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Quantum search algorithms

Birgit Hein

Published Oct 15, 2010• 0 citations

Abstract

In this thesis two quantum search algorithms on two different graphs, a hypercube and a d-dimensional square lattice, are analysed and some applications of the lattice search are discussed.\nThe approach in this thesis generalises a picture drawn by Shenvi, Kempe and Whaley, which was later adapted by Ambainis, Kempe and Rivosh. It defines a one parameter family of unitary operators U_λ with parameter λ. It will be shown that two eigenvalues of U_λ form an avoided crossing at the λ-value where U_λ is equal to the old search operator.\nThis generalised picture opens the way for a construction of two approximate eigen- vectors at the crossing and gives rise to a 2×2 model Hamiltonian that is used to approximate the operator U_λ near the crossing. The thus defined Hamiltonian can be used to calculate the leading order of search time and success probability for the search. To the best of my knowledge only the scaling of these quantities has been known.\nFor the algorithm searching the regular lattice, a generalisation of the model Hamiltonian for m target vertices is constructed. This algorithm can be used to send a signal from one vertex of the graph to a set of vertices. The signal is transmitted between these vertices exclusively and is localised only on the sender and the receiving vertices while the probability to measure the signal at one of the remaining vertices is significantly smaller.\nHowever, this effect can be used to introduce an additional sender to search settings and send a continuous signal to all target vertices where the signal will localise. This effect is an improvement compared to the original search algorithm as it does not need to know the number of target vertices.

Related papers

Quantum search algorithms on the hypercube

Birgit Hein, Gregor Tanner

We investigate a set of discrete-time quantum search algorithms on the n -dimensional hypercube following a proposal by Shenvi et al (2003 Phys. Rev. A 67 052307). We show that there exists a whole class of quantum search algorithms in the symmetry-reduced space which perform a search of a marked vertex in time of order where N = 2 n , the number of vertices. In analogy to Grover's algorithm, the spatial search is effectively facilitated through a rotation in a two-level subspace of the full Hilbert space. In the hypercube, these two-level systems are introduced through avoided crossings. We give estimates on the quantum states forming the two-level subspaces at the avoided crossings and derive improved estimates on the search times.

arXiv:0906.3094•Journal of Physics A Mathematical and Theoretical•Jan 30, 2009•26 citations

Quantum search algorithms on a regular lattice

Birgit Hein, Gregor Tanner

Quantum algorithms for searching for one or more marked items on a $d$-dimensional lattice provide an extension of Grover's search algorithm including a spatial component. We demonstrate that these lattice search algorithms can be viewed in terms of the level dynamics near an avoided crossing of a one-parameter family of quantum random walks. We give approximations for both the level splitting at the avoided crossing and the effectively two-dimensional subspace of the full Hilbert space spanning the level crossing. This makes it possible to give the leading order behavior for the search time and the localization probability in the limit of large lattice size including the leading order coefficients. For $d=2$ and $d=3$, these coefficients are calculated explicitly. Closed form expressions are given for higher dimensions.

arXiv:1005.3676•Physical Review A•Jul 23, 2010•40 citations

Algorithme de recherche par marche quantique sur hypercube : analyse propre et calcul de la probabilité de succès en temps polynomial

Hugo Pillin

Discrete time quantum walk algorithms are the direct analogs of classical random walk algorithms, an important family in theoretical informatics. They can solve a variety of problems, in particular the NPcomplete SAT problem, in a relatively short time. Thus, the study of quantum walks could lead to the conception of new quantum algorithms. Among the quantum algorithms, one of the most extensively studied is Grover’s searchalgorithm, an iterative process that can find one element among N that meet a certain criterion, with a quadratic time gain over any classical algorithm. In this thesis, we study the variant of Grover’s search algorithm proposed by Shenvi, Kempe and Whaley, based on walks on hypercube graphs.As search algorithms are iterative processes, one has to compute the correct number of steps R before running the search, as the probability of success decreases past a certain point. The value of R depends on the number of elements N, but also the number of solutions M. For Grover's algorithm, the optimal number of iterations is well known, but this is not the case for the search on the hypercube as soon as there are multiple solutions. In this thesis, we propose an extensive eigenanalysis of the hypercube search algorithm, which leads to a procedure that allows us to compute the probability of success of the search algorithm on the hypercube without running it, in polynomial time.

•Apr 5, 2024•0 citations

Quantum Search on Strongly Regular Graphs

Jonatan Janmark

In 1996, Farhi & Gutmann presented a continuous time analogue of Grover’s algorithm which was later interpreted as search by a quantum random walk on the complete graph. Similar work has since then been done on hypercubes and periodic lattices. We investigate the same search algorithm but for a new type of graphs called strongly regular graphs. The problem is solved by a, in this context, new technique involving degenerate perturbation theory. The result is that the search is optimal on all strongly regular graphs when the number of vertices N is large. However, different kinds of strongly regular graphs perform different when N is not large. Analyzing this provides some insight into what properties of graphs that are important for quantum search. Apart from the results, an important takeaway is the degenerate perturbation theory technique that was used to solve the problem and might be useful in solving other similar problems. In the thesis, an alternative solution to the complete graph case is given using this method. Also, the fact that strongly regular graphs proved to be convenient to analyze in these kinds of settings suggests that they can be used in other similar problems.

