We introduce a family of diagrammatical equational theories unifying two research programs: categorical quantum mechanics and graphical linear algebra. We prove their completeness with respect to denotational semantics described in terms of relations between vector spaces equipped with symplectic structure. This provides versatile graphical languages encompassing both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser and Gaussian quantum circuits. Terms are described by labelled graphs with input and output interfaces, and the languages are equipped with equational theories amenable to standard graph rewriting techniques. In order to reason about large composite systems, we introduce a compact scalable notation where the vertices are themselves labelled by graphs. This notation allows us to state new and powerful rewrite rules which operate on diagrams at a large scale. We also show how this notation neatly captures some important constructions, such as graph states of quantum computing and the impedance and admittance matrices of electrical networks.
We introduce a family of diagrammatical equational theories unifying two research programs: categorical quantum mechanics and graphical linear algebra. We prove their completeness with respect to denotational semantics described in terms of relations between vector spaces equipped with symplectic structure. This provides versatile graphical languages encompassing both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser and Gaussian quantum circuits. Terms are described by labelled graphs with input and output interfaces, and the languages are equipped with equational theories amenable to standard graph rewriting techniques. In order to reason about large composite systems, we introduce a compact scalable notation where the vertices are themselves labelled by graphs. This notation allows us to state new and powerful rewrite rules which operate on diagrams at a large scale. We also show how this notation neatly captures some important constructions, such as graph states of quantum computing and the impedance and admittance matrices of electrical networks.
Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.
We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties. This framework provides clear graphical representations of algebraic structures -- such as polynomials, ideals, and varieties -- enabling intuitive yet rigorous diagrammatic reasoning. We showcase two practical viewpoints on GAG. First, we show that instances of counting constraint satisfaction problem (#CSP) are recast as rewrite problems of closed diagrams in GAG. This means that deciding rewritability in GAG is #P-hard, and GAG can be viewed as a complete and compositional rewrite system for networks of polynomial constraints. Second, we characterize the qudit ZH calculus, a diagrammatic language for quantum computation, as an extension of Graphical Algebraic Geometry. This establishes the correspondence that Graphical Algebraic Geometry is to the ZH calculus what Graphical Linear Algebra is to the ZX calculus. Using this construction, we show that computing amplitudes in qudit ZH requires only a constant number of queries to a GAG oracle.
We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties. This framework provides clear graphical representations of algebraic structures - such as polynomials, ideals, and varieties - enabling intuitive yet rigorous diagrammatic reasoning. We showcase two practical viewpoints on GAG. First, we show that instances of counting constraint satisfaction problem (#CSP) are recast as rewrite problems of closed diagrams in GAG. This means that deciding rewritability in GAG is #P-hard, and GAG can be viewed as a complete and compositional rewrite system for networks of polynomial constraints. Second, we characterize the qudit ZH calculus, a diagrammatic language for quantum computation, as an extension of Graphical Algebraic Geometry. This establishes the correspondence that Graphical Algebraic Geometry is to the ZH calculus what Graphical Linear Algebra is to the ZX calculus. Using this construction, we show that computing amplitudes in qudit ZH requires only a constant number of queries to a GAG oracle.
In recent years, diagrammatic languages have been shown to be a powerful and expressive tool for reasoning about physical, logical, and semantic processes represented as morphisms in a monoidal category. In particular, categorical quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of quantum theory into abstract structural properties, expressed in the form of diagrammatic identities. One way we search for these properties is to start with a concrete model (e.g. a set of linear maps or finite relations) and start composing generators into diagrams and looking for graphical identities. Naively, we could automate this procedure by enumerating all diagrams up to a given size and check for equalities, but this is intractable in practice because it produces far too many equations. Luckily, many of these identities are not primitive, but rather derivable from simpler ones. In 2010, Johansson, Dixon, and Bundy developed a technique called conjecture synthesis for automatically generating conjectured term equations to feed into an inductive theorem prover. In this extended abstract, we adapt this technique to diagrammatic theories, expressed as graph rewrite systems, and demonstrate its application by synthesising a graphical theory for studying entangled quantum states.
Quantum control is an important logical primitive of quantum computing programs, and an important concept for equational reasoning in quantum graphical calculi. We show that controlled diagrams in the ZXW-calculus admit rich algebraic structure. The perspective of the higher-order map Ctrl recovers the standard notion of quantum controlled gates, while respecting sequential and parallel composition and multiple-control.
In this work, we prove that controlled square matrices form a ring and therefore satisfy powerful rewrite rules. We also show that controlled states form a ring isomorphic to multilinear polynomials. Putting these together, we have completeness for polynomials over same-size square matrices. These properties supply new rewrite rules that make factorisation of arbitrary qubit Hamiltonians achievable inside a single graphical calculus.
This thesis is about the application of graphical languages to quantum computing. By graphical language, we mean the use of diagrams, similar to circuits, representing the evolution of quantum systems. The thesis introduces those languages in the formalism of category theory and focuses mainly on one language: the ZX-calculus, and its close relatives, the ZW-calculus and ZH-calculus. The notion of flexsymmetry is introduced, describing diagrams whose inputs and outputs are all interchangeable. This notion is used to classify all languages similar to the ZX-calculus. It is shown that the only admissible languages are the ZX-calculus, the ZH-calculus, and the ZW-calculus. Then is tackled the question of extending those languages to mixed-state quantum mechanics. A general categorical construction is proposed and is applied to provide extensions of the different languages. Finally, the thesis introduces notations allowing to handle in a compact way quantum algorithms relying on arbitrary large diagrams. To challenge their efficiency, those notations are used to show the correction of various quantum algorithms.