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arXiv:quant-ph/0106152

A Novel Approach to Quantum Heuristics for Structured Database Search

Murphy, Brian

Published Jun 27, 2001• 0 citations
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Abstract

An algorithm for structured database searching is presented and used to solve the set partition problem. O(n) oracle calls are required in order to obtain a solution, but the probability that this solution is optimal decreases exponentially with problem size. Each oracle call is followed by a measurement, implying that it is necessary to maintain quantum coherence for only one oracle call at a time.

DOI
10.48550/arxiv.quant-ph/0106152

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An algorithm for structured database searching is presented and used to solve the set partition problem. O(n) oracle calls are required in order to obtain a solution, but the probability that this solution is optimal decreases exponentially with problem size. Each oracle call is followed by a measurement, implying that it is necessary to maintain quantum coherence for only one oracle call at a time.
We introduce a structured quantum search algorithm that leverages entanglement maps and a fixed-point method to minimize oracle query complexity in unsorted datasets. By partitioning qubits into rows based on their entanglement order, the algorithm enables parallel subspace searches, achieving solution identification with at most two oracle calls per row. Experimental results on IBM Kyiv hardware demonstrate successful searches in datasets with up to 5 TB of unsorted data. Our findings indicate that with optimal encoding, the quantum search complexity becomes O(1)\mathcal{O}(1)O(1), that is, independent of the dataset size NNN, surpassing both classical O(N)\mathcal{O}(N)O(N) and Grover's O(N)\mathcal{O}(\sqrt{N})O(N​) scaling. Furthermore, the letter hypothesizes a scalable simulation of the said algorithm using classical means.
IntroductionFor a search problem associated with a unsorted database, the remarkable Grover's quantum algorithm (1) provides a quadratic speedup over its classical counterpart.The search problem can be described as follows: for a given function f , there exists one unknown element in the set {0, 1, ..., N -1} that satisfies f (x)=-1, say x = τ, whereas the other N -1o n e sg i v e f (x)=1.How many times of evaluations of f are required to determine the element τ for f (τ)=-1?Through a conventional algorithm, one needs O(N) trials to achieve this aim.How about the utility of quantum algorithm for the search?For the scenario in the quantum world, the search problem can be rephrased in the quantum mechanical language: for a given unitary operator I τ , that is sometimes called the oracle operator, and a set of state vectors (orthonormal basis): s = {|0 , |1 , ..., |N -1 }, I τ |x = |x for all states in the set except I τ |x = -|x for x = τ.H owma n yqueriesofI τ are required to determine |τ ?By Grover's algorithm, one needs only O( √ N) quantum mechanical steps to find the marked state |τ out.It has been shown that Grover's algorithm is optimal since it needs minimal oracle calls to perform a quantum search (2).The quantum searching process will be briefly reviewed as follows.The first step of Grover's algorithm is to prepare a superposition state of all elements with uniform probability amplitude:Then apply the Grover kernel G = -I η I τ to |s ,w h er eI η is a unitary operator and contains no bias against the marked state.For large N,a f t e ra b o u tm = π √ N/4 repetitions of G operations, the probability to observe |τ is close to one, i.e., G m |s ∼ |τ .( 2 ) Since every single G involves one query of I τ ,onlyO( √ N) searching steps are required for a quantum search task.In what follows, we will first investigate on the general SU(2) formulation for the kernel of Grover's searching operator G.The discussions of quantum searching certainty, robustness, and the analog analogue version of the Grover's algorithm will be given afterwards.
We develop three new quantum algorithms for searching the desired target state in the unstructured database of size N. The first algorithm requires Log N iterative steps. It constructs two quantum bags of equal size in terms of two quantum states, out of which exactly one quantum state will have nonzero overlap with the target state. This determination of overlap is done by taking the inner product, in Log N time [2], of the implicitly known target state with any one of these two quantum states. The second algorithm requires just one single step which uses a new suitable operator and the choice of this operator is problem dependent, i.e. it depends upon the number of qubits required to be used to represent an element in the index set. The third algorithm again requires only a single step and this algorithm makes use of a fixed (same) operator. It is known that algorithm for unstructured database search can be easily adaptable for solving NP-Complete problems. However, the computational complexity of NP-Complete problems after the adaptations of both the classical as well as quantum [1] search algorithms remains of the exponential order as the exponent for quantum [1] algorithm changes only to one-half times the exponent for classical algorithm. But for our quantum algorithms the exponent falls substantially so that our new quantum algorithms for unstructured search are capable if reducing the computational complexity of NP-Complete problems to polynomial order!
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.
Database search has wide applications and is used as a subroutine in many important algorithms. In this paper, we will consider a database with a single target item. Quantum algorithm (Grover) locates the target item faster than any classical algorithm. In addition to a full (Grover) search, it frequently occurs that one is looking for a group of items (a block) containing the target item, rather than the target item itself. This problem is known as partial search. As a generalization of the full search, partial search is of particular importance in practice. Partial search trades accuracy for speed, i.e., it works faster than a full search. There exists different versions of partial search. We will study the optimized version of the algorithm discovered by Grover and Radhakrishnan and call it GRK. GRK can be applied successively (in a sequence). First, the database is partitioned into blocks and GRK is applied to find the target block. This target block is then partitioned into subblocks and GRK is used again to find the target subblock. This procedure can be repeated if the database is large enough. (This sequence of GRK's is called a hierarchy.) Another possibility is to partition the database into subblocks directly and use GRK to find the target subblock once. In this paper, we will prove that the latter is faster (makes less queries to the Oracle).