Bounds on the power of proofs and advice in general physical theories

Lee, Ciarán M.; Hoban, Matty J.

doi:10.1098/rspa.2016.0076

Cited by 21 publications

(23 citation statements)

References 66 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This, in conjunction with interference being a key resource in the quantum speed-up, suggests that post-quantum interference may allow for a speed-up over quantum computation.Surprisingly, we show that this is not the case-at least from the point of view of the search problem. We consider this problem within the framework of generalised probabilistic theories, which is suitable for describing arbitrary operationally defined theories [5,8,9,13,16,17]. Classical probability theory, quantum theory, Spekkenʼs toy model [15,31], and the theory of PR boxes [25] all provide examples of theories in this framework.…”

mentioning

confidence: 99%

Deriving Grover's lower bound from simple physical principles

Lee

Selby

2016

New J. Phys.

View full text Add to dashboard Cite

Groverʼs algorithm constitutes the optimal quantum solution to the search problem and provides a quadratic speed-up over all possible classical search algorithms. Quantum interference between computational paths has been posited as a key resource behind this computational speed-up. However there is a limit to this interference, at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power? Sorkin has defined a hierarchy of possible interference behaviours-currently under experimental investigation-where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. In this work, we consider how Groverʼs speed-up depends on the order of interference in a theory. Surprisingly, we show that the quadratic lower bound holds regardless of the order of interference. Thus, at least from the point of view of the search problem, post-quantum interference does not imply a computational speed-up over quantum theory. Groverʼs algorithm [12] provides the optimal quantum solution to the search problem and is one of the most versatile and influential quantum algorithms. The search problem-in its simplest form-asks one to find a single 'marked' item from an unstructured list of N elements by querying an oracle which can recognise the marked item. The importance of Groverʼs algorithm stems from the ubiquitous nature of the search problem and its relation to solving NP-complete problems [6]. Classical computers require ( ) O N queries to solve this problem, but quantum computers-using Groverʼs algorithm-only require ( ) O N queries. Quantum interference between computational paths has been posited [32] as a key resource behind this computational 'speed-up'. However, as first noted by Sorkin [29,30], there is a limit to this interference-at most pairs of paths can ever interact in a fundamental way. Could more interference imply more computational power?Sorkin has defined a hierarchy of possible interference behaviours-currently under experimental investigation [24, 27, 28]-where classical theory is at the first level of the hierarchy and quantum theory belongs to the second. Informally, the order in the hierarchy corresponds to the number of paths that have an irreducible interaction in a multi-slit experiment. To get a greater understanding of the role of interference in computation, we consider how Groverʼs speed-up depends on the order of interference in a theory.Restriction to the second level of this hierarchy implies many 'quantum-like' features, which, at first glance, appear to be unrelated to interference. For example, such interference behaviour restricts correlations [11] to the 'almost quantum correlations' discussed in [21], and bounds contextuality in a manner similar to quantum theory [14,23]. This, in conjunction with interference being a key resource in the quantum speed-up, suggests that...

show abstract

mentioning

confidence: 99%

Deriving Grover's lower bound from simple physical principles

Lee

Selby

2016

New J. Phys.

View full text Add to dashboard Cite

show abstract

“…If such non-local states existed, it has been shown that they would make certain tasks involving distributed computing trivial [19,20]. Moreover, recent studies have explored the possibility of computation in postquantum theories [21][22][23][24][25], particularly showing that non-trivial interference in a theory may be used as a resource for computation beyond the known classical limit [25].…”

Section: Introductionmentioning

confidence: 99%

Interferometric Computation Beyond Quantum Theory

Garner

2018

Found Phys

View full text Add to dashboard Cite

There are quantum solutions for computational problems that make use of interference at some stage in the algorithm. These stages can be mapped into the physical setting of a single particle travelling through a many-armed interferometer. There has been recent foundational interest in theories beyond quantum theory. Here, we present a generalized formulation of computation in the context of a manyarmed interferometer, and explore how theories can differ from quantum theory and still perform distributed calculations in this set-up. We shall see that quaternionic quantum theory proves a suitable candidate, whereas box-world does not. We also find that a classical hidden variable model first presented by Spekkens a can also be used for this type of computation due to the epistemic restriction placed on the hidden variable. a Phys. Rev. A, 75:3

show abstract

“…Recently a circuit-based model of computation has been defined and studied in a broad operationally-defined framework for physical theories [21,23,24]. Informally, a theory in this framework specifies a set of laboratory devices that can be connected together to form experiments, and assigns probabilities to experimental outcomes.…”

mentioning

confidence: 99%

The computational landscape of general physical theories

et al. 2019

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Bounds on the power of proofs and advice in general physical theories

Cited by 21 publications

References 66 publications

Deriving Grover's lower bound from simple physical principles

Deriving Grover's lower bound from simple physical principles

Interferometric Computation Beyond Quantum Theory

The computational landscape of general physical theories

Contact Info

Product

Resources

About