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.
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.
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.
Compared with classical search algorithms, Grover quantum algorithm [L. K. Grover, Phys. Rev. Lett. 79, 325 (1997)] achieves quadratic speedup and Brüschweiler hybrid quantum algorithm [R. Brüschweiler, Phys. Rev. Lett. 85, 4815 (2000)] achieves an exponential speedup. In this paper, we report the experimental realization of the Brüschweiler algorithm in a three-qubit nuclear magnetic resonance ensemble system. The pulse sequences are used for the algorithms and the measurement method used here is improved on that used by Brüschweiler, namely, instead of quantitatively measuring the spin projection of the ancilla bit, we utilize the shape of the ancilla bit spectrum. By simply judging the downwardness or upwardness of the corresponding peaks in an ancilla bit spectrum, the bit value of the marked state can be read out, especially the geometric nature of this read-out can make the results more robust against errors.
The Br\"uschweiler quantum search [Phys. Rev. Lett. 85, 4815 (2000)] offers an exponential speedup over the corresponding classical search. In the Br\"uschweiler search, the data bits and measurement bit (the ancillary bit) are different bits. We propose a modified method, in which no ancillary bit is needed to reach exponential speedup. The modified Br\"uschweiler search can be easier to design and realize. We also demonstrate the modified search in a three-qubit molecular system using nuclear magnetic resonance.
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 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.
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.