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Quantum information-flow, concretely, abstractly

Bob Coecke

Published Jan 1, 2004• 6 citations
OpenAlex

Abstract

These `lecture notes' are based on joint work with Samson Abramsky. I will survey and informally discuss the results of [3, 4, 5, 12, 13] in a pedestrian not too technical way. These include: .

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This paper surveys joint work with Samson Abramsky. I will somewhat informally discuss the main results of a series of papers in a pedestrian not too technical way. These include: . 'The logic of entanglement', that is, the identification and abstract axiomatization of the 'quantum information-flow' which enables protocols such as quantum teleportation. To this means we define strongly compact closed categories which abstractly capture the behavioral properties of quantum entanglement. . 'Postulates for an abstract quantum formalism' in which classical information-flow (e.g. token exchange) is part of the formalism. As an example, we provide a purely formal description of quantum teleportation and prove correctness in abstract generality. In this formalism types reflect kinds contra the essentially typeless von Neumann formalism. Hence even concretely this formalism manifestly improves on the usual one. .'Towards a high-level approach to quantum informatics'. Indeed the above discussed work can be conceived as aiming to solve: ???/von Neumann quantum formalism ≈ high-level language/low-level language
These lecture notes survey some joint work with Samson Abramsky as it was presented by me at several conferences in the summer of 2005. It concerns ‘doing quantum mechanics using only pictures of lines, squares, triangles and diamonds’. This picture calculus can be seen as a very substantial extension of Dirac’s notation, and has a purely algebraic counterpart in terms of so‐called Strongly Compact Closed Categories (introduced by Abramsky and I in [3, 4]) which subsumes my Logic of Entanglement [11]. For a survey on the ‘what’, the ‘why’ and the ‘hows’ I refer to a previous set of lecture notes [12, 13]. In a last section we provide some pointers to the body of technical literature on the subject.
This is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algorithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the "quantics" representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross-Pitaevskii) within the "quantics" representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements.