Efficient classical simulation of the Deutsch\u2013Jozsa and Simon\u2019s algorithms

Niklas Johansson,Jan-Åke Larsson

doi:10.1007/s11128-017-1679-7

Abstract

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.

Highlights

Quantum computational speed-up has motivated much research to build quantum computers, to find new algorithms, to quantify the speed-up, and to separate classical from quantum computation
We will attempt to point out one such resource, sufficient for some of the known algorithms, and we will do this by simulation of the quantum algorithms in a classical Turing machine
There are quantum algorithms that solve these problems efficiently, and here we present a framework that can simulate these quantum algorithms, Quantum Simulation Logic (QSL), that itself can be efficiently simulated on a classical probabilistic Turing machine

Summary

Introduction

Quantum computational speed-up has motivated much research to build quantum computers, to find new algorithms, to quantify the speed-up, and to separate classical from quantum computation. In this paper we consider the possibility that it might not be necessary to reproduce the complete quantum-mechanical behaviour to efficiently simulate some quantum computation algorithms. There are quantum algorithms that solve these problems efficiently, and here we present a framework that can simulate these quantum algorithms, Quantum Simulation Logic (QSL), that itself can be efficiently simulated on a classical probabilistic Turing machine. We construct a simulation of the quantum algorithm, in classical reversible logic This is followed by a brief discussion of the black-box oracle paradigm, stressing that the simulation completely reproduces the behaviour of the Deutsch–Jozsa quantum algorithm in this paradigm.

The Deutsch–Jozsa problem and quantum algorithm

Quantum Simulation Logic

Simulation of the Deutsch–Jozsa algorithm in Quantum Simulation Logic

Equal requirements on quantum and QSL oracles

Simon’s problem

Conclusions