13 research outputs found
A Diagrammatic Axiomatisation for Qubit Entanglement
Diagrammatic techniques for reasoning about monoidal categories provide an
intuitive understanding of the symmetries and connections of interacting
computational processes. In the context of categorical quantum mechanics,
Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used
as the building blocks of a new graphical calculus, aimed at a diagrammatic
classification of multipartite qubit entanglement that would highlight the
communicational properties of quantum states, and their potential uses in
cryptographic schemes.
In this paper, we present a full graphical axiomatisation of the relations
between GHZ and W: the ZW calculus. This refines a version of the preexisting
ZX calculus, while keeping its most desirable characteristics: undirectedness,
a large degree of symmetry, and an algebraic underpinning. We prove that the ZW
calculus is complete for the category of free abelian groups on a power of two
generators - "qubits with integer coefficients" - and provide an explicit
normalisation procedure.Comment: 12 page
The algebra of entanglement and the geometry of composition
String diagrams turn algebraic equations into topological moves that have
recurring shapes, involving the sliding of one diagram past another. We
individuate, at the root of this fact, the dual nature of polygraphs as
presentations of higher algebraic theories, and as combinatorial descriptions
of "directed spaces". Operations of polygraphs modelled on operations of
topological spaces are used as the foundation of a compositional universal
algebra, where sliding moves arise from tensor products of polygraphs. We
reconstruct several higher algebraic theories in this framework.
In this regard, the standard formalism of polygraphs has some technical
problems. We propose a notion of regular polygraph, barring cell boundaries
that are not homeomorphic to a disk of the appropriate dimension. We define a
category of non-degenerate shapes, and show how to calculate their tensor
products. Then, we introduce a notion of weak unit to recover weakly degenerate
boundaries in low dimensions, and prove that the existence of weak units is
equivalent to a representability property.
We then turn to applications of diagrammatic algebra to quantum theory. We
re-evaluate the category of Hilbert spaces from the perspective of categorical
universal algebra, which leads to a bicategorical refinement. Then, we focus on
the axiomatics of fragments of quantum theory, and present the ZW calculus, the
first complete diagrammatic axiomatisation of the theory of qubits.
The ZW calculus has several advantages over ZX calculi, including a
computationally meaningful normal form, and a fragment whose diagrams can be
read as setups of fermionic oscillators. Moreover, its generators reflect an
operational classification of entangled states of 3 qubits. We conclude with
generalisations of the ZW calculus to higher-dimensional systems, including the
definition of a universal set of generators in each dimension.Comment: v2: changes to end of Chapter 3. v1: 214 pages, many figures;
University of Oxford doctoral thesi
Higher-dimensional subdiagram matching
Higher-dimensional rewriting is founded on a duality of rewrite systems and
cell complexes, connecting computational mathematics to higher categories and
homotopy theory: the two sides of a rewrite rule are two halves of the boundary
of an (n+1)-cell, which are diagrams of n-cells. We study higher-dimensional
diagram rewriting as a mechanism of computation, focussing on the matching
problem for rewritable subdiagrams within the combinatorial framework of
diagrammatic sets. We provide an algorithm for subdiagram matching in arbitrary
dimensions, based on new results on layerings of diagrams, and derive upper
bounds on its time complexity. We show that these superpolynomial bounds can be
improved to polynomial bounds under certain acyclicity conditions, and that
these conditions hold in general for diagrams up to dimension 3. We discuss the
challenges that arise in dimension 4.Comment: Accepted for LICS 2023. 13 pages + appendix 9 page
Data Structures for Topologically Sound Higher-Dimensional Diagram Rewriting
We present a computational implementation of diagrammatic sets, a model of
higher-dimensional diagram rewriting that is "topologically sound": diagrams
admit a functorial interpretation as homotopies in cell complexes. This has
potential applications both in the formalisation of higher algebra and category
theory and in computational algebraic topology. We describe data structures for
well-formed shapes of diagrams of arbitrary dimensions and provide a solution
to their isomorphism problem in time O(n^3 log n). On top of this, we define a
type theory for rewriting in diagrammatic sets and provide a semantic
characterisation of its syntactic category. All data structures and algorithms
are implemented in the Python library rewalt, which also supports various
visualisations of diagrams.Comment: In Proceedings ACT 2022, arXiv:2307.1551
A diagrammatic calculus of fermionic quantum circuits
We introduce the fermionic ZW calculus, a string-diagrammatic language for
fermionic quantum computing (FQC). After defining a fermionic circuit model, we
present the basic components of the calculus, together with their
interpretation, and show how the main physical gates of interest in FQC can be
represented in our language. We then list our axioms, and derive some
additional equations. We prove that the axioms provide a complete equational
axiomatisation of the monoidal category whose objects are systems of finitely
many local fermionic modes (LFMs), with maps that preserve or reverse the
parity of states, and the tensor product as monoidal product. We achieve this
through a procedure that rewrites any diagram in a normal form. As an example,
we show how the statistics of a fermionic Mach-Zehnder interferometer can be
calculated in the diagrammatic language. We conclude by giving a diagrammatic
treatment of the dual-rail encoding, a standard method in optical quantum
computing used to perform universal quantum computation
A Diagrammatic Axiomatisation of Fermionic Quantum Circuits
We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC). After defining a fermionic circuit model, we present the basic components of the calculus, together with their interpretation, and show how the main physical gates of interest in FQC can be represented in the language. We then list our axioms, and derive some additional equations. We prove that the axioms provide a complete equational axiomatisation of the monoidal category whose objects are quantum systems of finitely many local fermionic modes, with operations that preserve or reverse the parity (number of particles mod 2) of states, and the tensor product, corresponding to the composition of two systems, as monoidal product. We achieve this through a procedure that rewrites any diagram in a normal form. We conclude by showing, as an example, how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language