Generalised phase kick-back: the structure of computational algorithms from physical principles

Lee, Ciarán M.; Selby, John H.

doi:10.1088/1367-2630/18/3/033023

Cited by 54 publications

(63 citation statements)

References 33 publications

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“…The implication 'strong symmetry  permutability' follows immediately from the definitions. The implication 'strong symmetry  reversible controllability' was proved by Lee and Selby [63] using causality, purification, and the property that the product of two pure states is pure, which is guaranteed by our purity preservation axiom. Hence, we only need to prove the implications 'permutability  strong symmetry' and 'reversible controllability  strong symmetry'…”

Section: Appendix D Operational Features Of Doubled Quantum Theorymentioning

confidence: 93%

Microcanonical thermodynamics in general physical theories

Chiribella

Scandolo

2017

New J. Phys.

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Microcanonical thermodynamics studies the operations that can be performed on systems with welldefined energy. So far, this approach has been applied to classical and quantum systems. Here we extend it to arbitrary physical theories, proposing two requirements for the development of a general microcanonical framework. We then formulate three resource theories, corresponding to three different sets of basic operations: (i) random reversible operations, resulting from reversible dynamics with fluctuating parameters, (ii) noisy operations, generated by the interaction with ancillas in the microcanonical state, and (iii) unital operations, defined as the operations that preserve the microcanonical state. We focus our attention on a class of physical theories, called sharp theories with purification, where these three sets of operations exhibit remarkable properties. Firstly, each set is contained into the next. Secondly, the convertibility of states by unital operations is completely characterised by a majorisation criterion. Thirdly, the three sets are equivalent in terms of state convertibility if and only if the dynamics allowed by theory satisfy a suitable condition, which we call unrestricted reversibility. Under this condition, we derive a duality between the resource theories of microcanonical thermodynamics and the resource theory of pure bipartite entanglement.1. random unitary channels [43][44][45], arising from unitary dynamics with randomly fluctuating parameters;2. noisy operations [46][47][48], generated by preparing ancillas in the microcanonical state, turning on a unitary dynamics, and discarding the ancillas;3. unital channels [49,50], defined as the quantum processes that preserve the microcanonical state.These three sets are strictly different: the set of random unitary channels is strictly contained in the set of noisy operations [51], and the latter is strictly contained in the set of unital channels [52]. In spite of this, the three sets are equivalent in terms of state convertibility [48]. This means that all the natural candidates for the sets of free operations induce the same notion of resource. This resource is generally called purity, and plays a fundamental role in many quantum protocols [53].In this paper we extend the paradigm of microcanonical thermodynamics from quantum theory to arbitrary physical theories [54][55][56][57][58][59][60][61]. We propose two minimal requirements a probabilistic theory must satisfy in order to support a microcanonical description, and, when these requirements are satisfied, we provide a general operational definition of random reversible, noisy, and unital operations. We then focus on a special class of theories, called sharp theories with purification, which are appealing for the foundations of thermodynamics [62], and have also been studied for their computational power [63, 64] and interference properties [65]. In sharp theories with purification, we show that the three sets of operations satisfy the same inclusion relations as in quantum theory, with...

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Section: Appendix D Operational Features Of Doubled Quantum Theorymentioning

confidence: 93%

Microcanonical thermodynamics in general physical theories

Chiribella

Scandolo

2017

New J. Phys.

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“…Previous work [18] has shown that the conjunction of principles 1, 2, 3 and 5 implies the existence of reversible controlled transformations. These can be used to define oracles in a manner analogous to quantum theory [18].…”

Section: Setting Up the Problemmentioning

confidence: 95%

“…As such, one can classify theories according to their maximal order of interference, h. For example quantum theory lies at = h 2 and classical theory at = h 1. Higher order interference was initially formalised by Sorkin in the framework of quantum measure theory [29] but has more recently been adapted to the setting of generalised probabilistic theories in [3,18,19,33]. The most direct translation to this setting describes the order of interference in terms of probability distributions corresponding to the different experimental setups (which slits are open, etc) [18].…”

Section: Higher-order Interferencementioning

confidence: 99%

Deriving Grover's lower bound from simple physical principles

Lee

Selby

2016

New J. Phys.

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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...

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“…It is at the basis of an enormous variety of present and potential future applications [1], such as quantum communication [2,3], quantum computation [4][5][6] and protocols like entanglement swapping [7] or teleportation [8]. However, all these applications ultimately rely on interference and entanglement, which can be alternatively explained by theories sharing only some fundamental features with quantum mechanics, such as the superposition principle or probabilistic predictions for outcomes, and yet differing from it in other aspects [9][10][11][12]. In order to distinguish between quantum theory and such alternatives, one needs to design dedicated experiments.…”

Section: Introductionmentioning

confidence: 99%

Obtaining tight bounds on higher-order interferences with a 5-path interferometer

et al. 2017

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Within the established theoretical framework of quantum mechanics, interference always occurs between pairs of paths through an interferometer. Higher order interferences with multiple constituents are excluded by Born's rule and can only exist in generalized probabilistic theories. Thus, high-precision experiments searching for such higher order interferences are a powerful method to distinguish between quantum mechanics and more general theories. Here, we perform such a test in an optical multi-path interferometer, which avoids crucial systematic errors, has access to the entire phase space and is more stable than previous experiments. Our results are in accordance with quantum mechanics and rule out the existence of higher order interference terms in optical interferometry to an extent that is more than four orders of magnitude smaller than the expected pairwise interference, refining previous bounds by two orders of magnitude.

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Generalised phase kick-back: the structure of computational algorithms from physical principles

Cited by 54 publications

References 33 publications

Microcanonical thermodynamics in general physical theories

Microcanonical thermodynamics in general physical theories

Deriving Grover's lower bound from simple physical principles

Obtaining tight bounds on higher-order interferences with a 5-path interferometer

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