scholarly journals Improving Quantum Search on Simple Graphs by Pretty Good Structured Oracles

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 96
Author(s):  
Matteo G. A. Paris ◽  
Claudia Benedetti ◽  
Stefano Olivares
Keyword(s):  
Search Algorithm ◽  
Quantum Walks ◽  
Quantum Search ◽  
Complete Graphs ◽  
Combinatorial Search ◽  
Simple Graphs ◽  
Superconducting Circuits ◽  
Time Quantum ◽  
Speed Up ◽  
Quantum Technology

Quantum search algorithms provide a way to speed up combinatorial search, and have found several applications in modern quantum technology. In particular, spatial search on graphs, based on continuous-time quantum walks (CTQW), represents a promising platform for the implementation of quantum search in condensed matter systems. CTQW-based algorithms, however, work exactly on complete graphs, while they are known to perform poorly on realistic graphs with low connectivity. In this paper, we put forward an alternative search algorithm, based on structuring the oracle operator, which allows one to improve the localization properties of the walker by tuning only the on-site energies of the graph, i.e., without altering its topology. As such, the proposed algorithm is suitable for implementation in systems with low connectivity, e.g., rings of quantum dots or superconducting circuits. Oracle parameters are determined by Hamiltonian constraints, without the need for numerical optimization.

Abnormal Quantum State Search Based on Parallel Phase Comparison

2021 ◽  
Author(s):  
Guanlei Xu ◽  
Xiaogang Xu ◽  
Xiaotong Wang
Keyword(s):  
Quantum State ◽  
Search Algorithm ◽  
Quantum States ◽  
Quantum Search ◽  
Superposition State ◽  
Quantum Operator ◽  
Abnormal State ◽  
Number Formula ◽  
Speed Up ◽  
Grover Search

We discuss the problem of filtering out abnormal states from a larger number of quantum states. For this type of problem with [Formula: see text] items to be searched, both the traditional search by enumeration and classical Grover search algorithm have the complexity about [Formula: see text]. In this letter a novel quantum search scheme with exponential speed up is proposed for abnormal states. First, a new comprehensive quantum operator is well-designed to extract the superposition state containing all abnormal states with unknown number [Formula: see text] with complexity [Formula: see text] in probability 1 via well-designed parallel phase comparison. Then, every abnormal state is achieved respectively from [Formula: see text] abnormal states via [Formula: see text] times’ measurement. Finally, a numerical example is given to show the efficiency of the proposed scheme.

Speed-up and entanglement in quantum Searching

2002 ◽  
Vol 2 (5) ◽  
pp. 399-409
Author(s):  
S.L. Braunstein ◽  
A.K. Pati
Keyword(s):  
Pure State ◽  
Search Algorithm ◽  
Quantum Search ◽  
Grover’S Algorithm ◽  
Active Molecule ◽  
Quantum Search Algorithm ◽  
Grover's Algorithm ◽  
Speed Up ◽  
Target States

We investigate the issue of speed-up and the necessity of entanglement in Grover's quantum search algorithm. We find that in a pure state implementation of Grover's algorithm entanglement is present even though the initial and target states are product states. In pseudo-pure state implementations, the separability of the states involved defines an entanglement boundary in terms of a bound on the purity parameter. Using this bound we investigate the necessity of entanglement in quantum searching for these pseudo-pure state implementations. If every active molecule involved in the ensemble is `charged for' then in existing machines speed-up without entanglement is not possible.

scholarly journals Search via quantum walks with intermediate measurements

2015 ◽  
Vol 29 (19) ◽  
pp. 1550127
Author(s):  
Efrain Buksman ◽  
André L. Fonseca de Oliveira ◽  
Jesús García López de Lacalle
Keyword(s):  
Close Relationships ◽  
Search Algorithm ◽  
Quantum Algorithms ◽  
Quantum Walks ◽  
Quantum Search ◽  
Brute Force ◽  
Correlation Measure ◽  
Quantum Search Algorithm ◽  
Control Qubit ◽  
Modified Algorithm

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.

scholarly journals Non-Markovianity is not a resource for quantum spatial search on a star graph subject to generalized percolation

2018 ◽  
Vol 5 (1) ◽  
pp. 40-49 ◽  
Author(s):  
Matteo A. C. Rossi ◽  
Marco Cattaneo ◽  
Matteo G. A. Paris ◽  
Sabrina Maniscalco
Keyword(s):  
Continuous Time ◽  
Quantum Algorithm ◽  
Quantum Walks ◽  
Quantum Search ◽  
Star Graph ◽  
Correlated Noise ◽  
Random Telegraph Noise ◽  
Spatial Search ◽  
Telegraph Noise ◽  
Time Quantum

Abstract Continuous-time quantum walks may be exploited to enhance spatial search, i.e., for finding a marked element in a database structured as a complex network. However, in practical implementations, the environmental noise has detrimental effects, and a question arises on whether noise engineering may be helpful in mitigating those effects on the performance of the quantum algorithm. Here we study whether time-correlated noise inducing non-Markovianity may represent a resource for quantum search. In particular, we consider quantum search on a star graph, which has been proven to be optimal in the noiseless case, and analyze the effects of independent random telegraph noise (RTN) disturbing each link of the graph. Upon exploiting an exact code for the noisy dynamics, we evaluate the quantum non-Markovianity of the evolution, and show that it cannot be considered as a resource for this algorithm, since its presence is correlated with lower probabilities of success of the search.

