Quantum computation is the unique reversible circuit model for which bits are balls

Marius Krumm,Markus P Müller

doi:10.1038/s41534-018-0123-x

Abstract

The computational efficiency of quantum mechanics can be characterized in terms of the qubit circuit model, which is defined by a few simple properties: each computational gate is a reversible transformation in a connected matrix group; single wires carry quantum bits, i.e. states of a three-dimensional Bloch ball; states on two or more wires are uniquely determined by local measurement statistics and their correlations. In this paper, we ask whether other types of computation are possible if we relax one of those characteristics (and keep all others), namely, if we allow wires to be described by d-dimensional Bloch balls, where d is different from three. Theories of this kind have previously been proposed as possible generalizations of quantum physics, and it has been conjectured that some of them allow for interesting multipartite reversible transformations that cannot be realized within quantum theory. However, here we show that all such potential beyond-quantum models of computation are trivial: if d is not three, then the set of reversible transformations consists entirely of single-bit gates, and not even classical computation is possible. In this sense, qubit quantum computation is an island in theoryspace.

Highlights

Since the discovery of quantum algorithms that outperform all known classical ones in certain tasks,[1] improving our understanding of the possibilities and limitations of quantum computation has become one of the central goals of quantum information theory
It has been shown that availability of closed timelike curves leads to implausible computational power,[10] that stronger-than-quantum nonlocality reduces the set of available transformations,[11,12,13,14] that tomographic locality forces computations to be contained in a class called AWPP,[15,16] and that in some theories higher-order interference[17] does not lead to a speed-up in Grover’s algorithm.[18]
We consider a specific modification of the quantum formalism that is arguably among the simplest and most conservative possibilities. This modification dates back to ideas by Jordan et al.,[23] and it has several independent motivations as we will explain further below. This generalization keeps all characteristic properties of quantum computation unchanged, but modifies a single aspect: namely, it allows the quantum bit to have any number of d ≥ 2 degrees of freedom, instead of standard quantum theory’s d = 3

Summary

INTRODUCTION

Since the discovery of quantum algorithms that outperform all known classical ones in certain tasks,[1] improving our understanding of the possibilities and limitations of quantum computation has become one of the central goals of quantum information theory. This modification dates back to ideas by Jordan et al.,[23] and it has several independent motivations as we will explain further below This generalization keeps all characteristic properties of quantum computation unchanged, but modifies a single aspect: namely, it allows the quantum bit to have any number of d ≥ 2 degrees of freedom, instead of standard quantum theory’s d = 3 (or the classical bit’s d = 1). It has been conjectured[24] that the resulting theories allow for interesting “beyond quantum” reversible multipartite dynamics, which would make the corresponding models of computation highly relevant objects of study within the research program mentioned above.

RESULTS

DISCUSSION

METHODS