•Jan 1, 2013•1 citation

The quantum walk search algorithm: factors affecting efficiency

NEIL B. LOVETT, MATTHEW EVERITT, ROBERT M. HEATH +1 more

We carry out a numerical study of the quantum walk search algorithm of Shenvi, Kempe and Whaley Shenvi et al. (2003) and the factors that affect its efficiency in finding an individual state from an unsorted set. Previous work has focused purely on the effects of the dimensionality of the dataset to be searched. In the current paper we consider the effects of interpolating between dimensions, the connectivity of the dataset and the possibility of disorder in the underlying substrate: all these factors affect the efficiency of the search algorithm. We show that in addition to the strong dependence on the spatial dimension of the structure to be searched, there are also secondary dependencies on the connectivity and symmetry of the lattice, with greater connectivity providing a more efficient algorithm. We also show that the algorithm can tolerate a non-trivial level of disorder in the underlying substrate.

arXiv:1110.4366•Mathematical Structures in Computer Science•May 16, 2018•10 citations

Quantum walk search algorithm for multi-objective searching with iteration auto-controlling on hypercube

Yao-Yao Jiang, Peng-Cheng Chu, Wenbin Zhang +1 more

Shenvi et al. have proposed a quantum algorithm based on quantum walking called Shenvi–Kempe–Whaley (SKW) algorithm, but this search algorithm can only search one target state and use a specific search target state vector. Therefore, when there are more than two target nodes in the search space, the algorithm has certain limitations. Even though a multi-objective SKW search algorithm was proposed later, when the number of target nodes is more than two, the SKW search algorithm cannot be mapped to the same quotient graph. In addition, the calculation of the optimal target state depends on the number of target states m. In previous studies, quantum computing and testing algorithms were used to solve this problem. But these solutions require more Oracle calls and cannot get a high accuracy rate. Therefore, to solve the above problems, we improve the multi-target quantum walk search algorithm, and construct a controllable quantum walk search algorithm under the condition of unknown number of target states. By dividing the Hilbert space into multiple subspaces, the accuracy of the search algorithm is improved from p c = (1/2) – O(1/n) to p c = 1 – O(1/n). And by adding detection gate phase, the algorithm can stop when the amplitude of the target state becomes the maximum for the first time, and the algorithm can always maintain the optimal number of iterations, so as to reduce the number of unnecessary iterations in the algorithm process and make the number of iterations reach <?CDATA ${t}_{{\rm{f}}}=(\pi /2)\sqrt{{2}^{n-2}}$?> t f = ( π / 2 ) 2 n − 2 .

Chinese Physics B•Sep 18, 2021•1 citation

In this thesis two quantum search algorithms on two different graphs, a hypercube and a d-dimensional square lattice, are analysed and some applications of the lattice search are discussed.\nThe approach in this thesis generalises a picture drawn by Shenvi, Kempe and Whaley, which was later adapted by Ambainis, Kempe and Rivosh. It defines a one parameter family of unitary operators U_λ with parameter λ. It will be shown that two eigenvalues of U_λ form an avoided crossing at the λ-value where U_λ is equal to the old search operator.\nThis generalised picture opens the way for a construction of two approximate eigen- vectors at the crossing and gives rise to a 2×2 model Hamiltonian that is used to approximate the operator U_λ near the crossing. The thus defined Hamiltonian can be used to calculate the leading order of search time and success probability for the search. To the best of my knowledge only the scaling of these quantities has been known.\nFor the algorithm searching the regular lattice, a generalisation of the model Hamiltonian for m target vertices is constructed. This algorithm can be used to send a signal from one vertex of the graph to a set of vertices. The signal is transmitted between these vertices exclusively and is localised only on the sender and the receiving vertices while the probability to measure the signal at one of the remaining vertices is significantly smaller.\nHowever, this effect can be used to introduce an additional sender to search settings and send a continuous signal to all target vertices where the signal will localise. This effect is an improvement compared to the original search algorithm as it does not need to know the number of target vertices.

We investigate a set of discrete-time quantum search algorithms on the n -dimensional hypercube following a proposal by Shenvi et al (2003 Phys. Rev. A 67 052307). We show that there exists a whole class of quantum search algorithms in the symmetry-reduced space which perform a search of a marked vertex in time of order where N = 2 n , the number of vertices. In analogy to Grover's algorithm, the spatial search is effectively facilitated through a rotation in a two-level subspace of the full Hilbert space. In the hypercube, these two-level systems are introduced through avoided crossings. We give estimates on the quantum states forming the two-level subspaces at the avoided crossings and derive improved estimates on the search times.

