[Paper Review] Quantum computation is an island in theoryspace

Marius Krumm, Markus P. Mueller|arXiv (Cornell University)|Apr 16, 2018

Quantum Computing Algorithms and Architecture3 citations

TL;DR

This paper investigates whether quantum computation could be generalized by allowing qubits to exist on d-dimensional Bloch spheres instead of the standard three-dimensional ones. Using the qubit circuit model as a framework, it shows that only when d = 3 do nontrivial reversible gates and universal computation emerge—making standard quantum computation an isolated 'island' in the broader space of possible physical theories.

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 continuous 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.

Motivation & Objective

• To explore whether generalizations of quantum mechanics with d-dimensional Bloch spheres (d ≠ 3) could support nontrivial computation.
• To assess whether such theories could realize multipartite reversible transformations beyond standard quantum theory.
• To determine whether computational universality or even classical computation is possible in these generalized theories.
• To identify the conditions under which reversible transformations remain nontrivial in alternative physical theories.
• To establish the uniqueness of the standard qubit model (d = 3) in enabling nontrivial computation within the circuit model framework.

Proposed method

• Formalizing computation within the qubit circuit model, assuming all properties except the dimension d of the Bloch sphere are preserved.
• Analyzing the group structure of reversible transformations on d-dimensional Bloch spheres using continuous matrix groups.
• Deriving constraints on the set of allowed reversible gates based on the dimension d of the state space.
• Using representation theory and symmetry arguments to show that for d ≠ 3, only single-bit gates are allowed.
• Applying the principle that correlations and local measurement statistics uniquely determine multi-wire states, as in quantum theory.
• Demonstrating that for d ≠ 3, the only consistent reversible operations are local and fail to generate entanglement or universal computation.

Experimental results

Research questions

• RQ1 Can nontrivial reversible transformations be realized in a generalized quantum theory where qubits live on d-dimensional Bloch spheres with d ≠ 3?
• RQ2 Do such generalized theories allow for multipartite entanglement or computational universality?
• RQ3 What constraints does the requirement of reversible, continuous matrix-group transformations impose on the dimension d of the Bloch sphere?
• RQ4 Is classical computation possible in theories with d ≠ 3, assuming all other features of the quantum circuit model are retained?
• RQ5 Why is the standard qubit model (d = 3) uniquely capable of supporting nontrivial computation in this framework?

Key findings

• For d ≠ 3, the set of reversible transformations on d-dimensional Bloch spheres reduces to only single-bit gates.
• No nontrivial multipartite entanglement or universal quantum computation is possible in theories with d ≠ 3.
• Even classical computation cannot be realized in such models, as no nontrivial correlations can be generated.
• The only consistent physical theories allowing nontrivial computation under the given axioms are those with d = 3.
• This establishes that standard qubit quantum computation is an isolated point—'an island'—in the broader space of possible physical theories.
• The result implies that the three-dimensional nature of the Bloch sphere is essential for computational richness in the circuit model framework.