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Physical Review A

Modification and realization of Brüschweiler’s search

Xiaodong Yang, Daxiu Wei, J. L. Luo, Xijia Miao

Published Oct 10, 2002• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.66.042305

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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.
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.
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.
With a three-bit nuclear magnetic resonance quantum computer, we have experimentally realized a quantum search algorithm that can take a desired function value (or target) t as its input and give the corresponding function argument x0{x}_{0}x0​ such that f(x0)=t{f(x}_{0})=tf(x0​)=t as a result. The function value is not retrieved from an existing table, but evaluated by the search algorithm in the manner of a typical algorithmic search. A single-query version of the quantum search algorithm is used to search for one of two function arguments.
Quantum algorithms are conventionally formulated for implementation on a single system of qubits amenable to projective measurements. However, in expectation value quantum computation, such as nuclear magnetic resonance realizations, the computer consists of an ensemble of identical qubit-systems amenable only to expectation value measurements. The prevalent strategy in such expectation value implementations of quantum algorithms has been to retain the conventional formulation's unitary operations but modify its initialization and measurement steps appropriately. This naive approach is not optimal for Grover's algorithm and a shortened version for expectation value quantum computers is presented.
Quantum computing by nuclear magnetic resonance using pseudopure spin states is bound by the maximal speed of quantum computing algorithms operating on pure states. In contrast to these quantum computing algorithms, a novel algorithm for searching an unsorted database is presented here that operates on truly mixed states in spin Liouville space. It provides an exponential speedup over Grover's quantum search algorithm with the sensitivity scaling exponentially with the number of spins, as for pseudopure state implementations. The minimal decoherence time required is exponentially shorter than that for Grover's algorithm.
Nuclear magnetic resonance offers an appealing prospect for implementation of quantum computers, because of the long coherence times associated with nuclear spins, and extensive laboratory experience in manipulating the spins with radio frequency pulses. Existing proposals, however, suffer from a signal-to-noise ratio that decays exponentially in the number of qubits in the quantum computer. This places a severe limit on the size of the computations that can be performed by such a computer; estimates of that limit are well within the range in which a conventional computer taking exponentially more steps would still be practical. We give an NMR implementation in which the signal-to-noise ratio depends only on features of NMR technology, not the size of the computer. This provides a means for NMR computation techniques to scale to sizes at which the exponential speedup enables quantum computation to solve problems beyond the capabilities of classical computers.