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Efficient Simulation of the Deutsch-Jozsa Algorithm

Niklas Johansson

Published Jan 1, 2015• 1 citation
OpenAlex

Abstract

We provide a framework wherein one can simulate the Deutsch-Jozsa quantum algorithm on a regular computer within polynomial time, and with linear memory consumption. Under certain reasonable assumptions the simulation solves the problem with a bounded error of probability with only one function evaluation, which is comparable with the efficiency of the quantum algorithm. The provided framework lies within a slight extension of the toy model purposed by Robert W. Spekkens Phys. Rev. A 75 (2007), and consists of transformations that are reminiscent of transformations in quantum mechanics.

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We provide a framework wherein one can simulate the Deutsch-Jozsa quantum algorithm on a regular computer within polynomial time, and with linear memory consumption. Under certain reasonable assumptions the simulation solves the problem with a bounded error of probability with only one function evaluation, which is comparable with the efficiency of the quantum algorithm. The provided framework lies within a slight extension of the toy model purposed by Robert W. Spekkens Phys. Rev. A 75 (2007), and consists of transformations that are reminiscent of transformations in quantum mechanics.
It is shown that Deutsch’s algorithm and the Deutsch-Jozsa algorithm for two and three qubits can be efficiently implemented on a classical computer. This is done by analysing the structure of the oracles contained in these algorithms, followed by an implementation in the toy theory proposed by Robert W. Spekkens which uses 2 classical bits to simulate a qubit. Further discussion is given about how to analyse, and possible implementations, of the Deutsch-Jozsa for higher number of qubits in the toy theory.
In 1985, David Deutsch challenged the Church-Turing thesis by stating that his quantum model of computation "could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine". While this is thought to be true in general, there is usually no way of knowing that the corresponding classical algorithms are the best possible solutions. Here we provide an efficient classical simulation of the Deutsch-Jozsa algorithm, which was one of the first examples of quantum computational speed-up. Our conclusion is that the Deutsch-Jozsa quantum algorithm owes its speed-up to resources that are not necessarily quantum-mechanical, and when compared with the classical simulation offers no speed-up at all.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch–Jozsa and Simon’s problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch–Jozsa problem with probability 1 using only one oracle query, and Simon’s problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch–Jozsa and Simon’s problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms and calls for further study of oracle separation between quantum and classical computation.
A long-standing aim of quantum information research is to understand what gives quantum computers their advantage. This requires separating problems that need genuinely quantum resources from those for which classical resources are enough. Two examples of quantum speed-up are the Deutsch-Jozsa and Simon's problem, both efficiently solvable on a quantum Turing machine, and both believed to lack efficient classical solutions. Here we present a framework that can simulate both quantum algorithms efficiently, solving the Deutsch-Jozsa problem with probability 1 using only one oracle query, and Simon's problem using linearly many oracle queries, just as expected of an ideal quantum computer. The presented simulation framework is in turn efficiently simulatable in a classical probabilistic Turing machine. This shows that the Deutsch-Jozsa and Simon's problem do not require any genuinely quantum resources, and that the quantum algorithms show no speed-up when compared with their corresponding classical simulation. Finally, this gives insight into what properties are needed in the two algorithms, and calls for further study of oracle separation between quantum and classical computation.
The Deutsch and Deutsch-Jozsa algorithms are foundational in the study of quantum computing, providing early and elegant demonstrations of quantum speedup over classical computation. This paper offers a pedagogical exposition of both algorithms, beginning with the necessary background in quantum computation and culminating in detailed, step-by-step derivations of the algorithms, complete with mathematical formulations and circuit diagrams. Simulations using quantum gates are also discussed to enhance understanding. The aim is to make these seminal algorithms accessible to readers with a basic background in linear algebra and quantum mechanics.
We are now ready to assemble quantum subroutines into quantum algorithms. A reasonable first question is: Are there problems which can be solved efficiently on a quantum computer? In his analysis of a universal quantum computer [25], Deutsch illustrated quantum parallelism by constructing a procedure to compute the parity f (0) ⊕ f (1) of a function on one bit. In a generalization of this context, Deutsch and Jozsa [27] defined a quantum algorithm that could solve a problem more efficiently than is possible on a classical computer. This problem helped motivate the search for algorithms that could solve “real” problems, and Simon [67] provided an example that inspired Shor’s subsequent work on the factoring problem and on the discrete logarithm problem. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.