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arXiv:1412.3127

Very ample line bundles, contextuality and quantum computation

Raouf Dridi

Published Dec 9, 2014• 0 citations
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Abstract

I relate contextuality to line bundles. Line bundles are important in algebraic geometry, they determine through their global sections rational maps to projective spaces. I explain how such maps, if they exist, relate rationally the input and output of measurement based computation (MBQC) and show geometrically that, indeed, contextuality is a necessary resource for the computational advantage in MBQC. I also leverage the definition of MBQC to category theory and present it as a "subfunctor" of the spectral presheaf. In general, the MBQC functor is pointless whereas the computation is trivial.

DOI
10.48550/arxiv.1412.3127

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I relate contextuality to line bundles. Line bundles are important in algebraic geometry, they determine through their global sections rational maps to projective spaces. I explain how such maps, if they exist, relate rationally the input and output of measurement based computation (MBQC) and show geometrically that, indeed, contextuality is a necessary resource for the computational advantage in MBQC. I also leverage the definition of MBQC to category theory and present it as a "subfunctor" of the spectral presheaf. In general, the MBQC functor is pointless whereas the computation is trivial.
Abstract For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where nonlocality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.
Inspired by the recast of the quantum mechanics in a toposical framework, we develop a contextual quantum mechanics via the geometric mathematics to propose a quantum contextuality adaptable in every topos. \nThe contextuality adopted corresponds to the belief that the quantum world must only be seen from the classical viewpoints à la Bohr consequently putting forth the notion of a context, while retaining a realist understanding. Mathematically, the cardinal object is a spectral Stone bundle Σ → B (between stably-compact locales) permitting a treatment of the kinematics, fibre by fibre and fully point-free. In leading naturally to a new notion of points, the geometricity permits to understand those of the base space B as the contexts C — the commutative C*–algebras of a incommutative C*–algebras — and those of the spectral locale Σ as the couples (C, ψ), with ψ a state of the system from the perspective of such a C. The contexts are furnished with a natural order, the aggregation order which is installed as the specialization on B and Σ thanks to (one part of) the Priestley's duality adapted geometrically as well as to the effectuality of the lax descent of the Stone bundles along the perfect maps.
Contextuality—the obstruction to describing quantum mechanics in a classical statistical way—has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate nonlinear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality—the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits , which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for state-independent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation (MBC) that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality , which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit MBC are of the same underlying nature, avoiding the pathologies associated with qubit contextuality.
We develop a resource theory of contextuality within the framework of symmetric monoidal categories, extending recent simplicial approaches to quantum contextuality. Building on the theory of simplicial distributions, which integrates homotopy-theoretic structures with probability, we introduce event scenarios as a functorial generalization of presheaf-theoretic measurement scenarios and prove their equivalence to bundle scenarios via the Grothendieck construction. We define symmetric monoidal structures on these categories and extend the distribution functor to a stochastic setting, yielding a resource theory that generalizes the presheaf-theoretic notion of simulations. Our main result characterizes convex maps between simplicial distributions in terms of non-contextual distributions on a corresponding mapping scenario, enhancing and extending prior results in categorical quantum foundations.
Okay, Roberts, Bartlett and Raussendorf recently introduced a new cohomological approach to contextuality in measurement based quantum computation. We give an abstract description of their obstruction and the algebraic structure it exploits, using the sheaf theoretic framework of Abramsky and Brandenburger. At this level of generality we contrast their approach to the Cech cohomology obstruction of Abramsky, Mansfield and Barbosa and give a direct proof that Cech cohomology is at least as powerful.