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arXiv:quant-ph/9807012 · Physical Review A

Deutsch-Jozsa algorithm as a test of quantum computation

David C. Collins, K. W. Kim, William C. Holton

Published Sep 1, 1998• 140 citations
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Abstract

A redundancy in the existing Deutsch-Jozsa quantum algorithm is removed and a refined algorithm, which reduces the size of the register and simplifies the function evaluation, is proposed. The refined version allows a simpler analysis of the use of entanglement between the qubits in the algorithm and provides criteria for deciding when the Deutsch-Jozsa algorithm constitutes a meaningful test of quantum computation.

DOI
10.1103/physreva.58.r1633

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A redundancy in the existing Deutsch-Jozsa quantum algorithm is removed and a refined algorithm, which reduces the size of the register and simplifies the function evaluation, is proposed. The refined version allows a simpler analysis of the use of entanglement between the qubits in the algorithm and provides criteria for deciding when the Deutsch-Jozsa algorithm constitutes a meaningful test of quantum computation.
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In this work we implement the Deutsch-Jozsa algorithm with one qubit by removing a redundancy on it, which reduces the size of the register and simpli es the function evaluation. In order to do this, we used a pair of photons produced by down convertion, where one of them is used as triguer and the other passes trougth a double slit; as result its output is in a superposition of paths, i.e., one qubit. Using a spatial light modulator (SLM) that puts phases on the paths of the photons, and detecting in coincidence, we demonstrate the Deutsch-Jozsa algorithm with photons, and using only one qubit.
In this paper, we briefly review the basic concepts of quantum computation, entanglement, quantum cryptography and quantum fourier transform. Quantum algorithms like Deutsch Jozsa, Shor’s factorization and Grover’s data search are developed using fourier transform and quantum computation concepts to build quantum computers. Researchers are finding a way to build quantum computer that works more efficiently than classical computer. Among the standard well known algorithms in the field of quantum computation and communication we describe mathematically Deutsch Jozsa algorithm in detail for 2 and 3 qubits. Calculation of balanced and unbalanced states is shown in the mathematical description of the algorithm.
Quantum correlations have been pointed out as the most likely source of the speed-up in quantum computation. Here we analyzed the presence of quantum correlations in the implementation of Deutsch-Jozsa algorithm running in the DQC1 and DQCp models of quantum computing. For some balanced functions, the qubits in DQC1 model are quantum correlated just in the intermediate steps of the algorithm for a given decomposition into one and two qubits gates. In the DQCp model the final state is strongly quantum correlated for some balanced functions, so that the pairwise entanglement between blocks scales with the system size. Although the Deutsch-Jozsa algorithm is efficiently implemented in both models of computation, the presence of quantum correlations is not a sufficient property for computational gain in this case, since the performance of the classical probabilistic algorithm is better than the quantum ones. The measurement of other qubits than the control one showed to be inefficient to turn the algorithm deterministic.