ which require $n+1$ queries classically and only $1$ query quantumly.

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Daniel Copeland, Jamie Pommersheim. 2018-12-22. Quantum query complexity of symmetric oracle problems. https://doi.org/10.22331/q-2021-03-07-403

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