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 unitary operator U∈S:={U1, U2} under some probability distribution over S, the goal is to decide whether U=U1 or U=U2. 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 $\lceil{\sqrt{6}\theta_{\rm cover}^{-1}}\rceil$ queries to the black box in the worst case, i.e., under any probability distribution over S, where the parameter θcover, which is determined by the eigenvalues of $U_1^\dagger {U_2}$ , represents the “closeness” between U1 and U2. We also show that this upper bound is essentially tight: we prove that for every θcover > 0 there exist operators U1 and U2 such that any quantum algorithm solving this problem with bounded error probability requires at least $\lceil{\frac{2}{3 \theta_{\rm cover}}}\rceil$ queries under uniform distribution over S.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 unitary operator U∈S:={U1, U2} under some probability distribution over S, the goal is to decide whether U=U1 or U=U2. 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 $\lceil{\sqrt{6}\theta_{\rm cover}^{-1}}\rceil$ queries to the black box in the worst case, i.e., under any probability distribution over S, where the parameter θcover, which is determined by the eigenvalues of $U_1^\dagger {U_2}$, represents the “closeness” between U1 and U2. We also show that this upper bound is essentially tight: we prove that for every θcover > 0 there exist operators U1 and U2 such that any quantum algorithm solving this problem with bounded error probability requires at least $\lceil{\frac{2}{3 \theta_{\rm cover}}}\rceil$ queries under uniform distribution over S.

IEICE Transactions on Information and SystemsFeb 28, 20193 citations

Local discrimination scheme for some unitary operations

Tian-Qing Cao, Fei Gao, Guojing Tian +2 more

Science China Physics Mechanics and AstronomyJul 25, 20165 citations

Query Complexity in Expectation

Jędrzej Kaniewski, Troy Lee, Ronald de Wolf

Lecture notes in computer scienceJan 1, 201510 citations

On the Query Complexity of Perfect Gate Discrimination

Giulio Chiribella, Giacomo Mauro D’Ariano, Martin Roetteler

We investigate the problem of finding the minimum number of queries needed to perfectly identify an unknown quantum gate within a finite set of alternatives, considering both deterministic strategies. For unambiguous gate discrimination, where errors are not tolerated but inconclusive outcomes are allowed, we prove that parallel strategies are sufficient to identify the unknown gate with minimum number of queries and we use this fact to provide upper and lower bounds on the query complexity. In addition, we introduce the notion of generalized $t$ -designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized $t$ -design we prove that there is no difference between perfect probabilistic and perfect deterministic gate discrimination. Hence, evaluating of the query complexity of perfect discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous discrimination.We investigate the problem of finding the minimum number of queries needed to perfectly identify an unknown quantum gate within a finite set of alternatives, considering both deterministic strategies. For unambiguous gate discrimination, where errors are not tolerated but inconclusive outcomes are allowed, we prove that parallel strategies are sufficient to identify the unknown gate with minimum number of queries and we use this fact to provide upper and lower bounds on the query complexity. In addition, we introduce the notion of generalized $t$-designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized $t$-design we prove that there is no difference between perfect probabilistic and perfect deterministic gate discrimination. Hence, evaluating of the query complexity of perfect discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous discrimination.

Leibniz international proceedings in informaticsJan 1, 20130 citations

Query complexity of generalized Simon's problem

Zekun Ye, Yunqi Huang, Lvzhou Li +1 more

arXiv:1907.07367Information and ComputationAug 27, 20217 citations

\lceil{\sqrt{6}\theta_{\rm cover}^{-1}}\rceil

queries to the black box in the worst case, i.e., under any probability distribution over S, where the parameter θcover, which is determined by the eigenvalues of

U_1^\dagger {U_2}

, represents the “closeness” between U1 and U2. We also show that this upper bound is essentially tight: we prove that for every θcover > 0 there exist operators U1 and U2 such that any quantum algorithm solving this problem with bounded error probability requires at least

\lceil{\frac{2}{3 \theta_{\rm cover}}}\rceil

queries under uniform distribution over S.

t

-designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized

t

-design we prove that there is no difference between perfect probabilistic and perfect deterministic gate discrimination. Hence, evaluating of the query complexity of perfect discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous discrimination.