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

Magic-State Functional Units: Mapping and Scheduling Multi-Level Distillation Circuits for Fault-Tolerant Quantum Architectures

Quantum computers have recently made great strides and are on a long-term path towards useful fault-tolerant computation. A dominant overhead in fault-tolerant quantum computation is the production of high-fidelity encoded qubits, called magic states, which enable reliable error-corrected computation. We present the first detailed designs of hardware functional units that implement space-time optimized magic-state factories for surface code error-corrected machines. Interactions among distant qubits require surface code braids (physical pathways on chip) which must be routed. Magic-state factories are circuits comprised of a complex set of braids that is more difficult to route than quantum circuits considered in previous work [1]. This paper explores the impact of scheduling techniques, such as gate reordering and qubit renaming, and we propose two novel mapping techniques: braid repulsion and dipole moment braid rotation. We combine these techniques with graph partitioning and community detection algorithms, and further introduce a stitching algorithm for mapping subgraphs onto a physical machine. Our results show a factor of 5.64 reduction in space-time volume compared to the best-known previous designs for magic-state factories.Comment: 13 pages, 10 figure

Almost isomorphism for countable state Markov shifts

Countable state Markov shifts are a natural generalization of the well-known subshifts of finite type. They are the subject of current research both for their own sake and as models for smooth dynamical systems. In this paper, we investigate their almost isomorphism and entropy conjugacy and obtain a complete classification for the especially important class of strongly positive recurrent Markov shifts. This gives a complete classification up to entropy conjugacy of the natural extensions of smooth entropy expanding maps, including all smooth interval maps with non-zero topological entropy

Nonlocal Games and Quantum Permutation Groups

We present a strong connection between quantum information and quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs $X$ and $Y$ are quantum isomorphic if and only if there exists $x \in V(X)$ and $y \in V(Y)$ that are in the same orbit of the quantum automorphism group of the disjoint union of $X$ and $Y$. This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. We exploit this link by using ideas and results from quantum information in order to prove new results about quantum automorphism groups, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum group literature. Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other.Comment: 39 page