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arXiv:quant-ph/0112161 · The Journal of Chemical Physics

Experimental realization of the Brüschweiler’s algorithm in a homonuclear system

Xiao Li, Gui‐Lu Long, Haiyang Yan, Yang Sun

Published Aug 15, 2002• 11 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.1494784

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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.
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 Fourier transform is a key factor in achieving exponential speedup relative to classical algorithms. We implemented the phase estimation algorithm, which is the very basic application example of the quantum Fourier transform, on a three-bit nuclear magnetic resonance quantum-information processor. The algorithm was applied to one-bit Grover operators to estimate eigenvalues. We also demonstrated the counting algorithm with the same operators to obtain the number of the marked states. The results of projective measurements required by the algorithms were obtained from the split peak positions in the spectra.
A generalized quantum search algorithm, where phase inversions for the marked state and the prepared state are replaced by π/2 phase rotations, is realized in a 2-qubit NMR heteronuclear system. The quantum algorithm searches a marked state with a smaller step compared to standard Grover algorithm. Phase matching requirement in quantum searching is demonstrated by comparing it with another generalized algorithm where the two phase rotations are π/2 and 3π/2 respectively. Pulse sequences which include non 90 1 degree pulses are given. Typeset using REVTEX 2 Grover’s quantum search algorithm is one of the most important development in quantum computation [1]. It achieves quadratic speedup in searching a marked state in an unordered list over classical searching algorithms. It has many potential applications in various fields of interests, for instance in deciphering the DES encryption code [2] and algorithms that need searching. Typical examples of the algorithms that need searching are the Simon
Liouville space search algorithm [Bruschweiler, Phys. Rev. Lett. {\bf 85}, 4815(2000).] utilizes mixed initial states of the ensemble, and has been successfully implemented earlier in weakly coupled spins, in which a spin can be identified as a qubit. It has recently been demonstrated that n-strongly coupled spins can be collectively treated as an n-qubit system. Application of algorithms in such systems, requires new approaches using transition selective pulses rather than qubit selective pulses. This work develops a modified version of Liouville space search algorithm, which is applicable for strongly as well as weakly coupled spins. All the steps of the algorithm, can be implemented by using transition selective pulses. Experimental implementation is carried out on a strongly dipolar coupled four qubit system.
Quantum information processing has been effectively demonstrated on a small number of qubits by nuclear magnetic resonance. An important subroutine in any computing is the readout of the output. "Spectral implementation" originally suggested by Z. L. Madi, R. Bruschweiler, and R. R. Ernst [J. Chem. Phys. 109, 10603 (1999)], provides an elegant method of readout with the use of an extra "observer" qubit. At the end of computation, detection of the observer qubit provides the output via the multiplet structure of its spectrum. In spectral implementation by two-dimensional experiment the observer qubit retains the memory of input state during computation, thereby providing correlated information on input and output, in the same spectrum. Spectral implementation of Grover's search algorithm, approximate quantum counting, a modified version of Berstein-Vazirani problem, and Hogg's algorithm are demonstrated here in three- and four-qubit systems.