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

TL;DR

This paper demonstrates that quantum computation's unique properties are tied to three-dimensional Bloch balls, and generalizations with different dimensions do not support complex reversible transformations, making qubit quantum computation fundamentally special.

Contribution

The paper proves that only three-dimensional Bloch balls support non-trivial reversible quantum transformations, establishing the uniqueness of qubit quantum computation.

Findings

Reversible transformations are trivial for dimensions other than three.

No classical or complex computation is possible outside the three-dimensional case.

Quantum computation's structure is uniquely tied to the three-dimensional Bloch ball.

Abstract

The computational efficiency of quantum mechanics can be defined in terms of the qubit circuit model, which is characterized 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.