Quantum algorithms for searching for one or more marked items on a

d

-dimensional lattice provide an extension of Grover's search algorithm including a spatial component. We demonstrate that these lattice search algorithms can be viewed in terms of the level dynamics near an avoided crossing of a one-parameter family of quantum random walks. We give approximations for both the level splitting at the avoided crossing and the effectively two-dimensional subspace of the full Hilbert space spanning the level crossing. This makes it possible to give the leading order behavior for the search time and the localization probability in the limit of large lattice size including the leading order coefficients. For

d=2

and

d=3

, these coefficients are calculated explicitly. Closed form expressions are given for higher dimensions.

Discrete time quantum walk algorithms are the direct analogs of classical random walk algorithms, an important family in theoretical informatics. They can solve a variety of problems, in particular the NPcomplete SAT problem, in a relatively short time. Thus, the study of quantum walks could lead to the conception of new quantum algorithms. Among the quantum algorithms, one of the most extensively studied is Grover’s searchalgorithm, an iterative process that can find one element among N that meet a certain criterion, with a quadratic time gain over any classical algorithm. In this thesis, we study the variant of Grover’s search algorithm proposed by Shenvi, Kempe and Whaley, based on walks on hypercube graphs.As search algorithms are iterative processes, one has to compute the correct number of steps R before running the search, as the probability of success decreases past a certain point. The value of R depends on the number of elements N, but also the number of solutions M. For Grover's algorithm, the optimal number of iterations is well known, but this is not the case for the search on the hypercube as soon as there are multiple solutions. In this thesis, we propose an extensive eigenanalysis of the hypercube search algorithm, which leads to a procedure that allows us to compute the probability of success of the search algorithm on the hypercube without running it, in polynomial time.

In 1996, Farhi & Gutmann presented a continuous time analogue of Grover’s algorithm which was later interpreted as search by a quantum random walk on the complete graph. Similar work has since then been done on hypercubes and periodic lattices. We investigate the same search algorithm but for a new type of graphs called strongly regular graphs. The problem is solved by a, in this context, new technique involving degenerate perturbation theory. The result is that the search is optimal on all strongly regular graphs when the number of vertices N is large. However, different kinds of strongly regular graphs perform different when N is not large. Analyzing this provides some insight into what properties of graphs that are important for quantum search. Apart from the results, an important takeaway is the degenerate perturbation theory technique that was used to solve the problem and might be useful in solving other similar problems. In the thesis, an alternative solution to the complete graph case is given using this method. Also, the fact that strongly regular graphs proved to be convenient to analyze in these kinds of settings suggests that they can be used in other similar problems.

We carry out a numerical study of the quantum walk search algorithm of Shenvi, Kempe and Whaley Shenvi et al. (2003) and the factors that affect its efficiency in finding an individual state from an unsorted set. Previous work has focused purely on the effects of the dimensionality of the dataset to be searched. In the current paper we consider the effects of interpolating between dimensions, the connectivity of the dataset and the possibility of disorder in the underlying substrate: all these factors affect the efficiency of the search algorithm. We show that in addition to the strong dependence on the spatial dimension of the structure to be searched, there are also secondary dependencies on the connectivity and symmetry of the lattice, with greater connectivity providing a more efficient algorithm. We also show that the algorithm can tolerate a non-trivial level of disorder in the underlying substrate.

Shenvi et al. have proposed a quantum algorithm based on quantum walking called Shenvi–Kempe–Whaley (SKW) algorithm, but this search algorithm can only search one target state and use a specific search target state vector. Therefore, when there are more than two target nodes in the search space, the algorithm has certain limitations. Even though a multi-objective SKW search algorithm was proposed later, when the number of target nodes is more than two, the SKW search algorithm cannot be mapped to the same quotient graph. In addition, the calculation of the optimal target state depends on the number of target states m. In previous studies, quantum computing and testing algorithms were used to solve this problem. But these solutions require more Oracle calls and cannot get a high accuracy rate. Therefore, to solve the above problems, we improve the multi-target quantum walk search algorithm, and construct a controllable quantum walk search algorithm under the condition of unknown number of target states. By dividing the Hilbert space into multiple subspaces, the accuracy of the search algorithm is improved from p c = (1/2) – O(1/n) to p c = 1 – O(1/n). And by adding detection gate phase, the algorithm can stop when the amplitude of the target state becomes the maximum for the first time, and the algorithm can always maintain the optimal number of iterations, so as to reduce the number of unnecessary iterations in the algorithm process and make the number of iterations reach <?CDATA

{t}_{{\rm{f}}}=(\pi /2)\sqrt{{2}^{n-2}}

?> t f = ( π / 2 ) 2 n − 2 .