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

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 threedimensional 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.that tomographic locality forces computations to be contained in a class called AWPP [15,16], and that in some theories (satisfying additional axioms) higher-order interference does not lead to a speed-up in Grover's algorithm [17]. Further examples can be found e.g. in [18][19][20][21].

Advantage of Quantum Theory over Nonclassical Models of Communication

“…Answer to this question is partially known from the perspective of computational power of a physical theory. [ 24,25 ] It has been shown that several beyond‐quantum models of computation are trivial, that is, the set of reversible transformations consists entirely of single‐bit gates, and not even classical computation is possible. [ 25 ] However, it is known that the class of functions computable with classical physics exactly coincides with the class computable quantum mechanically, and the quantum exponential speed‐up over classical computation for a range of problems, such as factoring, is based upon the strong believe about persistence of polynomial hierarchy.…”

How dynamics constrains probabilities in general probabilistic theories

“…Examples of GPTs (excepting classical and quantum theory) include Boxworld [15,[39][40][41][42], quantum theory over the field of real numbers [43][44][45] or quaternions [46], theories based on Euclidean Jordan algebras [35], quartic quantum theory [47], d-balls [16,48,49], density cubes [50] and quantum systems with modified measurements [51]. Amongst these, only Boxworld, quantum theory over real or quaternionic fields and theories based on Euclidean Jordan algebras are full theories, in that they have non-trivial composites.…”

Compositional resource theories of coherence

“…This last point is useful as it provides the ability to characterise coherence in operational and physical terms, rather than via specific features of the mathematical formalism of quantum theory-such as Hilbert spaces and complex numbers. There is some existing work on studying resource theories in generalised theories such as [46,14,15,86,4] and this general approach has also been of use to deepen our understanding of computation [52,47,9,31,6,54,55,50,51], cryptography [77,74,48,78,7,5,53,49] and much more.…”