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Leibniz international proceedings in informatics

A 2-Categorical Approach to Composing Quantum Structures

David Reutter, Jamie Vicary

Published Jan 1, 2017• 2 citations
Publisher

Abstract

We present an infinite number of construction schemes for quantum structures, including unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on the type structure of biunitary connections, 2-categorical structures which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method.

DOI
10.4230/lipics.calco.2017.20

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We present an infinite number of construction schemes for quantum structures, including unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on the type structure of biunitary connections, 2-categorical structures which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method.
We present an infinite number of construction schemes involving unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on biunitary connections, algebraic objects which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method.
Unitary error bases have a great number of applications across quantum information and quantum computation, and are fundamentally linked to quantum teleportation, dense coding and quantum error correction. Werner's combinatorial construction builds a unitary error basis from a family of Hadamard matrices and a Latin square. In this dissertation, I give a new categorical axiomatisation of Latin squares, and use this to give a fully graphical presentation and proof of the correctness of Werner's construction. The categorical approach makes clear that some of the Latin square axioms are unnecessary for the construction to go through, and I propose a generalised construction scheme with the potential to create new classes of unitary error bases.
In this paper we introduce quantum Latin squares, combinatorial quantum objects which generalize classical Latin squares, and investigate their applications in quantum computer science. Our main results are on applications to unitary error bases (UEBs), basic structures in quantum information which lie at the heart of procedures such as teleportation, dense coding and error correction. We present a new method for constructing a UEB from a quantum Latin square equipped with extra data. Developing construction techniques for UEBs has been a major activity in quantum computation, with three primary methods proposed: shift-and-multiply, Hadamard, and grouptheoretic. We show that our new approach simultaneously generalizes the shift-andmultiply and Hadamard methods. Furthermore, we explicitly construct a UEB using our technique which we prove cannot be obtained from any of these existing methods.
In this paper we introduce quantum Latin squares, combinatorial quantum objects which generalize classical Latin squares, and investigate their applications in quantum computer science. Our main results are on applications to unitary error bases (UEBs), basic structures in quantum information which lie at the heart of procedures such as teleportation, dense coding and error correction. We present a new method for constructing a UEB from a quantum Latin square equipped with extra data. Developing construction techniques for UEBs has been a major activity in quantum computation, with three primary methods proposed: shift-and-multiply, Hadamard, and group theoretic. We show that our new approach simultaneously generalizes the shift-and multiply and Hadamard methods. Furthermore, we explicitly construct a UEB using our technique which we prove cannot be obtained from any of these existing methods.
A central theme of this thesis is the quantization and generalization of objects from areas of mathematics such as set theory and combinatorics with application to quantum information. We represent mathematical objects such as Latin squares or functions as string diagrams over the category of finite-dimensional Hilbert spaces obeying various diagrammatic axioms. This leads to a direct understanding of how these objects arise in quantum mechanics and gives insight into how quantum analogues can be defined. In Part I of this thesis we introduce quantum Latin squares (QLS), quantum objects which generalize the classical Latin squares from combinatorics. We present a new method for constructing a unitary error basis (UEB) from a quantum Latin square equipped with extra data, which we show simultaneously generalizes the existing shift-and-multiply and Hadamard methods. We introduce two different notions of orthogonality for QLS which we use to construct families of mutually unbiased bases and perfect tensors respectively. We also introduce a further generalization of Latin squares called quantum Latin isometry squares. We use these to produce quantum error codes and to give a new way of characterizing UEBs. In Part II we show that maximal families of mutually unbiased bases (MUBs) are characterized in all dimensions by partitioned UEBs, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned UEBs, and thus maximal families of MUBs, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of MUBs, partitioned UEBs, and finite fields as algebraic structures defined over Hilbert spaces. In Part III we introduce quantum functions and quantum sets, which quantize the classical notions. We show that these structures form a 2-category QSet. We extend this framework to introduce quantum graphs and quantum homomorphisms which form a 2-category QGraph. We show that these 2-categories capture several different notions of quantum morphism and noncommutative graph from various previous papers in noncommutative topology, quantum non-local games and quantum information. We later use the correspondence between our quantum graph morphisms and those from quantum non-local games to show that pairs of quantum isomorphic graphs with multiple connected components are built up of quantum isomorphisms on those components.
Quantum Latin Squares have emerged as mathematical structures of interest in quantum information theory. This paper provides a comprehensive overview of quantum Latin squares. We aim to synthesize the current knowledge for the benefit of researchers in quantum information theory. We begin by establishing the theoretical foundation linking classical Latin squares with their quantum counterparts. This provides insight into their potential applications, particularly in the realm of quantum error correction. Through a detailed examination, we outline known constructions and elucidate the properties of quantum Latin squares, highlighting their role within quantum computing frameworks. We then explore the relationships of quantum Latin squares with mutually unbiased bases and unitary error bases. This underscores their importance in the construction of quantum error correcting codes using mutually orthogonal quantum Latin squares. We give significant focus to a comparative analysis between classical and quantum Latin squares, particularly in their applications to error correcting codes. The applications are analogous to the application of classical Latin squares to classical error correcting codes. This overview aims to not only compile existing research but also to inspire further exploration into Quantum Latin squares and their applications in quantum information theory.