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Physical Review A

Phase-creation algorithm to solve an extended Deutsch problem by a quantum computer

Hiroshi Ozawa

Published Apr 18, 2001• 0 citations
Publisher

Abstract

The Deutsch-Jozsa algorithm to implement an f-controlled-NOT transformation on a quantum computer was generalized to create arbitrary phases. This phase-creation algorithm was applied to solve an extended Deutsch problem for a function that maps ${0,1{}}^{n}$ to ${0,1{}}^{m}.$ Implementation of the algorithm on a NMR quantum computer is discussed.

DOI
10.1103/physreva.63.052312

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The Deutsch-Jozsa algorithm to implement an f-controlled-NOT transformation on a quantum computer was generalized to create arbitrary phases. This phase-creation algorithm was applied to solve an extended Deutsch problem for a function that maps 0,1n{0,1{}}^{n}0,1n to 0,1m.{0,1{}}^{m}.0,1m. Implementation of the algorithm on a NMR quantum computer is discussed.
The Deutsch-Jozsa algorithm to implement an fff -controUed-NOT transformation on a quantum computer was generalized to create arbitrary phases.This phase-creation algorithm was applied to solve an extended Deutsch problem for afunction that maps {0,1}n\{0, 1\}^{n}{0,1}n to {0,1}m\{0, 1\}^{m}{0,1}m .Implementation of the algorithm on an NMR quantum computer is discussed.
We propose a scalable method on the basis of nth-order coupling operators to construct f-dependent phase transformations in the n-qubit modified Deutsch-Jozsa (D-J) quantum algorithm. The novel n-qubit entangling transformations are easily implemented via J-couplings between neighboring spins. The seven-qubit modified D-J quantum algorithm and seventh-order coupling transformations are then experimentally demonstrated with liquid state nuclear magnetic resonance (NMR) techniques. The method may offer the possibility of creating generally entangled states of n qubits and simulating n-body interactions on n-qubit NMR quantum computers.
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.
A detailed description of the development of a three qubit NMR realization of the Deutsch-Jozsa algorithm [Collins et.al., Phys. Rev. A 62, 022304 (2000)] is provided. The theoretical and experimental techniques used for the reduction of the algorithm's evolution steps into a sequence of NMR pulses are discussed at length. This includes the description of general pulse sequence compilation techniques, various schemes for indirectly coupled gate realizations, experimental pulse parameterization techniques and bookkeeping methods for pulse phases.