Quantifying computational advantage of Grover’s algorithm with the trace speed

AbstractDespite intensive research, the physical origin of the speed-up offered by quantum algorithms remains mysterious. No general physical quantity, like, for instance, entanglement, can be singled out as the essential useful resource. Here we report a close connection between the trace speed and the quantum speed-up in Grover’s search algorithm implemented with pure and pseudo-pure states. For a noiseless algorithm, we find a one-to-one correspondence between the quantum speed-up and the polarization of the pseudo-pure state, which can be connected to a wide class of quantum statistical speeds. For time-dependent partial depolarization and for interrupted Grover searches, the speed-up is specifically bounded by the maximal trace speed that occurs during the algorithm operations. Our results quantify the quantum speed-up with a physical resource that is experimentally measurable and related to multipartite entanglement and quantum coherence.

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Classical and Quantum Algorithms for Assembling a Text from a Dictionary

Nonlinear Phenomena in Complex Systems ◽

10.33581/1561-4085-2021-24-3-207-221 ◽

2021 ◽

Vol 24 (3) ◽

pp. 207-221

Author(s):

Kamil Khadiev ◽

Vladislav Remidovskii

Keyword(s):

Deoxyribonucleic Acid ◽

Quantum Algorithms ◽

Randomized Algorithm ◽

Quantum Algorithm ◽

Deterministic Algorithm ◽

Constant Length ◽

Running Time ◽

Assembly Method ◽

Speed Up ◽

Quantum Speed Up

We study algorithms for solving the problem of assembling a text (long string) from a dictionary (a sequence of small strings). The problem has an application in bioinformatics and has a connection with the sequence assembly method for reconstructing a long deoxyribonucleic-acid (DNA) sequence from small fragments. The problem is assembling a string t of length n from strings s1,...,sm. Firstly, we provide a classical (randomized) algorithm with running time Õ(nL0.5 + L) where L is the sum of lengths of s1,...,sm. Secondly, we provide a quantum algorithm with running time Õ(nL0.25 + √mL). Thirdly, we show the lower bound for a classical (randomized or deterministic) algorithm that is Ω(n+L). So, we obtain the quadratic quantum speed-up with respect to the parameter L; and our quantum algorithm have smaller running time comparing to any classical (randomized or deterministic) algorithm in the case of non-constant length of strings in the dictionary.

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Low-rank density-matrix evolution for noisy quantum circuits

npj Quantum Information ◽

10.1038/s41534-021-00392-4 ◽

2021 ◽

Vol 7 (1) ◽

Author(s):

Yi-Ting Chen ◽

Collin Farquhar ◽

Robert M. Parrish

Keyword(s):

Density Matrix ◽

Search Algorithm ◽

Full Rank ◽

Quantum Circuits ◽

Low Rank ◽

Simulation Step ◽

Iterative Compression ◽

Speed Up ◽

Grover’S Search Algorithm ◽

Matrix Evolution

AbstractIn this work, we present an efficient rank-compression approach for the classical simulation of Kraus decoherence channels in noisy quantum circuits. The approximation is achieved through iterative compression of the density matrix based on its leading eigenbasis during each simulation step without the need to store, manipulate, or diagonalize the full matrix. We implement this algorithm using an in-house simulator and show that the low-rank algorithm speeds up simulations by more than two orders of magnitude over existing implementations of full-rank simulators, and with negligible error in the noise effect and final observables. Finally, we demonstrate the utility of the low-rank method as applied to representative problems of interest by using the algorithm to speed up noisy simulations of Grover’s search algorithm and quantum chemistry solvers.

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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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QUANTUM COMPUTING, GROVER'S SEARCH ALGORITHM AND ITS APPLICATIONS IN BIOINFORMATICS

COSMOS ◽

10.1142/s0219607706000171 ◽

2006 ◽

Vol 02 (01) ◽

pp. 71-80

Author(s):

K. W. CHOO

Keyword(s):

Quantum Computing ◽

Mass Spectra ◽

Search Algorithm ◽

Quantum Algorithms ◽

Quantum Algorithm ◽

Protein Mass ◽

Quantum Counting ◽

Grover’S Search Algorithm ◽

Counting Algorithm

This article reviews quantum computing and quantum algorithms. Some insights into its potential in speeding up computations are covered, with emphasis on the use of Grover's Search. In the last section, we discuss applications of quantum algorithm to bioinformatics. In particular, the extension of quantum counting algorithm to protein mass spectra counting is proposed.

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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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Quantum discord and entanglement in grover search algorithm

Open Physics ◽

10.1515/phys-2016-0020 ◽

2016 ◽

Vol 14 (1) ◽

pp. 171-176 ◽

Cited By ~ 1

Author(s):

Bin Ye ◽

Tingzhong Zhang ◽

Liang Qiu ◽

Xuesong Wang

Keyword(s):

Quantum Entanglement ◽

Quantum Discord ◽

Search Algorithm ◽

Quantum Algorithms ◽

Quantum Correlations ◽

Quantum Computers ◽

Grover Algorithm ◽

Grover Search ◽

Grover’S Search Algorithm ◽

The Impact

AbstractImperfections and noise in realistic quantum computers may seriously affect the accuracy of quantum algorithms. In this article we explore the impact of static imperfections on quantum entanglement as well as non-entangled quantum correlations in Grover’s search algorithm. Using the metrics of concurrence and geometric quantum discord, we show that both the evolution of entanglement and quantum discord in Grover algorithm can be restrained with the increasing strength of static imperfections. For very weak imperfections, the quantum entanglement and discord exhibit periodic behavior, while the periodicity will most certainly be destroyed with stronger imperfections. Moreover, entanglement sudden death may occur when the strength of static imperfections is greater than a certain threshold.

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Practical Security of RSA Against NTC-Architecture Quantum Computing Attacks

International Journal of Theoretical Physics ◽

10.1007/s10773-021-04789-x ◽

2021 ◽

Author(s):

Kai Li ◽

Qing-yu Cai

Keyword(s):

Quantum Computing ◽

Ion Trap ◽

Quantum Algorithms ◽

Coherence Time ◽

Quantum Computers ◽

Quantum Time ◽

Speed Up ◽

Key Length ◽

Concurrent Architecture ◽

Qubit Gate

AbstractQuantum algorithms can greatly speed up computation in solving some classical problems, while the computational power of quantum computers should also be restricted by laws of physics. Due to quantum time-energy uncertainty relation, there is a lower limit of the evolution time for a given quantum operation, and therefore the time complexity must be considered when the number of serial quantum operations is particularly large. When the key length is about at the level of KB (encryption and decryption can be completed in a few minutes by using standard programs), it will take at least 50-100 years for NTC (Neighbor-only, Two-qubit gate, Concurrent) architecture ion-trap quantum computers to execute Shor’s algorithm. For NTC architecture superconducting quantum computers with a code distance 27 for error-correcting, when the key length increased to 16 KB, the cracking time will also increase to 100 years that far exceeds the coherence time. This shows the robustness of the updated RSA against practical quantum computing attacks.

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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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