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Recursive quantum spatial search does not need ancilla qubit

Avatar Tulsi

Published Aug 2, 2015• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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.
Grover's quantum search algorithm drives a quantum computer from a prepared initial state to a desired final state by using selective transformations of these states. Here, we analyze a framework when one of the selective transformations is replaced by a more general unitary transformation. Our framework encapsulates several previous generalizations of the Grover's algorithm. We show that the general quantum search algorithm can be improved by controlling the transformations through an ancilla qubit. As a special case of this improvement, we get a faster quantum algorithm for the two-dimensional spatial search.
This paper deals with the problem of the requirements for quantum systems that enable one to design efficient quantum algorithms. We rise the issue of the possibility to utilise the non-complete networks for algorithmic purposes. In particular we consider applications for the spatial search problems. We focus on showing that the asymptotic complexity widely discussed in the related work is not enough tool for determining the potential of the network. We provide an example of a network where the asymptotic complexity is the same for a variety of cases and yet it is not always possible to implement successful search procedure within the quantum walk scheme. The examples are based on an Apollonian network which models a variety of iteratively generated planar networks. The network is planar, exhibits linear growth of edges number, consists nodes of different degrees and has the small-world and scale-free properties. This motivates its analysis due to the simplicity in terms of connections density and potential for quantum phenomena due to the nodes diversity.
In the sub-field of quantum algorithms, physicists and computer scientist take classical computing algorithms and principles and see if there is a more efficient or faster approach implementable on a quantum computer, i.e. a ”quantum advantage”. We take random walks, a widely applicable group of classical algorithms, and move them into the quantum computing paradigm. Additionally, an introduction to a popular quantum search algorithm called Grover’s search is included to guide the reader to the development of a quantum search algorithm using quantum random walks. To close the gap between algorithm and hardware, we will look at using neutral-atom (also known as cold-atom) quantum computing to physically implement two types of quantum random walks and Grover’s algorithm.