Logical paradoxes in quantum computation

Abstract

The precise features of quantum theory enabling quantum computational power are unclear. Contextuality-the denial of a notion of classical physical reality-has emerged as a promising hypothesis: e.g. Howard et al. showed that the magic states needed to practically achieve quantum computation are contextual.Strong contextuality, as defined by Abramsky-Brandenburger, is an extremal form of contextuality describing systems that exhibit logically paradoxical behaviour.After introducing number-theoretic techniques for constructing exotic quantum paradoxes, we present large families of strongly contextual magic states that are computationally optimal in the sense that they enable universal quantum computation via deterministic injection of gates of the Clifford hierarchy. We thereby bolster a refinement of the resource theory of contextuality that emphasises the computational power of logical paradoxes. IntroductionIssues. Identification of the precise features of quantum theory accounting for quantum advantages over classical devices, and of the mechanisms by which they do so, remains an open problem in quantum computation. Physicists and quantum information theorists seek high-level principles behind quantum advantages in order to optimise resources and serve as guideposts towards novel protocols and algorithms.A promising, emerging hypothesis is contextuality: a concept from the foundations of quantum mechanics first articulated by Bell-Kochen-Specker [13,47] that is commonly understood as the denial of a classical notion of reality. Essentially, contextuality is the failure of correlated probabilistic data to be reproducible by a classical probability model. It subsumes nonlocality-the failure of separated systems to respect a strong classical assumption about causality [14]-as a special case. With these concepts, tensions between classical and quantum physics, e.g. the Einstein-Podolsky-Rosen paradox [28], are formally articulated to a degree that experiments (e.g. [10,39]) can be performed to refute classical ontological assumptions.An important lesson of recent studies of contextuality is that it is a logical phenomenon that is most clearly understood independently of quantum physics, the setting in which it first appeared

Contextuality and the fundamental theorems of quantum mechanics

“…It has been argued that contextuality is a resource for quantum computation. In particular, for the quantum computing architecture known as measurement-based quantum computation (MBQC) it has been shown that contextuality allows to outperform certain non-contextual settings [5,51,19,30]. Furthermore, in [36] it is proven that contextuality is essential also for magic state injection in the stabiliser formalism.…”

The Poincaré Half-Plane for Informationally-Complete POVMs

“…Remarkably, it will be shown that subsets of projectors within the IC-POVMs below may sometimes be used to derive proofs of KS theorem (in Dimensions 4, 8 and 9). Contextuality and negativity of the Wigner function happen to play a fundamental role in universal schemes of quantum computation (e.g., [ 10 , 20 , 30 , 31 ]).…”

Non-locality, contextuality and valuation algebras: a general theory of disagreement

“…In 2011, Abramsky & Brandenburger introduced an abstract mathematical framework based on sheaf theory to describe these phenomena in a unified treatment, thereby providing a common general theory for the study of non-locality and contextuality, which had been carried out in a rather concrete, example-driven fashion until then [3]. This high-level description shows that contextuality is not a feature specific to quantum theory, but rather a general Notable examples of the sheaf-theoretic notion of contextuality have been found in connection with relational database theory [4], robust constraint satisfaction [5], natural language semantics [6] and logical paradoxes [7][8][9]. This profusion of instances motivates the search for a general theory of contextual semantics, an all-comprehensive approach able to capture the essence and structure of contextual behaviour.…”

“…All rights reserved. Notable examples of the sheaf-theoretic notion of contextuality have been found in connection with relational database theory [4], robust constraint satisfaction [5], natural language semantics [6] and logical paradoxes [7][8][9]. This profusion of instances motivates the search for a general theory of contextual semantics, an all-comprehensive approach able to capture the essence and structure of contextual behaviour.…”