Graphical Reasoning in Compact Closed Categories for Quantum Computation

Compact closed categories provide a foundational formalism for a variety of important domains, including quantum computation. These categories have a natural visualisation as a form of graphs. We present a formalism for equational reasoning about such graphs and develop this into a generic proof system with a fixed logical kernel for equational reasoning about compact closed categories. Automating this reasoning process is motivated by the slow and error prone nature of manual graph manipulation. A salient feature of our system is that it provides a formal and declarative account of derived results that can include `ellipses'-style notation. We illustrate the framework by instantiating it for a graphical language of quantum computation and show how this can be used to perform symbolic computation.Comment: 21 pages, 9 figures. This is the journal version of the paper published at AIS

Quantum picturalism for topological cluster-state computing

Topological quantum computing is a way of allowing precise quantum computations to run on noisy and imperfect hardware. One implementation uses surface codes created by forming defects in a highly-entangled cluster state. Such a method of computing is a leading candidate for large-scale quantum computing. However, there has been a lack of sufficiently powerful high-level languages to describe computing in this form without resorting to single-qubit operations, which quickly become prohibitively complex as the system size increases. In this paper we apply the category-theoretic work of Abramsky and Coecke to the topological cluster-state model of quantum computing to give a high-level graphical language that enables direct translation between quantum processes and physical patterns of measurement in a computer - a "compiler language". We give the equivalence between the graphical and topological information flows, and show the applicable rewrite algebra for this computing model. We show that this gives us a native graphical language for the design and analysis of topological quantum algorithms, and finish by discussing the possibilities for automating this process on a large scale.Comment: 18 pages, 21 figures. Published in New J. Phys. special issue on topological quantum computin

Quantum Picturalism

The quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. In this review we present steps towards a diagrammatic `high-level' alternative for the Hilbert space formalism, one which appeals to our intuition. It allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the no-cloning theorem, and phenomena such as quantum teleportation. As a logic, it supports `automation'. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required step-stone towards a deeper conceptual understanding of quantum theory, as well as its unification with other physical theories. Specific applications discussed here are purely diagrammatic proofs of several quantum computational schemes, as well as an analysis of the structural origin of quantum non-locality. The underlying mathematical foundation of this high-level diagrammatic formalism relies on so-called monoidal categories, a product of a fairly recent development in mathematics. These monoidal categories do not only provide a natural foundation for physical theories, but also for proof theory, logic, programming languages, biology, cooking, ... The challenge is to discover the necessary additional pieces of structure that allow us to predict genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures, some colo

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

Teleportation, Braid Group and Temperley--Lieb Algebra

We explore algebraic and topological structures underlying the quantum teleportation phenomena by applying the braid group and Temperley--Lieb algebra. We realize the braid teleportation configuration, teleportation swapping and virtual braid representation in the standard description of the teleportation. We devise diagrammatic rules for quantum circuits involving maximally entangled states and apply them to three sorts of descriptions of the teleportation: the transfer operator, quantum measurements and characteristic equations, and further propose the Temperley--Lieb algebra under local unitary transformations to be a mathematical structure underlying the teleportation. We compare our diagrammatical approach with two known recipes to the quantum information flow: the teleportation topology and strongly compact closed category, in order to explain our diagrammatic rules to be a natural diagrammatic language for the teleportation.Comment: 33 pages, 19 figures, latex. The present article is a short version of the preprint, quant-ph/0601050, which includes details of calculation, more topics such as topological diagrammatical operations and entanglement swapping, and calls the Temperley--Lieb category for the collection of all the Temperley--Lieb algebra with physical operations like local unitary transformation

Causal categories: relativistically interacting processes

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a `causal category'. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.Comment: 43 pages, lots of figure

The Measurement Calculus

Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model which is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns, and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models: Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.Comment: 46 pages, 2 figures, Replacement of quant-ph/0412135v1, the new version also include formalization of several other measurement-based models: Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. To appear in Journal of AC

Theory and practice of social norms interventions: eight common pitfalls.

BACKGROUND: Recently, Global Health practitioners, scholars, and donors have expressed increased interest in "changing social norms" as a strategy to promote health and well-being in low and mid-income countries (LMIC). Despite this burgeoning interest, the ability of practitioners to use social norm theory to inform health interventions varies widely. MAIN BODY: Here, we identify eight pitfalls that practitioners must avoid as they plan to integrate a social norms perspective in their interventions, as well as eight learnings. These learnings are: 1) Social norms and attitudes are different; 2) Social norms and attitudes can coincide; 3) Protective norms can offer important resources for achieving effective social improvement in people's health-related practices; 4) Harmful practices are sustained by a matrix of factors that need to be understood in their interactions; 5) The prevalence of a norm is not necessarily a sign of its strength; 6) Social norms can exert both direct and indirect influence; 7) Publicising the prevalence of a harmful practice can make things worse; 8) People-led social norm change is both the right and the smart thing to do. CONCLUSIONS: As the understanding of how norms evolve in LMIC advances, practitioners will develop greater understanding of what works to help people lead change in harmful norms within their contexts. Awareness of these pitfalls has helped several of them increase the effectiveness of their interventions addressing social norms in the field. We are confident that others will benefit from these reflections as well

A Categorical Quantum Logic

We define a sequent calculus corresponding to the logic of strongly compact closed categories with biproducts. Based on this calculus, we define a proof-net syntax with a strongly normalising cut-elimination. This syntax encodes abstract qualitative and quantitative information about the behaviour of quantum processes