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arXiv:quant-ph/0603231

Deutsch and Jozsa's Algorithm Revisited

J. Cooper

Published Mar 27, 2006• 0 citations
arXivHTMLPDF

Abstract

A classical analogue of Deutsch and Jozsa's algorithm is given and its implications on quantum computing is discussed
DOI
10.48550/arxiv.quant-ph/0603231

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The original Deutsch–Jozsa (oDJ) problem is for an oracle (realized here as a database) of size N , where, according to their claim, the deterministic solution of the problem on a classical Turing computer requires O( N ) computational complexity. They produced the famous Deutsch–Jozsa quantum algorithm that offered an exponential speed-up over the classical computer, namely O[log( N )] complexity for the solution in a quantum computer. In this paper, the problem is implemented on an instantaneous noise-based logic processor. It is shown that, similarly to the quantum algorithm, the oDJ problem can deterministically be solved with O[log( N )] complexity. The implication is that by adding a truly random coin to a classical Turing machine and using this classical-physical algorithm can also speed up the deterministic solution of the Deutsch–Jozsa problem exponentially, similarly to the quantum algorithm. Then it is realized that the same database and the solution of the Deutsch–Jozsa problem can also be realized by using an identical algorithmic structure in a simpler way, even without noise/random coin. The only lost function in this new system, as compared with noise-based logic, is the ability to do generic parallel logic operations over the whole database. As the latter feature is not needed for the oDJ problem, it is concluded that the problem can be solved on a classical computer with O[log( N )] complexity even without a random coin. Therefore, while the oDJ algorithm is a historically important step in the developments of quantum computers, it is insufficient to prove quantum supremacy. Note, there is also a simplified Deutsch–Jozsa problem proposed later, which is more popular in the field; however, it is irrelevant for the present paper.
Recent advances in quantum computational science promise substantial improvements in the speed with which certain classes of problems can be computed. Various algorithms that utilize the distinctively non-classical characteristics of quantum mechanics have been formulated to take advantage of this promising new approach to computation. One such algorithm was formulated by David Deutsch and Richard Jozsa. By measuring the output of a quantum network that implements this algorithm, it is possible to determine with N – 1 measurements certain global properties of a function f(x), where N is the number of network inputs. Classically, it may not be possible to determine these same properties without evaluating f(x) a number of times that rises exponentially as N increases. Hitherto, the potential power of this algorithm has been explored in the context of qubits, the quantum computational analogue of classical bits. However, just as one can conceive of classical computation in the context of non-binary logic, such as ternary or quaternary logic, so also can one conceive of corresponding higher-order quantum computational equivalents.This thesis investigates the behaviour of the Deutsch-Jozsa algorithm in the context of these higher-order quantum computational forms of logic and explores potential applications for this algorithm. An important conclusion reached is that, not only can the Deutsch-Jozsa algorithm’s known computational advantages be formulated in more general terms, but also a new algorithmic property is revealed with potential practical applications.