Graph-theoretic strengths of contextuality

Cabello-Severini-Winter and Abramsky-Hardy (building on the framework of Abramsky-Brandenburger) both provide classes of Bell and contextuality inequalities for very general experimental scenarios using vastly different mathematical techniques. We review both approaches, carefully detail the links between them, and give simple, graph-theoretic methods for finding inequality-free proofs of nonlocality and contextuality and for finding states exhibiting strong nonlocality and/or contextuality. Finally, we apply these methods to concrete examples in stabilizer quantum mechanics relevant to understanding contextuality as a resource in quantum computation.Comment: 13 pages; significantly rewritte

From Topology to Noncommutative Geometry: KK-theory

We associate to each unital $C^*$-algebra $A$ a geometric object---a diagram of topological spaces representing quotient spaces of the noncommutative space underlying $A$---meant to serve the role of a generalized Gel'fand spectrum. After showing that any functor $F$ from compact Hausdorff spaces to a suitable target category can be applied directly to these geometric objects to automatically yield an extension $\tilde{F}$ which acts on all unital $C^*$-algebras, we compare a novel formulation of the operator $K_0$ functor to the extension $\tilde K$ of the topological $K$-functor.Comment: 14 page

Partial and Total Ideals of Von Neumann Algebras

A notion of partial ideal for an operator algebra is a weakening the notion of ideal where the defining algebraic conditions are enforced only in the commutative subalgebras. We show that, in a von Neumann algebra, the ultraweakly closed two-sided ideals, which we call total ideals, correspond to the unitarily invariant partial ideals. The result also admits an equivalent formulation in terms of central projections. We place this result in the context of an investigation into notions of spectrum of noncommutative $C^*$-algebras.Comment: 14 page

Bases for optimising stabiliser decompositions of quantum states

Stabiliser states play a central role in the theory of quantum computation. For example, they are used to encode data in quantum error correction schemes. Arbitrary quantum states admit many stabiliser decompositions: ways of being expressed as a superposition of stabiliser states. Understanding the structure of stabiliser decompositions has applications in verifying and simulating near-term quantum computers. We introduce and study the vector space of linear dependencies of $n$-qubit stabiliser states. These spaces have canonical bases containing vectors whose size grows exponentially in $n$. We construct elegant bases of linear dependencies of constant size three. We apply our methods to computing the stabiliser extent of large states and suggest potential future applications to improving bounds on the stabiliser rank of magic states

Fast algorithms for classical specifications of stabiliser states and Clifford gates

The stabiliser formalism plays a central role in quantum computing, error correction, and fault-tolerance. Stabiliser states are used to encode quantum data. Clifford gates are those which can be easily performed fault-tolerantly in the most common error correction schemes. Their mathematical properties are the subject of significant research interest. Numerical experiments are critical to formulating and testing conjectures involving the stabiliser formalism. Conversions between different specifications of stabiliser states and Clifford gates are also important components of classical algorithms for simulating quantum circuits. In this note, we provide fast methods for verifying that a vector is a stabiliser state, and interconverting between its specification as amplitudes, a quadratic form, and a check matrix. We use these to rapidly check if a given unitary matrix is a Clifford gate and to convert between the matrix of a Clifford gate and its compact specification as a stabiliser tableau. For example, we extract the stabiliser tableau of a Clifford gate matrix with $N^2$ entries, which naively requires $O(N^3 \log N)$ time, in time $O(N \log N)$. Our methods outperform the best-known brute force methods by some orders of magnitude with asymptotic improvements that are exponential in the number of qubits. We provide example implementations of our algorithms in Python.Comment: Python implementations available at https://github.com/ndesilva/stabiliser-tools. New in v2: new algorithm for extracting the stabiliser tableau of a Clifford gate matrix that is exponentially faster compared to v1, more thorough complexity analyse

The Quantum Monad on Relational Structures

Homomorphisms between relational structures play a central role in finite model theory, constraint satisfaction, and database theory. A central theme in quantum computation is to show how quantum resources can be used to gain advantage in information processing tasks. In particular, non-local games have been used to exhibit quantum advantage in boolean constraint satisfaction, and to obtain quantum versions of graph invariants such as the chromatic number. We show how quantum strategies for homomorphism games between relational structures can be viewed as Kleisli morphisms for a quantum monad on the (classical) category of relational structures and homomorphisms. We use these results to exhibit a wide range of examples of contextuality-powered quantum advantage, and to unify several apparently diverse strands of previous work

Contextuality and noncommutative geometry in quantum mechanics

It is argued that the geometric dual of a noncommutative operator algebra represents a notion of quantum state space which differs from existing notions by representing observables as maps from states to outcomes rather than from states to distributions on outcomes. A program of solving for an explicitly geometric manifestation of quantum state space by adapting the spectral presheaf, a construction meant to analyze contextuality in quantum mechanics, to derive simple reconstructions of noncommutative topological tools from their topological prototypes is presented. We associate to each unital C*-algebra A a geometric object--a diagram of topological spaces representing quotient spaces of the noncommutative space underlying A—meant to serve the role of a generalized Gel'fand spectrum. After showing that any functor F from compact Hausdorff spaces to a suitable target category C can be applied directly to these geometric objects to automatically yield an extension F∼ which acts on all unital C*-algebras, we compare a novel formulation of the operator K0 functor to the extension K∼ of the topological K-functor. We then conjecture that the extension of the functor assigning a topological space its topological lattice assigns a unital C*-algebra the topological lattice of its primary ideal spectrum and prove the von Neumann algebraic analogue of this conjecture