The GHZ/W-calculus contains rational arithmetic

Graphical calculi for representing interacting quantum systems serve a number of purposes: compositionally, intuitive graphical reasoning, and a logical underpinning for automation. The power of these calculi stems from the fact that they embody generalized symmetries of the structure of quantum operations, which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One such calculus takes the GHZ and W states as its basic generators. Here we show that this language allows one to encode standard rational calculus, with the GHZ state as multiplication, the W state as addition, the Pauli X gate as multiplicative inversion, and the Pauli Z gate as additive inversion.Comment: In Proceedings HPC 2010, arXiv:1103.226

If physics is an information science, what is an observer?

Interpretations of quantum theory have traditionally assumed a "Galilean" observer, a bare "point of view" implemented physically by a quantum system. This paper investigates the consequences of replacing such an informationally-impoverished observer with an observer that satisfies the requirements of classical automata theory, i.e. an observer that encodes sufficient prior information to identify the system being observed and recognize its acceptable states. It shows that with reasonable assumptions about the physical dynamics of information channels, the observations recorded by such an observer will display the typical characteristics predicted by quantum theory, without requiring any specific assumptions about the observer's physical implementation.Comment: 30 pages, comments welcome; v2 significant revisions - results unchange

Generalized Color Codes Supporting Non-Abelian Anyons

We propose a generalization of the color codes based on finite groups $G$. For non-abelian groups, the resulting model supports non-abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.Comment: 17 pages, 8 figures; references added, typos remove

Towards Verifying Nonlinear Integer Arithmetic

We eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using CDCL SAT solvers, by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits. Such short proofs were conjectured not to exist. More precisely, we give n^{O(1)} size regular resolution proofs for arbitrary degree 2 identities on array, diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result

Non-Pauli topological stabilizer codes from twisted quantum doubles

It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by commuting non-Pauli stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models

Narrowing-based Optimization of Rewrite Theories

Partial evaluation has been never investigated in the context of rewrite theories that allow concurrent systems to be specified by means of rules, with an underlying equational theory being used to model system states as terms of an algebraic data type. In this paper, we develop a symbolic, narrowing-driven partial evaluation framework for rewrite theories that supports sorts, subsort overloading, rules, equations, and algebraic axioms. Our partial evaluation scheme allows a rewrite theory to be optimized by specializing the plugged equational theory with respect to the rewrite rules that define the system dynamics. This can be particularly useful for automatically optimizing rewrite theories that contain overly general equational theories which perform unnecessary computations involving matching modulo axioms, because some of the axioms may be blown away after the transformation. The specialization is done by using appropriate unfolding and abstraction algorithms that achieve significant specialization while ensuring the correctness and termination of the specialization. Our preliminary results demonstrate that our transformation can speed up a number of benchmarks that are difficult to optimize otherwise.This work has been partially supported by the EU (FEDER) and the Spanish MCIU under grant RTI2018094403-B-C32,andbyGeneralitatValencianaundergrantPROMETEO/2019/098. JuliaSapiñahasbeensupported by the Generalitat Valenciana APOSTD/2019/127 grantAlpuente Frasnedo, M.; Ballis, D.; Escobar Román, S.; Sapiña Sanchis, J. (2020). Narrowing-based Optimization of Rewrite Theories. Universitat Politècnica de València. http://hdl.handle.net/10251/14557

Communication protocols and quantum error-correcting codes from the perspective of topological quantum field theory

Topological quantum field theories (TQFTs) provide a general, minimal-assumption language for describing quantum-state preparation and measurement. They therefore provide a general language in which to express multi-agent communication protocols, e.g. local operations, classical communication (LOCC) protocols. Here we construct LOCC protocols using TQFT, and show that LOCC protocols induce quantum error-correcting codes (QECCs) on the agent-environment boundary. Such QECCs can be regarded as implementing, or inducing the emergence of, spacetimes on such boundaries. We investigate this connection between inter-agent communication and spacetime using BF and Chern-Simons theories, and then using topological M-theory.Comment: 52 page

Enumerating Independent Linear Inferences

A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. Equivalently, it is a linear rewrite rule on Boolean terms that constitutes a valid implication. Linear inferences have played a significant role in structural proof theory, in particular in models of substructural logics and in normalisation arguments for deep inference proof systems. In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is capable of more efficiently searching for switch-medial-independent inferences. We use it to find four `minimal' 8-variable independent inferences and also prove that no smaller ones exist; in contrast, a previous approach based directly on formulae reached computational limits already at 7 variables. Two of these new inferences derive some previously found independent linear inferences. The other two (which are dual) exhibit structure seemingly beyond the scope of previous approaches we are aware of; in particular, their existence contradicts a conjecture of Das and Strassburger. We were also able to identify 10 minimal 9-variable linear inferences independent of all the aforementioned inferences, comprising 5 dual pairs, and present applications of our implementation to recent `graph logics'.Comment: 33 pages, 3 figure