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arXiv:quant-ph/0006103 · Physical Review A

Implementation of a Deutsch-like quantum algorithm utilizing entanglement at the two-qubit level on an NMR quantum-information processor

Kavita Dorai, Arvind, Anil Kumar

Published Feb 9, 2001• 26 citations
arXivHTMLPDF

Abstract

We describe the NMR implementation of a recently proposed quantum algorithm involving quantum entanglement at the level of two qubits. The algorithm solves a generalization of the Deutsch problem, and distinguishes between even and odd functions using fewer function calls than is possible classically. The manipulation of entangled states of the two qubits is essential here, unlike the Deutsch-Jozsa algorithm and Grover's search algorithm for two bits.

DOI
10.1103/physreva.63.034101

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We describe the NMR implementation of a recently proposed quantum algorithm involving quantum entanglement at the level of two qubits. The algorithm solves a generalization of the Deutsch problem, and distinguishes between even and odd functions using fewer function calls than is possible classically. The manipulation of entangled states of the two qubits is essential here, unlike the Deutsch-Jozsa algorithm and Grover's search algorithm for two bits.
A scheme to execute an \\eta-bit Deutsch-Jozsa (DJ) algorithm using \\eta qubits has been implemented for up to three qubits on an NMR quantum computer. For the one- and the two-bit Deutsch problem, the qubits do not get entangled, and the NMR implementation is achieved without using spin-spin interactions. It is for the three-bit case, that the manipulation of entangled states becomes essential. The interactions through scalar J-couplings in NMR spin systems have been exploited to implement entangling transformations required for the three bit DJ algorithm.
The present work, briefly described below, consists of implementation of several quantum algorithms in an NMR Quantum Information Processor. Game theory gives us mathematical tools to analyze situations of conflict between two or more players who take decisions that influence their welfare. Classical game theory has been applied to various fields such as market strategy, communication theory, biological processes, foreign policies. It is interesting to study the behaviour of the games when the players share certain quantum correlations such as entanglement. Various games have been studied under the quantum regime with the hope of obtaining some insight into designing new quantum algorithms. Chapter 2 presents the NMR implementation of three such algorithms. Experimental NMR implementation given in this chapter are: (i) Three qubit ‘Dilemma’ game with corrupt sources’. The Dilemma game deals with the situation where three players have to choose between going/not going to a bar with a seating capacity of two. It is seen that in the players have a higher payoff if they share quantum correlations. However, the pay-off falls rapidly with increasing corruption in the source qubits. Here we report the experimental NMR implementation of the quantum version of the Dilemma game with and without corruption in the source qubits. (ii) Two qubit ‘Ulam’s game’. This is a two player game where one player has to find out the binary number thought by the other player. This problem can be solved with one query if quantum resources are used. This game has been implemented in a two qubit system in an NMR quantum information processor. (iii) Two qubit ‘Battle of Sexes’ game. This game deal with a situation where two players have conflicting choices but a deep desire to be together. This leads to a dilemma in the classical case. Quantum mechanically this dilemma is resolved and a unique solution emerges. The NMR implementation of the quantum version of this game is also given in this chapter. Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. In some cases, such as the adiabatic versions of Grover’s search algorithm and Deutsch-Jozsa algorithm, applying the global adiabatic evolution yields a complexity similar to their classical algorithms. However, if one uses local adiabatic evolutions, their complexity is of the order √N (where N=2n) [37, 38]. In Chapter 3, the NMR implementation of (i) the Deutsch-Jozsa and the (ii) Grover’s search algorithm using local adiabatic evolution has been presented. In adiabatic algorithm, the system is first prepared in the equal superposition of all the possible states which is the ground state of the beginning Hamiltonian. The solution is encoded in the ground state of the final Hamiltonian. The system is evolved under a linear combination of the beginning and the final Hamiltonian. During each step of the…
Quantum adiabatic algorithms is a method of solving computational problems by evolving the ground state of a slowly varying Hamiltonian. The technique uses evolution of the ground state of a slowly varying Hamiltonian to reach the required output state. The adiabatic Deutsch-Jozsa algorithm given be Saurya Das, Randy Kobes and Gabor Kunstatter [Phys. Rev. A. 65, 062301 (2002)], shows how the Deutsch-Jozsa algorithm can be implemented in N^{1/2} steps, where N=2^n and n is the number of qubits, using local adiabatic evolution. In this paper we report the experimental implementation of the adiabatic Deutsch-Jozsa algorithm in a two qubit quantum information processor by Nuclear Magnetic Resonance.
A quantum computer is a device that processes information in a quantum-mechanically coherent fashion. In principle, it can exploit coherent quantum interference and entanglement to perform computations, such as factoring large numbers or searching an unsorted database, more rapidly than classical computers. Noise, decoherence, and manufacturing problems make constructing large scale quantum computers difficult. Ion traps and optical cavities offer promising experimental approaches, but no quantum algorithm has yet been implemented with those systems. On the other hand, because of their natural isolation from the environment, nuclear spins are particularly good 'quantum bits', and their use for quantum computation is possible by applying nuclear magnetic resonance (NMR) techniques in an unconventional manner. Here, we report on the experimental implementation of a quantum algorithm using NMR to solve a purely mathematical problem in fewer steps than is possible classically. Our simple quantum computer can determine the type of an unknown function using fewer function 'calls' than is possible using a classical computer.
The existence of entangled quantum states gives extra power to quantum computers over their classical counterparts. Quantum entanglement shows up qualitatively at the level of two qubits. We show that if no entanglement is envolved then whatever one can do with qubits can also be done with classical optical systems. We demonstrate that the one- and the two-bit Deutsch-Jozsa algorithm does not require entanglement and can be mapped onto a classical optical scheme. It is only for three and more input bits that the DJ algorithm requires the implementation of entangling transformations and in these cases it is impossible to implement this algorithm classically. 1