On quantum algorithms

Cleve, Richard; Ekert, Artur; Henderson, Leah; Macchiavello, Chiara; Mosca, Michele

doi:10.1002/(sici)1099-0526(199809/10)4:1<33::aid-cplx10>3.0.co;2-u

Cited by 47 publications

(50 citation statements)

References 21 publications

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“…The problem is to determine the function class with a minimum number of oracle invocations. A classical algorithm that solves with certainty requires 2 n−1 + 1 oracle invocations in the worst case [1,9]. A quantum algorithm for solving this exists [1,9] and, in its modified form [5], uses the oracle defined on computational basis elements aŝ…”

Section: Application To the Deutsch-jozsa Algorithmmentioning

confidence: 99%

Discrimination of Unitary Transformations and Quantum Algorithms

Collins¹

2009

AIP Conference Proceedings

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Abstract. Quantum algorithms are typically understood in terms of the evolution of a multiqubit quantum system under a prescribed sequence of unitary transformations. The input to the algorithm prescribes some of the unitary transformations in the sequence with others remaining fixed. For oracle query algorithms, the input determines the oracle unitary transformation. Such algorithms can be regarded as devices for discriminating amongst a set of unitary transformations. In quantum algorithms a given oracle, f , is invoked via unitary transformation,Û f , whose structure depends on the nature of the oracle. For example, in both the DeutschJozsa algorithm [5] and Grover's algorithm the oracle is defined on computational basis states, |x ≡ |x n . . . x 1 with x i ∈ {0, 1} , to beÛ f |x = (−1) f (x) |x and this is extended linearly to superpositions of computational basis states. The general structure of such oracle algorithms is encapsulated in an algorithm unitarŷwhereV 0 , . . . ,V M are oracle independent unitary transformations and the oracle is invoked M times. This is applied to a quantum system in an oracle independent initial state |Ψ i , giving an oracle dependent final state Ψ f =Û alg |Ψ i , upon which a computational basis measurement is performed. It is important to note that the input to the algorithm is the oracle unitary and not the initial state. The output from the algorithm potentially identifies the oracle unitary or a class of oracle unitaries. For example, in Grover's algorithm for searching a database one marked item at location s, f (x) = 0 whenever x = s and f (s) = 1. The standard Grover's algorithm terminates in a compu-

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Section: Application To the Deutsch-jozsa Algorithmmentioning

confidence: 99%

Discrimination of Unitary Transformations and Quantum Algorithms

Collins¹

2009

AIP Conference Proceedings

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“…The corresponding circuit working on qubit systems is already known (see, e.g., Ref. [14]). Therefore in the case of a two dimensional system (that is, K = 1 in Eq.…”

Section: Quantum Circuit For the Optimal Strategymentioning

confidence: 99%

“…The circuit corresponding toÛ DFT is found in Ref. [14]. Therefore the optimal estimation is realized by applying the unitary transform (Î ⊗Û DFT ) ·T (4) on the input state |ψ(θ) ⊗4 and then by measuring the last 3 qubits of the transformed state in the basis {|L 3 }.…”

Section: Fig 4 the Circuit Which Converts The Basis Statesmentioning

confidence: 99%

Optimal phase estimation and square root measurement

Sasaki

Carlini

Chefles

2001

J. Phys. A: Math. Gen.

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We present an optimal strategy having finite outcomes for estimating a single parameter of the displacement operator on an arbitrary finite dimensional system using a finite number of identical samples. Assuming the uniform a priori distribution for the displacement parameter, an optimal strategy can be constructed by making the square root measurement based on uniformly distributed sample points. This type of measurement automatically ensures the global maximality of the figure of merit, that is, the so called average score or fidelity. Quantum circuit implementations for the optimal strategies are provided in the case of a two dimensional system.

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“…In this paper, we show some practical examples how our quantum Hoare logic verifies properties of algorithms, quantum teleportation [6], Shor's prime factorization [7], a solution of the Deutsch problem [8], and quantum coin tossing [9].…”

Section: Introductionmentioning

confidence: 99%

A Logic for Formal Verification of Quantum Programs

Kakutani

2009

Advances in Computer Science - ASIAN 2009. Information Security and Privacy

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Abstract. This paper provides a Hoare-style logic for quantum computation. While the usual Hoare logic helps us to verify classical deterministic programs, our logic supports quantum probabilistic programs. Our target programming language is QPL defined by Selinger, and our logic is an extension of the probabilistic Hoare-style logic defined by den Hartog. In this paper, we demonstrate how the quantum Hoare-style logic proves properties of well-known algorithms.

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On quantum algorithms

Cited by 47 publications

References 21 publications

Discrimination of Unitary Transformations and Quantum Algorithms

Discrimination of Unitary Transformations and Quantum Algorithms

Optimal phase estimation and square root measurement

A Logic for Formal Verification of Quantum Programs

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