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Conference on Theory of Quantum Computation, Communication and Cryptography

How many copies are needed for gate discrimination

Giulio Chiribella, Giacomo Mauro D’Ariano, Martin Roetteler

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

We address the problem of finding the minimum number of queries needed to perfectly identify an unknown gate within a finite set of alternatives. We provide a series of upper and lower bounds, by establishing a connection with the task of unambiguous discrimination, where one does not tolerate errors but allows for an inconclusive result. Our bounds show that the minimum number of queries needed for perfect discrimination is significantly smaller than the number required by the best known strategy based on pairwise tests.

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We address the problem of finding the minimum number of queries needed to perfectly identify an unknown gate within a finite set of alternatives. We provide a series of upper and lower bounds, by establishing a connection with the task of unambiguous discrimination, where one does not tolerate errors but allows for an inconclusive result. Our bounds show that the minimum number of queries needed for perfect discrimination is significantly smaller than the number required by the best known strategy based on pairwise tests.
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 ttt-designs, which includes unitary t-designs and group representations as special cases. For gates forming a generalized ttt-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 assess the resources needed to identify a reversible quantum gate among a finite set of alternatives, including in our analysis both deterministic and probabilistic strategies. Among the probabilistic strategies, we consider unambiguous gate discrimination—where errors are not tolerated but inconclusive outcomes are allowed—and we prove that parallel strategies are sufficient to unambiguously identify the unknown gate with minimum number of queries. This result is used to provide upper and lower bounds on the query complexity and on the minimum ancilla dimension. 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 give an explicit expression for the maximum probability of correct gate identification and we prove that there is no gap between the performances of deterministic strategies and those of probabilistic strategies. Hence, evaluating of the query complexity of perfect deterministic discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous discrimination. Finally, we consider discrimination strategies where the use of ancillas is forbidden, providing upper bounds on the number of additional queries needed to make up for the lack of entanglement with the ancillas.
We assess the resources needed to identify a reversible quantum gate among a finite set of alternatives, including in our analysis both deterministic and probabilistic strategies. Among the probabilistic strategies we consider unambiguous gate discrimination, where errors are not tolerated but inconclusive outcomes are allowed, and we prove that parallel strategies are sufficient to unambiguously identify the unknown gate with minimum number of queries. This result is used to provide upper and lower bounds on the query complexity and on the minimum ancilla dimension. 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 give an explicit expression for the maximum probability of correct gate identification and we prove that there is no gap between the performances of deterministic strategies an those of probabilistic strategies. Hence, evaluating of the query complexity of perfect deterministic discrimination is reduced to the easier problem of evaluating the query complexity of unambiguous discrimination. Finally, we consider discrimination strategies where the use of ancillas is forbidden, providing upper bounds on the number of additional queries needed to make up for the lack of entanglement with the ancillas.
The goal of quantum channel discrimination and estimation is to determine the identity of an unknown channel from a discrete or continuous set, respectively. The query complexity of these tasks is equal to the minimum number of times one must call an unknown channel to identify it within a desired threshold on the error probability. In this paper, we establish lower bounds on the query complexities of channel discrimination and estimation, in both the parallel and adaptive access models. We do so by establishing new or applying known upper bounds on the squared Bures distance and symmetric logarithmic derivative Fisher information of channels. Phrasing our statements and proofs in terms of isometric extensions of quantum channels allows us to give conceptually simple proofs for both novel and known bounds. We also provide alternative proofs for several established results in an effort to present a consistent and unified framework for quantum channel discrimination and estimation, which we believe will be helpful in addressing future questions in the field.
Quantum hypothesis testing is an important tool for quantum information processing.Two main strategies have been widely adopted: in a minimum error discrimination strategy, the average error probability is minimized; while in an unambiguous discrimination strategy, an inconclusive decision (abstention) is allowed to vanish any possibility of errors when a conclusive result is obtained.In both scenarios, the testing between quantum states is relatively well understood, for example, the ultimate limits of the performance are established decades ago; however, the testing between quantum channels is less understood.Although the ultimate limit of minimum error discrimination between channels has been explored recently, the corresponding limit of unambiguous discrimination is unknown.In this paper, we formulate an approximate unambiguous discrimination scenario, and derive the ultimate limits of the performance for both states and channels.In particular, in the channel case, our lower bound of the inconclusive probability holds for arbitrary adaptive sensing protocols.For the special class of "teleportation-covariant" channels, the lower bound is achievable with maximum entangled inputs and no adaptive strategy is necessary.
We prove a lower bound on the qqq-maximal fidelities between two quantum channels $\E_0$ and $\E_1$ and an upper bound on the qqq-maximal fidelities between a quantum channel $\E$ and an identity $\I$. Then we apply these two bounds to provide a simple sufficient and necessary condition for sequential perfect distinguishability between $\E$ and $\I$ and provide both a lower bound and an upper bound on the minimum number of queries required to sequentially perfectly discriminating $\E$ and $\I$. Interestingly, in the 222-dimensional case, both bounds coincide. Based on the optimal perfect discrimination protocol presented in \cite{DFY09}, we can further generalize the lower bound and upper bound to the minimum number of queries to perfectly discriminating $\E$ and III over all possible discrimination schemes. Finally the two lower bounds are shown remain working for perfectly discriminating general two quantum channels $\E_0$ and $\E_1$ in sequential scheme and over all possible discrimination schemes respectively.