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arXiv:1508.00207

Robust quantum spatial search

Avatar Tulsi

Published Aug 2, 2015• 0 citations
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Abstract

Quantum spatial search has been widely studied with most of the study focusing on quantum walk algorithms. We show that quantum walk algorithms are extremely sensitive to systematic errors. We present a recursive algorithm which offers significant robustness to certain systematic errors. To search N items, our recursive algorithm can tolerate errors of size O(1/\sqrt{\ln N}) which is exponentially better than quantum walk algorithms for which tolerable error size is only O(\ln N/\sqrt{N}). Also, our algorithm does not need any ancilla qubit. Thus our algorithm is much easier to implement experimentally compared to quantum walk algorithms.

DOI
10.48550/arxiv.1508.00207

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Quantum spatial search has been widely studied with most of the study focusing on quantum walk algorithms. We show that quantum walk algorithms are extremely sensitive to systematic errors. We present a recursive algorithm which offers significant robustness to certain systematic errors. To search N items, our recursive algorithm can tolerate errors of size O(1/\sqrt{\ln N}) which is exponentially better than quantum walk algorithms for which tolerable error size is only O(\ln N/\sqrt{N}). Also, our algorithm does not need any ancilla qubit. Thus our algorithm is much easier to implement experimentally compared to quantum walk algorithms.
Spatial search is a widely studied application of quantum algorithms. Several algorithms have been proposed for spatial search without using ancilla qubits which are easier to implement physically. All such algorithms are based on the widely studied quantum walk approach. Here we consider another approach by Aaronson and Ambainis based on the recursive application of Grover's algorithm rather than quantum walks. Their algorithm needs one ancilla qubit for successful implementation. We present a different recursive algorithm which does not need any ancilla qubit and hence it offers physical implementation advantages.
We consider the problem of finding a desired item out of NNN items arranged on the sites of a two-dimensional lattice of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$. The previous quantum-walk based algorithms take O(N ln N)O(\sqrt{N}\text{ }\text{ln}\text{ }N)O(N​ ln N) steps to solve this problem, and it is an open question whether the performance can be improved. We present an algorithm which solves the problem in O(N ln N)O(\sqrt{N\text{ }\text{ln}\text{ }N})O(N ln N​) steps, thus giving an O(ln N)O(\sqrt{\text{ln}\text{ }N})O(ln N​) improvement over the known algorithms. The improvement is achieved by controlling the quantum walk on the lattice using an ancilla qubit.
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.
This study investigates the effects of systematic errors in phase inversions on the success rate and number of iterations in the optimized quantum random-walk search algorithm. Using the geometric description of this algorithm, a model of the algorithm with phase errors is established, and the relationship between the success rate of the algorithm, the database size, the number of iterations, and the phase error is determined. For a given database size, we obtain both the maximum success rate of the algorithm and the required number of iterations when phase errors are present in the algorithm. Analyses and numerical simulations show that the optimized quantum random-walk search algorithm is more robust against phase errors than Grover’s algorithm.
One of the most important algorithmic applications of quantum walks is to solve spatial search problems. A widely used quantum algorithm for this problem, introduced by Childs and Goldstone [Phys. Rev. A 70, 022314 (2004)], finds a marked node on a graph of nnn nodes via a continuous-time quantum walk. This algorithm is said to be optimal if it can find any of the nodes in O(n)O(\sqrt{n})O(n​) time. However, given a graph, no general conditions for the optimality of the algorithm are known and previous works demonstrating optimal quantum search for certain graphs required an instance-specific analysis. In fact, the demonstration of the necessary and sufficient conditions that a graph must fulfill for quantum search to be optimal has been a long-standing open problem. In this work we make significant progress towards solving this problem. We derive general expressions, depending on the spectral properties of the Hamiltonian driving the walk, that predict the performance of this quantum search algorithm provided certain spectral conditions are fulfilled. Our predictions are valid, for example, for (normalized) Hamiltonians whose spectral gap is considerably larger than ${n}^{\ensuremath{-}1/2}$. This allows us to derive necessary and sufficient conditions for optimal quantum search in this regime, as well as provide examples of graphs where quantum search is suboptimal. In addition, by extending this analysis, we are also able to show the optimality of quantum search for certain graphs with very small spectral gaps, such as graphs that can be efficiently partitioned into clusters. Our results imply that, to the best of our knowledge, all prior results analytically demonstrating the optimality of this algorithm for specific graphs can be recovered from our general results.
Quantum random walks on graphs have been shown to display many interesting properties, including exponentially fast hitting times when compared with their classical counterparts. However, it is still unclear how to use these novel properties to gain an algorithmic speedup over classical algorithms. In this paper, we present a quantum search algorithm based on the quantum random-walk architecture that provides such a speedup. It will be shown that this algorithm performs an oracle search on a database of N items with O(N)O(\sqrt{N})O(N​) calls to the oracle, yielding a speedup similar to other quantum search algorithms. It appears that the quantum random-walk formulation has considerable flexibility, presenting interesting opportunities for development of other, possibly novel quantum algorithms.