Optimal quantum discrimination of single-qubit unitary gates between two candidates
TL;DR: In this paper, the authors obtained the optimal protocol that maximizes the expected success probability of a single-qubit unitary gate with two candidates, assuming the Haar distribution for the candidates.
Abstract: We analyze a discrimination problem of a single-qubit unitary gate with two candidates, where the candidates are not provided with their classical description, but their quantum sample is. More precisely, there are three unitary quantum gates---one target and one sample for each of the two candidates---whose classical description is unknown except for their dimension. The target gate is chosen equally among the candidates. We obtain the optimal protocol that maximizes the expected success probability, assuming the Haar distribution for the candidates. This problem is originally introduced in Ref. [5] which provides a protocol achieving 7/8 in the expected success probability based on the ``unitary comparison'' protocol of Ref. [6]. The optimality of the protocol has been an open question since then. We prove the optimality of the comparison protocol, implying that only one of the two samples (one for each candidate) is needed to achieve an optimal discrimination. The optimization includes protocols outside the scope of quantum testers due to the dynamic ordering of the sample and target gates within a given protocol.
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TL;DR: In this article , the maximal probability of successfully discriminating any set of unitary channels with any number of copies for the most general strategies that are suitable for channel discrimination was derived. But this bound is tight since it is saturated by sets of units forming a group k-design.
Abstract: For minimum-error channel discrimination tasks that involve only unitary channels, we show that sequential strategies may outperform the parallel ones. Additionally, we show that general strategies that involve indefinite causal order are also advantageous for this task. However, for the task of discriminating a uniformly distributed set of unitary channels that forms a group, we show that parallel strategies are, indeed, optimal, even when compared to general strategies. We also show that strategies based on the quantum switch cannot outperform sequential strategies in the discrimination of unitary channels. Finally, we derive an absolute upper bound for the maximal probability of successfully discriminating any set of unitary channels with any number of copies for the most general strategies that are suitable for channel discrimination. Our bound is tight since it is saturated by sets of unitary channels forming a group k-design.
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TL;DR: Quantum metrology is the use of quantum techniques such as entanglement to yield higher statistical precision than purely classical approaches as discussed by the authors, where the central limit theorem implies that the reduction is proportional to the square root of the number of repetitions.
Abstract: The statistical error in any estimation can be reduced by repeating the measurement and averaging the results. The central limit theorem implies that the reduction is proportional to the square root of the number of repetitions. Quantum metrology is the use of quantum techniques such as entanglement to yield higher statistical precision than purely classical approaches. In this Review, we analyse some of the most promising recent developments of this research field and point out some of the new experiments. We then look at one of the major new trends of the field: analyses of the effects of noise and experimental imperfections.
2,977 citations
TL;DR: This book discusses classical models of computations, quantum formalism, symplecto-classical cases, and error correction in the computation process: general principles.
Abstract: Contents §0. Introduction §1. Abelian problem on the stabilizer §2. Classical models of computations2.1. Boolean schemes and sequences of operations2.2. Reversible computations §3. Quantum formalism3.1. Basic notions and notation3.2. Transformations of mixed states3.3. Accuracy §4. Quantum models of computations4.1. Definitions and basic properties4.2. Construction of various operators from the elements of a basis4.3. Generalized quantum control and universal schemes §5. Measurement operators §6. Polynomial quantum algorithm for the stabilizer problem §7. Computations with perturbations: the choice of a model §8. Quantum codes (definitions and general properties)8.1. Basic notions and ideas8.2. One-to-one codes8.3. Many-to-one codes §9. Symplectic (additive) codes9.1. Algebraic preparation9.2. The basic construction9.3. Error correction procedure9.4. Torus codes §10. Error correction in the computation process: general principles10.1. Definitions and results10.2. Proofs §11. Error correction: concrete procedures11.1. The symplecto-classical case11.2. The case of a complete basis Bibliography
1,235 citations
TL;DR: In this article, the authors show that fingerprints consisting of quantum information can be made exponentially smaller than the original strings without any correlations or entanglement between the parties, and they give a test that distinguishes any two unknown quantum fingerprints with high probability.
Abstract: Classical fingerprinting associates with each string a shorter string (its fingerprint), such that, with high probability, any two distinct strings can be distinguished by comparing their fingerprints alone The fingerprints can be exponentially smaller than the original strings if the parties preparing the fingerprints share a random key, but not if they only have access to uncorrelated random sources In this paper we show that fingerprints consisting of quantum information can be made exponentially smaller than the original strings without any correlations or entanglement between the parties: we give a scheme where the quantum fingerprints are exponentially shorter than the original strings and we give a test that distinguishes any two unknown quantum fingerprints with high probability Our scheme implies an exponential quantum/classical gap for the equality problem in the simultaneous message passing model of communication complexity We optimize several aspects of our scheme
760 citations
TL;DR: In this article, the first energy transition 1h − 1e as a function of x and the well width was calculated for cubic AlxGa1−xN/GaN/AlxGa 1 −xN quantum wells and the nearest neighbor sp 3 s ∗ empirical tight binding approximation, together with the Surface Green Function Matching method was used.
Abstract: For cubic AlxGa1−xN/GaN/AlxGa1−xN quantum wells we calculated the first energy transition 1h–1e as a function of x and the well width. The nearest neighbour sp 3 s ∗ empirical tight binding approximation, including spin-orbit interaction, together with the Surface Green Function Matching method is used.
543 citations
TL;DR: Connections between state discrimination, the manipulation of quantum entanglement and quantum cloning are described, and recent experimental work is discussed.
Abstract: There are fundamental limits to the accuracy with which one can determine the state of a quantum system. I give an overview of the main approaches to quantum state discrimination. Several strategies exist. In quantum hypothesis testing, a quantum system is prepared in a member of a known, finite set of states, and the aim is to guess which one with the minimum probability of error. Error free discrimination is also sometimes possible, if we allow for the possibility of obtaining inconclusive results. If no prior information about the state is provided, then it is impractical to try to determine it exactly, and it must be estimated instead. In addition to reviewing these various strategies, I describe connections between state discrimination, the manipulation of quantum entanglement and quantum cloning. Recent experimental work is also discussed.
425 citations