scholarly journals Continuous-time quantum search and time-dependent two-level quantum systems

2019 ◽  
Vol 17 (03) ◽  
pp. 1950025 ◽  
Author(s):  
Carlo Cafaro ◽  
Paul M. Alsing
Keyword(s):  
Continuous Time ◽  
Transition Probabilities ◽  
Search Algorithm ◽  
Quantum Systems ◽  
Time Dependent ◽  
Quantum Search ◽  
Quantum Search Algorithm ◽  
Time Quantum ◽  
Characteristic Features ◽  
External Time

It was recently emphasized by Byrnes, Forster and Tessler [Phys. Rev. Lett. 120 (2018) 060501] that the continuous-time formulation of Grover’s quantum search algorithm can be intuitively understood in terms of Rabi oscillations between the source and the target subspaces. In this work, motivated by this insightful remark and starting from the consideration of a time-independent generalized quantum search Hamiltonian as originally introduced by Bae and Kwon [Phys. Rev. A 66 (2002) 012314], we present a detailed investigation concerning the physical connection between quantum search Hamiltonians and exactly solvable time-dependent two-level quantum systems. Specifically, we compute in an exact analytical manner the transition probabilities from a source state to a target state in a number of physical scenarios specified by a spin-[Formula: see text] particle immersed in an external time-dependent magnetic field. In particular, we analyze both the periodic oscillatory as well as the monotonic temporal behaviors of such transition probabilities and, moreover, explore their analogy with characteristic features of Grover-like and fixed-point quantum search algorithms, respectively. Finally, we discuss from a physics standpoint the connection between the schedule of a search algorithm, in both adiabatic and nonadiabatic quantum mechanical evolutions, and the control fields in a time-dependent driving Hamiltonian.

ENTANGLEMENT PROPERTIES OF ADIABATIC QUANTUM ALGORITHMS

2009 ◽  
Vol 07 (08) ◽  
pp. 1531-1539 ◽  
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.

QUANTUM SEARCH ALGORITHMS

2009 ◽  
Vol 23 (31) ◽  
pp. 5727-5758 ◽  
Author(s):  
VLADIMIR E. KOREPIN ◽  
YING XU
Keyword(s):  
Target Item ◽  
Search Algorithm ◽  
Search Algorithms ◽  
Quantum Search ◽  
Large Database ◽  
Grover Algorithm ◽  
Speed Up ◽  
The Subject ◽  
Grover Search ◽  
Target Block

This article reviews recent progress in quantum database search algorithms. The subject is presented in a self-contained and pedagogical way. The problem of searching a large database (a Hilbert space) for a target item is performed by the famous Grover algorithm which locates the target item with high probability and a quadratic speed-up compared with the corresponding classical algorithm. If the database is partitioned into blocks and one is searching for the block containing the target item instead of the target item itself, then the problem is referred to as partial search. Partial search trades accuracy for speed and the most efficient version is the Grover–Radhakrishnan–Korepin (GRK) algorithm. The target block can be further partitioned into sub-blocks so that GRK's can be performed in a sequence called a hierarchy. We study the Grover search and GRK partial search in detail and prove that a GRK hierarchy is less efficient than a direct GRK partial search. Both the Grover search and the GRK partial search can be generalized to the case with several target items (or target blocks for a GRK). The GRK partial search algorithm can also be represented in terms of group theory.

On mixing in continuous-time quantum walks on some circulant graphs

2003 ◽  
Vol 3 (6) ◽  
pp. 611-618
Author(s):  
A. Ahmadi ◽  
R. Belk ◽  
C. Tamon ◽  
C. Wendler
Keyword(s):  
Continuous Time ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Complete Graphs ◽  
Circulant Graphs ◽  
Multipartite Graphs ◽  
Time Quantum ◽  
Complete Multipartite ◽  
Cycle Graph ◽  
Mixing Property

Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that the continuous-time quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous exactly uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs of the symmetric group lack this mixing property as well.

scholarly journals CONTINUOUS-TIME QUANTUM WALKS ON ULTRAMETRIC SPACES

2006 ◽  
Vol 04 (06) ◽  
pp. 1023-1035 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Continuous Time ◽  
Search Algorithm ◽  
Quantum Walk ◽  
Ultrametric Space ◽  
Quantum Walks ◽  
Ultrametric Spaces ◽  
Quantum Search Algorithm ◽  
Time Quantum ◽  
Cycle Graph ◽  
Striking Contrast

We introduce a continuous-time quantum walk on an ultrametric space corresponding to the set of p-adic integers and compute its time-averaged probability distribution. It is shown that localization occurs for any location of the ultrametric space for the walk. This result presents a striking contrast to the classical random walk case. Moreover, we clarify a difference between the ultrametric space and other graphs, such as cycle graph, line, hypercube and complete graph, for the localization of the quantum case. Our quantum walk may be useful for a quantum search algorithm on a tree-like hierarchical structure.

scholarly journals Robust Quantum Search with Uncertain Number of Target States

Entropy ◽  
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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