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arXiv:quant-ph/0306054 · Physical Review A

Spatial search by quantum walk

Andrew M. Childs, Jeffrey Goldstone

Published Aug 23, 2004• 726 citations
arXivHTMLPDF

Abstract

Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of $N$ items laid out in $d$ spatial dimensions can be searched in time of order $\sqrt{N}$ for $d>2$, and in time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$ for $d=2$. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that $\sqrt{N}$ speedup can also be achieved on the hypercube. We show that full $\sqrt{N}$ speedup can be achieved on a $d$-dimensional periodic lattice for $d>4$. In $d=4$, the quantum walk search algorithm takes time of order $\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)$, and in $d<4$, the algorithm does not provide substantial speedup.

DOI
10.1103/physreva.70.022314

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•Jan 1, 2013•1 citation
Grover's quantum search algorithm provides a way to speed up combinatorial search, but is not directly applicable to searching a physical database. Nevertheless, Aaronson and Ambainis showed that a database of NNN items laid out in ddd spatial dimensions can be searched in time of order N\sqrt{N}N​ for d>2d>2d>2, and in time of order Npoly(logN)\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)N​poly(logN) for d=2d=2d=2. We consider an alternative search algorithm based on a continuous-time quantum walk on a graph. The case of the complete graph gives the continuous-time search algorithm of Farhi and Gutmann, and other previously known results can be used to show that N\sqrt{N}N​ speedup can also be achieved on the hypercube. We show that full N\sqrt{N}N​ speedup can be achieved on a ddd-dimensional periodic lattice for d>4d>4d>4. In d=4d=4d=4, the quantum walk search algorithm takes time of order Npoly(logN)\sqrt{N}\phantom{\rule{0.3em}{0ex}}\text{poly}(\mathrm{log}\phantom{\rule{0.3em}{0ex}}N)N​poly(logN), and in d<4d<4d<4, the algorithm does not provide substantial speedup.
Many-body quantum systems with effective nonlinearities have been shown to speed up quantum search on the complete graph, \textit{i.e.}, the combinatorial version of Grover's algorithm, at the expense of the number of particles needed for the effective nonlinearity to hold. Physically, however, data may not be arranged in an all-to-all network, and the task of searching incomplete graphs is the spatial search problem. We explore spatial search using a continuous-time nonlinear quantum walk on a variety of graphs. First, we consider incomplete graphs that are ``sufficiently complete'' so as to asymptotically search like the complete graph under a continuous-time (linear) quantum walk, which includes strongly regular graphs such as Paley graphs, regular graphs such as hypercubes, and irregular graphs such as complete bipartite graphs. For these sufficiently complete graphs, we analytically prove nonlinear speedups for Paley graphs and for complete bipartite graphs whose two partite sets both have size Θ(N)Θ(N)Θ(N), for suitable cubic and cubic-quintic nonlinearities, and we give numerical evidence for stronger nonlinearities and for hypercubes. Second, we explore arbitrary-dimensional cubic lattices, and we numerically show that certain nonlinearities speed up search on sufficiently high dimensional lattices. Thus, nonlinear quantum search can remain viable even when the underlying graph is incomplete.
A quantum walk on a lattice is a paradigm of a quantum search in a database. The database qubit strings are the lattice sites, qubit rotations are tunneling events, and the target site is tagged by an energy shift. For quantum walks on a continuous time, the walker diffuses across the lattice and the search ends when it localizes at the target site. The search time T can exhibit Grover's optimal scaling with the lattice size N , namely, T ∼ N , on an all-connected, complete lattice. For finite-range tunneling between sites, instead, Grover's optimal scaling is warranted when the lattice is a hypercube of d > 4 dimensions. Here, we show that Grover's optimum can be reached in lower dimensions on lattices of long-range interacting particles, when the interaction strength scales algebraically with the distance r as 1 / r α and 0 < α < 3 d / 2 . For α < d the dynamics mimics the one of a globally connected graph. For d < α < d + 2 , the quantum search on the graph can be mapped to a short-range model on a hypercube with spatial dimension d s = 2 d / ( α − d ) , indicating that the search is optimal for d s > 4 . Our work identifies an exact relation between criticality of long-range and short-range systems, it provides a quantitative demonstration of the resources that long-range interactions provide for quantum technologies, and indicates when existing experimental platforms can implement efficient analog quantum search algorithms.
We consider the problem of searching a general ddd-dimensional lattice of NNN vertices for a single marked item using a continuous-time quantum walk. We demand locality, but allow the walk to vary periodically on a small scale. By constructing lattice Hamiltonians exhibiting Dirac points in their dispersion relations and exploiting the linear behavior near a Dirac point, we develop algorithms that solve the problem in a time of O(N)O(\sqrt{N})O(N​) for d>2d>2d>2 and O(NlogN)O(\sqrt{N}\phantom{\rule{0.16em}{0ex}}logN)O(N​logN) in d=2d=2d=2. In particular, we show that such algorithms exist even for hypercubic lattices in any dimension. Unlike previous continuous-time quantum walk algorithms on hypercubic lattices in low dimensions, our approach does not use external memory.
The task of finding an entry in an unsorted list of NNN elements famously takes O(N)O(N)O(N) queries to an oracle for a classical computer and O(N)O(\sqrt{N})O(N​) queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large NNN, the random walks converge to the same evolution, both taking Nln⁡(1/ϵ)N \ln(1/\epsilon)Nln(1/ϵ) time to reach a success probability of 1−ϵ1-\epsilon1−ϵ. In contrast, the discrete-time quantum walk asymptotically takes πN/22\pi\sqrt{N}/2\sqrt{2}πN​/22​ timesteps to reach a success probability of 1/21/21/2, while the continuous-time quantum walk takes πN/2\pi\sqrt{N}/2πN​/2 time to reach a success probability of 111.
We show how to search N items arranged on a N×N\sqrt{N}\times\sqrt{N}N​×N​ grid in time O(Nlog⁡N)O(\sqrt N \log N)O(N​logN), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time Ω(N)Ω(N)Ω(N) to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of O(N)O(\sqrt{N})O(N​) and give several extensions of quantum walk search algorithms for general graphs. The coin-flip operation needs to be chosen judiciously: we show that another ``natural'' choice of coin gives a walk that takes Ω(N)Ω(N)Ω(N) steps. We also show that in 2 dimensions it is sufficient to have a two-dimensional coin-space to achieve the time O(Nlog⁡N)O(\sqrt{N} \log N)O(N​logN).
One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a lattice or graph containing a marked site. In most conceivable physical applications, however, one would rather expect to have multiple interacting particles. In bosonic systems at zero temperature, the dynamics would be well-described by a discrete non-linear Schrodinger equation. In this thesis we investigate the role of non-linearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. Our analytical results indicate that the search time for this non-linear quantum search scales with size of the database N like square root of N, equivalent to linear spatial search time. The analytical results will be compared with numerical calculations of multiple interacting quantum walkers.