Abstract
Unitary operator discrimination is a fundamental problem in quantum information theory. The basic version of this problem can be described as follows: given a black box implementing a quantum operator U, and the promise that the black box implements either the unitary operator \(U_1\) or the unitary operator \(U_2\), the goal is to decide whether \(U=U_1\) or \(U=U_2\). In this paper, we consider the query complexity of this problem. We show that there exists a quantum algorithm that solves this problem with bounded-error probability using \(\left\lceil \frac{\pi }{3\theta _\mathrm{cover}} \right\rceil \) queries to the black-box, where \(\theta _\mathrm{cover}\) represents the “closeness” between \(U_1\) and \(U_2\) (this parameter is determined by the eigenvalues of the matrix \(U_1^\dag U_2\)). We also show that this upper bound is essentially tight: we prove that there exist operators \(U_1\) and \(U_2\) such that any quantum algorithm solving this problem with bounded-error probability requires at least \(\left\lceil \frac{2}{3\theta _\mathrm{cover}} \right\rceil \) queries.
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Acknowledgments
AK was partially supported by MEXT KAKENHI (24106009) and JSPS KAKENHI (16H01705, 17K12640). ST was supported in part by MEXT KAKENHI (24106003) and JSPS KAKENHI (26330011, 16H02782). FLG was partially supported by MEXT KAKENHI (24106009) and JSPS KAKENHI (16H01705, 16H05853). The authors are grateful to Akihito Soeda for helpful discussions.
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Kawachi, A., Kawano, K., Le Gall, F., Tamaki, S. (2017). Quantum Query Complexity of Unitary Operator Discrimination. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_26
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DOI: https://doi.org/10.1007/978-3-319-62389-4_26
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