Quantum Algorithms for Number Fields

Haase, Daniel; Maier, Helmut

doi:10.1002/9783527611065.ch18

Cited by 2 publications

(3 citation statements)

References 6 publications

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“…In the standard formulation, the initial distribution of amplitudes is taken in the form (3). Then the evolution of amplitudes can be described within the two-dimensional picture.…”

Section: Preliminariesmentioning

confidence: 99%

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Degradation of Grover’s search under collective phase flips in queries to the oracle

Rastegin

2018

Front. Phys.

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We address the case, when querying to the oracle in Grover's algorithm is exposed to noise including phase distortions. The oracle-box wires can be altered by an opposite party that tries to prevent correct receiving data from the oracle. This situation reflects an experienced truth that any access to prophetic knowledge cannot be common and direct. To study the problem, we introduce a simple model of collective phase distortions on the base of phase damping channel. In the model used, the success probability is not altered via the oracle-box wires per se. Phase distortions of the considered type can hardly be detected via any one-time query to the oracle. However, the success probability is significantly changed, when such errors are introduced as an intermediate step into the Grover iteration. We investigate the success probability with respect to variations the parameter that characterizes the amount of phase errors. It turns out that the success probability is decreased significantly even if the error amount is not very high. Moreover, this probability quickly reduces to the value of one half, which corresponds to the completely mixed state. We also study trade-off relations between quantum coherence and the success probability in the presence of noise of the considered type.

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“…In the standard formulation, the initial distribution of amplitudes is taken in the form (3). Then the evolution of amplitudes can be described within the two-dimensional picture.…”

Section: Preliminariesmentioning

confidence: 99%

“…In last decades, quantum effects have found a new field of applications for information processing [1]. Famous Shor's discovery [2] had lead to numerous quantum algorithms for algebraic problems [3][4][5]. Grover's search algorithm [6][7][8] is another fundamental result.…”

Section: Introductionmentioning

confidence: 99%

Degradation of Grover’s search under collective phase flips in queries to the oracle

Rastegin

2018

Front. Phys.

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“…The Grover search algorithm [1][2][3] is one of fundamental discoveries that motivate quantum computations. Celebrated Shor's results [4] have led to numerous quantum algorithms for algebraic problems [5][6][7]. The authors of [8] gave arguments that Grover's and Shor's algorithms are more closely related than one might expect at first.…”

Section: Introductionmentioning

confidence: 99%

On the role of dealing with quantum coherence in amplitude amplification

Rastegin

2018

Quantum Inf Process

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Amplitude amplification is one of primary tools in building algorithms for quantum computers. This technique generalizes key ideas of the Grover search algorithm. Potentially useful modifications are connected with changing phases in the rotation operations and replacing the intermediate Hadamard transform with arbitrary unitary one. In addition, arbitrary initial distribution of the amplitudes may be prepared. We examine trade-off relations between measures of quantum coherence and the success probability in amplitude amplification processes. As measures of coherence, the geometric coherence and the relative entropy of coherence are considered. In terms of the relative entropy of coherence, complementarity relations with the success probability seem to be the most expository. The general relations presented are illustrated within several model scenarios of amplitude amplification processes.

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Quantum Algorithms for Number Fields

Cited by 2 publications

References 6 publications

Degradation of Grover’s search under collective phase flips in queries to the oracle

Degradation of Grover’s search under collective phase flips in queries to the oracle

On the role of dealing with quantum coherence in amplitude amplification

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