The computational landscape of general physical theories

Abstract

The emergence of quantum computers has challenged long-held beliefs about what is efficiently computable given our current physical theories. However, going back to the work of Abrams and Lloyd, changing one aspect of quantum theory can result in yet more dramatic increases in computational power, as well as violations of fundamental physical principles. Here we focus on efficient computation within a framework of general physical theories that make good operational sense. In prior work, Lee and Barrett showed that in any theory satisfying the principle of tomographic locality (roughly, local measurements suffice for tomography of multipartite states) the complexity bound on efficient computation is AWPP. This bound holds independently of whether the principle of causality (roughly, no signalling from the future) is satisfied. In this work we show that this bound is tight: there exists a theory satisfying both the principles of tomographic locality and causality which can efficiently decide everything in AWPP, and in particular can simulate any efficient quantum computation. Thus the class AWPP has a natural physical interpretation: it is precisely the class of problems that can be solved efficiently in tomographically-local theories. This theory is built upon a model of computing involving Turing machines with quasi-probabilities, to wit, machines with transition weights that can be negative but sum to unity over all branches. In analogy with the study of non-local quantum correlations, this leads us to question what physical principles recover the power of quantum computing. Along this line, we give some computational complexity evidence that quantum computation does not achieve the bound of AWPP.

Quantum Theory is a Quasi-stochastic Process Theory

“…As shown by Abramsky and Brandenburger [1], any non-signalling correlations (including those that maximally violate the Bell inequality) can be represented by a hidden variable model if one allows negative probabilities. Furthermore it was shown that any tomographically local theory has a complexity bound of AWPP [15] which was later shown by the same authors to be achieved by a computational model based on a quasi-probabilistic Turing machine [5]. These results together suggest that the features of quantum theory that don't occur in classical epistricted theories can be explained by the presence of negative probabilities in quantum theory and that in fact the necessity of this negativity is the 'cause' of violating Bell inequalities and achieving a computational speed-up.…”

Oracles and Query Lower Bounds in Generalised Probabilistic Theories

“…Moreover, the unique deterministic effect allows one to define a notion of marginalisation for multi-partite states. 5 Operationally this can be seen as saying that one does not need to perform an infinite number of distinct experiments to characterise states 6 The process {Uj}j∈Y , where j index the positions of the classical pointer, is a coarse-graining of the process {Ei}i∈X if there is a disjoint partition {Xj }j∈Y of X such that Uj = i∈X j Ei. 7 or, more accurately, the real vector corresponding to the state.…”

“…This naturally leads to the question of what general relationships hold between computational power and physical principles. This question has recently been studied in the framework of generalised probabilistic theories [2,3,4,5,6,7], which contains operationally-defined physical theories that generalise the probabilistic formalism of quantum theory. By studying how computational power varies as the underlying physical theory is changed, one can determine the connection between physical principles and computational power in a manner not tied to the specific mathematical manifestation of a particular principle within a theory.…”

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

“…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.…”

“…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.…”