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arXiv:quant-ph/0301007 · Journal of Physics A Mathematical and General

Constant-time solution to the global optimization problem using Br schweiler s ensemble search algorithm

V. Protopopescu, C D Helon, J. Barhen

Published Jun 5, 2003• 6 citations
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Abstract

A constant-time solution of the continuous global optimization problem (GOP) is obtained by using an ensemble algorithm. We show that under certain assumptions, the solution can be guaranteed by mapping the GOP onto a discrete unsorted search problem, whereupon Brüschweiler's ensemble search algorithm is applied. For adequate sensitivities of the measurement technique, the query complexity of the ensemble search algorithm depends linearly on the size of the function's domain. Advantages and limitations of an eventual NMR implementation are discussed.

DOI
10.1088/0305-4470/36/24/104

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arXiv:quant-ph/0112162•Physical Review A•Nov 22, 2002•24 citations
A constant-time solution of the continuous global optimization problem (GOP) is obtained by using an ensemble algorithm. We show that under certain assumptions, the solution can be guaranteed by mapping the GOP onto a discrete unsorted search problem, whereupon Brüschweiler's ensemble search algorithm is applied. For adequate sensitivities of the measurement technique, the query complexity of the ensemble search algorithm depends linearly on the size of the function's domain. Advantages and limitations of an eventual NMR implementation are discussed.
An exponentially fast solution of the continuous Global Optimization Problem (GOP) is obtained by using an ensemble algorithm. We show that under certain assumptions, one can map the GOP onto a discrete unsorted search problem, whereupon Bruschweiler's ensemble search algorithm [Phys. Rev. Lett. 85 4815 (2000)] is applied. For adequate sensitivities of the measurement techniques, the query complexity of the ensemble search algorithm depends linearly - as opposed to exponentially - on the size of the function's domain. Opportunities and limitations of an eventual NMR implementation are discussed.
Abstract Brüschweiler's Ensemble algorithm requires n = log N queries to discover the wanted object from the database of the size N. In this paper, we use a new oracle query function g(.) to improve Brüschweiler's algorithm such that only one query is needed for searching the single object. Our algorithm extends from one ancillary qubit to n ancillary qubits. We then construct the oracle query function g(.) satisfying g(x) = 0 for all input x, except for one, say x = z, where g(z) = z. We first prepare 2n qubits, including n qubits for representing the elements and n ancillary qubits for storing the result of the oracle query function. After one query, we obtain z by measuring these ancillary qubits. Our algorithm can efficiently discover the wanted object if N ≤ 230 according to current NMR spectrometer technology.
In this paper, we present an ensemble algorithm for selection problem to find the k-th smallest element in the unsorted database. We will search the k-th smallest element by using "divide-and-conquer" strategy. We first divide D, the domain of the database, into two parts, determine which of the two parts the object element sought belongs to, and then concentrate on that part. We repeat divide that part until object element is found. The determination of which part depends on the output of ensemble counting scheme, which outputs the number of assignments satisfying the value of the oracle query function is set to one. Our algorithm modifies the ensemble counting scheme by constructing a new oracle query function g_y(j). We set g_y(j) to one if the j-th element is less than or equal to y. At first, we set y to the middle value of D and perform the ensemble counting scheme with the oracle query function g_y(.) to compute the number C, the number of j satisfying g_y(j)=1. If C>k, the object element lies in the first half of D. If C<=k, then it must be in the second half of D. We recursively apply this method by adapting y until the object element is found. Our algorithm thus requires O(ln|D|) oracle queries for adequate measure accuracy to find the k-th smallest element, where |D| denotes the size of D.
In recent years, quantum computing research has made big progress, which exploit quantum mechanical laws, such as interference, superposition and parallelism, to perform computing tasks. The most inducing thing is that the quantum computing can provide large rise to the speedup in quantum algorithm. Quantum computing can solve some problems, which are impossible or difficult for the classical computing. The problem of searching for a specific item in an unsorted database can be solved with certain quantum algorithm, for example, Grover quantum algorithm and Bruschweiler quantum algorithm. The former gives a quadratic speedup, and the latter gives an exponential speedup comparing with the corresponding classical algorithm. In Bruschweiler quantum searching algorithm, the data qubit and the read-out qubit (the ancilla qubit) are different qubits. We have studied Bruschweiler algorithm and proposed a modified version, in which no ancilla qubit is needed to reach exponential speedup in the searching, the data and the read-out qubit are the same qubits. The modified Bruschweiler algorithm can be easier to design and realize. We also demonstrate the modified Bruschweiler algorithm in a 3-qubit molecular system by Nuclear Magnetic Resonance (NMR) experiment.
The quantum search problem is an important problem due to the fact that a general NP problem can be solved efficiently by an unsorted quantum search algorithm. Here it has been shown that the quantum search problem could be solved in polynomial time on an NMR quantum computer. The NMR ensemble quantum computation is based on the quantum mechanical unitary dynamics that both a closed quantum system and its ensemble obey the same quantum mechanical unitary dynamics instead of on the pseudopure state or the effective pure state of the classical NMR quantum computation. Based on the new principle the conventional NMR multiple-quantum spectroscopy has been developed to solve experimentally the search problem. The solution information of the search problem is first loaded on the unitary evolution propagator which is constructed with the oracle unitary operation and oracle-independent unitary operations and is used to excite the multiple-quantum coherence in a spin ensemble. Then the multiple- quantum spectroscopy is used to extract experimentally the solution information. It has been discussed how to enhance the output NMR signal of the quantum search NMR multiple-quantum pulse sequence and some approaches to enhancing the NMR signal are also proposed. The present work could be helpful for conventional high field NMR machines to solve efficiently the quantum search problem.
The speedup of an unsorted database search has always been a challenging problem not only in classical computation but also in quantum computation. We propose a fetching algorithm for a Liouville space nuclear magnetic resonance (NMR) ensemble computing to fetch a marked item from an unsorted database directly. This algorithm achieves the absolute optimum in an unsorted database search problem: a single query. In fact, this algorithm also works well for fetching several unknown numbers of marked items. The computer simulation realization of the algorithm in a three-qubit homonuclear spin system is also presented. In addition, a compromised version of the algorithm, which reduces the demand on the structure of the molecules for the NMR experiment, is also devised.