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arXiv:2102.04814 · Frontiers of Physics

Categorical computation

Liang Kong, Hao Zheng

Published Feb 13, 2023• 0 citations
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Abstract

In quantum computing, the computation is achieved by linear operators in or between Hilbert spaces. In this work, we explore a new computation scheme, in which the linear operators in quantum computing are replaced by (higher) functors between two (higher) categories. If from Turing computing to quantum computing is the first quantization of computation, then this new scheme can be viewed as the second quantization of computation. The fundamental problem in realizing this idea is how to realize a (higher) functor physically. We provide a theoretical idea of realizing (higher) functors physically based on the physics of topological orders.

DOI
10.1007/s11467-022-1251-5

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In quantum computing, the computation is achieved by linear operators in or between Hilbert spaces. In this work, we explore a new computation scheme, in which the linear operators in quantum computing are replaced by (higher) functors between two (higher) categories. If from Turing computing to quantum computing is the first quantization of computation, then this new scheme can be viewed as the second quantization of computation. The fundamental problem in realizing this idea is how to realize a (higher) functor physically. We provide a theoretical idea of realizing (higher) functors physically based on the physics of topological orders.
Quantum computing is captured in the formalism of the monoidal subcategory of 𝐕 𝐞 𝐜 𝐭 ℂ generated by ℂ 2 — in particular, quantum circuits are diagrams in 𝐕 𝐞 𝐜 𝐭 ℂ — while topological quantum field theories , in the sense of Atiyah, are diagrams in 𝐕 𝐞 𝐜 𝐭 ℂ indexed by cobordisms. We initiate a program to formalize this connection. In doing so, we equip cobordisms with machinery for producing linear maps by parallel transport along curves under a connection and then assemble these structures into a higher category. Finite-dimensional complex vector spaces and linear maps between them are given a suitable higher categorical structure which we call 𝔽 ⁡ 𝐕 𝐞 𝐜 𝐭 ℂ . We realize quantum circuits as images of cobordisms under monoidal functors from these modified cobordisms to 𝔽 ⁡ 𝐕 𝐞 𝐜 𝐭 ℂ , which are computed by taking parallel transports of vectors and then combining the results in a pattern encoded in the domain category.
The report deals with the axiomatic approach to the phenomena of quantum computation and algorithms. It gives an overview of the classical approach (due to von Neumann) using the structures of Hilbert spaces, as well as mentions some major results in the field and discusses the flaws of the model and its current dominance. This leads to the introduction of a new axiomatization, first proposed by S. Abramsky and B. Coecke in 2007, which relies on the modern concepts of category theory. It is a field of mathematics originally created in the 1940s by specialists in algebraic topology but currently gaining popularity in other branches because of its abstract nature and emphasis on operations rather than objects. Most importantly, category theory is a powerful tool in functional analysis allowing mathematicians to reformulate the statements of Hilbert space theory in a more intuitive way, without matrix calculus. Finally, the logic behind quantum operations is discussed: it is revealed that the ideas of quantum algorithms have a strong connection to Jean-Yves Girard’s linear logic.
This paper presents first steps of an approach to quantum information processing in the framework of higher category theory from a noncommutative mathematics perspective. The aim is to provide a unifying theory for the structure and dynamics of composite quantum information processing systems, such that states, evolution, entanglement, decoherence are modeled by abstract categorical constructions and vice versa new mathematical structures arising from higher dimensional algebra could be "tested" as computational schemes and possibly realized by physical experiments.
Topological quantum computation offers a paradigm shift for fault-tolerant quantum computation. It proposes encoding qubits in the mutual states of quasiparticles (called anyons) of 2-dimensional topologically ordered matter. Unitary ribbon fusion categories describe the algebraic theory of anyons. Although the category of an anyonic theory specifies suitable computational spaces, qubits, and quantum gates, it does not precisely formalise the category of topological quantum computation (TQC). We describe the TQC category using the elements of categorical quantum mechanics (CQM). Given the qubit-based graphical calculus accompanying CQM, called the ZX-calculus, we present the details of the ZX-representation of Fibonacci and Ising anyons. Finally, we propose a bicategorical approach to convene anyonic theory and TQC in one (bi)category. In the second part of the thesis, we discuss how the bicategory of module categories formalises boundary types and their excitations. We show theorems that reduce the data to physically meaningful data. A 2-category capable of describing topological quantum computation with point-like and string-like excitations is speculated to be a fusion 2-category. We discuss the available definition in the literature. The paper particularly lacks a rigorous definition for additive 2-categories; in this thesis, we give a definition. We propose the idea of typing higher-ranked matrices in additive 2-categories. Lastly, we overview the complete definition of monoidal 2-categories and recover Kapranov and Voevodsky's definition from the algebraic definition of tricategory by Gurski. This definition and the detailed list of data are the complete ingredients physicists need for quantum computation with higher dimensional excitations.
The search for a semantics for higher-order quantum computation leads naturallyto the study of categories of normed cones. In the first part of this paper, we develop the theory of continuous normed cones, and prove some of their basic properties, includinga Hahn-Banach style theorem. We then describe two different concrete *-autonomous categories of normed cones. The first of these categories is built from completelypositive maps as in the author's semantics of first-order quantum computation. The second category is a reformulation of Girard's quantum coherent spaces. We also pointout why ultimately, neither of these categories is a satisfactory model of higher-order quantum computation.