Search via quantum walks with intermediate measurements

A modification of Tulsi's quantum search algorithm with intermediate measurements of the control qubit is presented. In order to analyze the effect of measurements in quantum searches, a different choice of the angular parameter is used. The study is performed for several values of time lapses between measurements, finding close relationships between probabilities and correlations (mutual information and cumulative correlation measure). The order of this modified algorithm is estimated, showing that for some time lapses the performance is improved, and becomes of order O(N) (classical brute-force search) when measurements are taken in every step. The results provide a possible way to analyze improvements to other quantum algorithms using one, or more, control qubits.

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How significant are the known collision and element distinctness quantum algorithms?

Quantum Information and Computation ◽

10.26421/qic4.3-5 ◽

2004 ◽

Vol 4 (3) ◽

pp. 201-206

Author(s):

L. Grover ◽

T. Rudolph

Keyword(s):

Search Algorithm ◽

Quantum Algorithms ◽

Search Space ◽

Space Complexity ◽

Quantum Search ◽

Quantum Search Algorithm ◽

Time Space ◽

Simple Quantum ◽

Structured Problems ◽

Better Than

Quantum search is a technique for searching $N$ possibilities for a desired target in $O(\sqrt{N})$ steps. It has been applied in the design of quantum algorithms for several structured problems. Many of these algorithms require significant amount of quantum hardware. In this paper we propose the criterion that an algorithm which requires $O(S)$ hardware should be considered significant if it produces a speedup of better than $O\left(\sqrt{S}\right)$ over a simple quantum search algorithm. This is because a speedup of $O\left(\sqrt{S}\right)$ can be trivially obtained by dividing the search space into $S$ separate parts and handing the problem to $S$ independent processors that do a quantum search (in this paper we drop all logarithmic factors when discussing time/space complexity). Known algorithms for collision and element distinctness exactly saturate the criterion.

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Superposition, entanglement and quantum computation

Quantum Information and Computation ◽

10.26421/qic2.2-1 ◽

2002 ◽

Vol 2 (2) ◽

pp. 97-116

Author(s):

T.M. Forcer ◽

A.J.G. Hey ◽

D.A. Ross ◽

P.G.R. Smith

Keyword(s):

Quantum Computing ◽

Quantum Computation ◽

Search Algorithm ◽

Quantum Algorithms ◽

Quantum Search ◽

Quantum Search Algorithm ◽

Electronic Implementation

The paper examines the roles played by superposition and entanglement in quantum computing. The analysis is illustrated by discussion of a "classical" electronic implementation of Grover's quantum search algorithm. It is shown explicitly that the absence of multi-particle entanglement leads to exponentially rising resources for implementing such quantum algorithms.

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ENTANGLEMENT PROPERTIES OF ADIABATIC QUANTUM ALGORITHMS

International Journal of Quantum Information ◽

10.1142/s0219749909006000 ◽

2009 ◽

Vol 07 (08) ◽

pp. 1531-1539 ◽

Cited By ~ 4

Author(s):

JIAYAN WEN ◽

YI HUANG ◽

DAOWEN QIU

Keyword(s):

Quantum System ◽

Search Algorithm ◽

Quantum Algorithms ◽

The Other ◽

Constant Time ◽

Quantum Search ◽

Adiabatic Evolution ◽

Algorithm Performance ◽

Quantum Search Algorithm ◽

Speed Up

In this paper, by constructing a more entangled quantum system, we shorten the adiabatic quantum search algorithm to constant time. On the other hand, we show that the speed-up of adiabatic quantum algorithms by selecting particular adiabatic evolution paths or injecting energy into the quantum system can be explained as a form of entanglement enlargement. These findings suggest that entanglement plays a fundamental role for the efficiency of algorithm performance.

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Robust Quantum Search with Uncertain Number of Target States

Entropy ◽

10.3390/e23121649 ◽

2021 ◽

Vol 23 (12) ◽

pp. 1649

Author(s):

Yuanye Zhu ◽

Zeguo Wang ◽

Bao Yan ◽

Shijie Wei

Keyword(s):

Success Rate ◽

Search Algorithm ◽

Quantum Algorithms ◽

Query Complexity ◽

Quantum Search ◽

Success Rates ◽

Quantum Search Algorithm ◽

Speed Up ◽

Target States ◽

Marked State

The quantum search algorithm is one of the milestones of quantum algorithms. Compared with classical algorithms, it shows quadratic speed-up when searching marked states in an unsorted database. However, the success rates of quantum search algorithms are sensitive to the number of marked states. In this paper, we study the relation between the success rate and the number of iterations in a quantum search algorithm of given λ=M/N, where M is the number of marked state and N is the number of items in the dataset. We develop a robust quantum search algorithm based on Grover–Long algorithm with some uncertainty in the number of marked states. The proposed algorithm has the same query complexity ON as the Grover’s algorithm, and shows high tolerance of the uncertainty in the ratio M/N. In particular, for a database with an uncertainty in the ratio M±MN, our algorithm will find the target states with a success rate no less than 96%.

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A Little Bit of Classical Magic to Achieve (Super-)Quantum Speedup

Foundations of Physics ◽

10.1007/s10701-021-00461-w ◽

2021 ◽

Vol 51 (3) ◽

Author(s):

Dagomir Kaszlikowski ◽

Paweł Kurzyński

Keyword(s):

Random Walk ◽

Probability Distribution ◽

Search Algorithm ◽

Quantum Algorithms ◽

Quantum Search ◽

Classical Dynamics ◽

Quantum Search Algorithm ◽

Operational Interpretation ◽

Simple Dynamical Model ◽

Quantum Speedup

AbstractWe introduce nebit, a classical bit with a signed probability distribution. We study its properties and basic transformations that can be applied to it. Then, we introduce a simple dynamical model – a classical random walk supplemented with nebits. We show that such a model exhibits some counterintuitive non-classical properties and that it can achieve or even exceed the speedup of Grover’s quantum search algorithm. The proposed classical dynamics never reveals negativity of nebits and thus we do not need any operational interpretation of negative probabilities. We argue that nebits can be useful as a measure of non-classicality as well as a tool to find new quantum algorithms.

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