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arXiv:quant-ph/0201152 · Physical Review A

Trade-offs in the quantum search algorithm

Lov K. Grover

Published Nov 21, 2002• 50 citations
arXivHTMLPDF

Abstract

Quantum search has been proved to be the best possible algorithm for the exhaustive search problem in the sense that the number of queries it requires cannot be reduced. However, the number of nonquery operations, and thus the total number of operations, can be reduced. The number of nonquery unitary operations can be reduced by a factor of $\mathrm{log}N/\ensuremath{\alpha}\mathrm{log}(\mathrm{log}N)$ while increasing the number of queries by a factor of only $[1+(\mathrm{log}{N)}^{\ensuremath{-}\ensuremath{\alpha}}].$ For example, by choosing \ensuremath{\alpha} to be $O(\mathrm{log}N/\mathrm{log}(\mathrm{log}N)),$ the number of nonquery unitary operations can be reduced by 40% while increasing the number of queries by just two.

DOI
10.1103/physreva.66.052314

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Quantum search has been proved to be the best possible algorithm for the exhaustive search problem in the sense that the number of queries it requires cannot be reduced. However, the number of nonquery operations, and thus the total number of operations, can be reduced. The number of nonquery unitary operations can be reduced by a factor of logN/\ensuremathαlog(logN)\mathrm{log}N/\ensuremath{\alpha}\mathrm{log}(\mathrm{log}N)logN/\ensuremathαlog(logN) while increasing the number of queries by a factor of only [1+(logN)\ensuremath−\ensuremathα].[1+(\mathrm{log}{N)}^{\ensuremath{-}\ensuremath{\alpha}}].[1+(logN)\ensuremath−\ensuremathα]. For example, by choosing \ensuremath{\alpha} to be O(logN/log(logN)),O(\mathrm{log}N/\mathrm{log}(\mathrm{log}N)),O(logN/log(logN)), the number of nonquery unitary operations can be reduced by 40% while increasing the number of queries by just two.
In its usual form, Grover’s quantum search algorithm uses O( √ N) queries and O( √ N log N) other elementary gates to find a solution in an N-bit database. Grover in 2002 showed how to reduce the number of other gates to O( √ N log log N) for the special case where the database has a unique solution, without significantly increasing the number of queries. We show how to reduce this further to O( √ N log(r) N) gates for every constant r, and sufficiently large N. This means that, on average, the circuits between two queries barely touch more than a constant number of the log N qubits on which the algorithm acts. For a very large N that is a power of 2, we can choose r such that the algorithm uses essentially the minimal number π 4 √ N of queries, and only O( √ N log(log? N)) other gates.
Query complexity is a model of computation in which we have to compute a function f(x1,…,xN)f(x_1, \ldots, x_N)f(x1​,…,xN​) of variables xix_ixi​ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we survey these results and related work, including: - the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring Ω(N)Ω(\sqrt{N})Ω(N​) queries classically); - the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with MMM queries quantumly but requiring Ω~(M2.5)\tildeΩ(M^{2.5})Ω~(M2.5) queries probabilistically); - the biggest probabilistic-vs-deterministic gap for total functions (a problem solvable with MMM queries probabilistically but requiring Ω~(M2)\tildeΩ(M^{2})Ω~(M2) queries deterministically); - the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions); - the connections between query algorithms and approximations by low-degree polynomials.
Grover's search algorithm drives a quantum system from an initial state $|s\ensuremath{\rangle}$ to a desired final state $|t\ensuremath{\rangle}$ by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of Is{I}_{s}Is​, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator Ds{D}_{s}Ds​ with the only restriction of having $|s\ensuremath{\rangle}$ as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of O(B)O(B)O(B), where BBB is a characteristic of the eigenspectrum of Ds{D}_{s}Ds​ and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still O(B)O(B)O(B) times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.
We consider the problem of finding one or more desired items out of an unsorted database. Patel has shown that if the database permits quantum queries, then mere digitization is sufficient for efficient search for one desired item. The algorithm, called factorized quantum search algorithm, presented by him can locate the desired item in an unsorted database using O(log4N)O(log_{4}N)O(log4​N) queries to factorized oracles. But the algorithm requires that all the property values must be distinct from each other. In this paper, we discuss how to make a database satisfy the requirements, and present a quantum search engine based on the algorithm. Our goal is achieved by introducing auxiliary files for the property values that are not distinct, and converting every complex query request into a sequence of calls to factorized quantum search algorithm. The query complexity of our algorithm is O(P∗Q∗M∗log4N)O(P*Q*M*log_{4}N)O(P∗Q∗M∗log4​N), where P is the number of the potential simple query requests in the complex query request, Q is the maximum number of calls to the factorized quantum search algorithm of the simple queries, M is the number of the auxiliary files for the property on which our algorithm are searching for desired items. This implies that to manage an unsorted database on an actual quantum computer is possible and efficient.
One of the most basic computational problems is the task of finding a desired item in an ordered list of NNN items. While the best classical algorithm for this problem uses log2N{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}Nlog2​N queries to the list, a quantum computer can solve the problem using a constant factor fewer queries. However, the precise value of this constant is unknown. By characterizing a class of quantum query algorithms for the ordered search problem in terms of a semidefinite program, we find quantum algorithms for small instances of the ordered search problem. Extending these algorithms to arbitrarily large instances using recursion, we show that there is an exact quantum ordered search algorithm using $4\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{605}\phantom{\rule{0.2em}{0ex}}N\ensuremath{\approx}0.433\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}N$ queries, which improves upon the previously best known exact algorithm.