ZX calculus provides a graphical formalism for reasoning about quantum processes, built from two interacting Frobenius algebras associated with the Z and X bases of a qubit. While it has found widespread application in quantum information and computing, its relationship to quantum field theory has only recently begun to be explored. In this work, we further develop this connection by providing a generalization of ZX calculus to two-dimensional Yang Mills theory with a compact gauge group. The key observation is that both frameworks can be organized around the Hopf Frobenius algebraic structure associated with a group algebra, which can in turn be described by the diagrammatics of two dimensional topological quantum field theory. Given the well known relationship between gauge theory and gravity in two and three dimensions, our work paves the way for applications of ZX to low dimensional gravity.
ZX calculus provides a graphical formalism for reasoning about quantum processes, built from two interacting Frobenius algebras associated with the Z and X bases of a qubit. While it has found widespread application in quantum information and computing, its relationship to quantum field theory has only recently begun to be explored. In this work, we further develop this connection by providing a generalization of ZX calculus to two-dimensional Yang Mills theory with a compact gauge group. The key observation is that both frameworks can be organized around the Hopf Frobenius algebraic structure associated with a group algebra, which can in turn be described by the diagrammatics of two dimensional topological quantum field theory. Given the well known relationship between gauge theory and gravity in two and three dimensions, our work paves the way for applications of ZX to low dimensional gravity.
Abstract We apply quantum group methods to quantum computing, starting with the notion of interacting Frobenius Hopf algebras for ZX calculus with noncommutative algebra and noncocommutative coalgebra. We introduce the notion of *-structures in ZX calculus at this algebraic level and construct examples based on the quantum group u q (sl 2) at a root of unity. We provide an abstract formulation of the Hadamard gate related to Hopf algebra self-duality. We then solve the problem of extending the notion of interacting Hopf algebras and ZX calculus to take place in a braided tensor category. In the ribbon case, the Hadamard gate coming from braided self-duality obeys a modular identity. We give the example of b q (sl 2), the self-dual braided version of u q (sl 2).
We introduce the Spin-ZX calculus as an elevation of Penrose's diagrams and associated binor calculus to the level of a formal diagrammatic language. The power of doing so is illustrated by the variety of scientific areas we apply it to: permutational quantum computing, quantum machine learning, condensed matter physics, and quantum gravity. Respectively, we analyse permutational computing transition amplitudes, evaluate barren plateaus for SU(2) symmetric ansätze, study properties of AKLT states, and derive the minimum quantised volume in loop quantum gravity.
Our starting point is the mixed-dimensional ZX calculus, a purely diagrammatic language that has been proven to be complete for finite-dimensional Hilbert spaces. That is, any equation that can be derived in the Hilbert space formalism, can also be derived in the mixed-dimensional ZX calculus. We embed the Spin-ZX calculus inside the mixed-dimensional ZX calculus, rendering it a quantum information flavoured diagrammatic language for the quantum theory of angular momentum, i.e. SU(2) representation theory. We diagrammatically derive the fundamental spin coupling objects - such as Clebsch-Gordan coefficients, symmetrising mappings between qubits and spin spaces, and spin Hamiltonians - under this embedding.
Our results establish the Spin-ZX calculus as a powerful tool for representing and computing with SU(2) systems graphically, offering new insights into foundational relationships and paving the way for new diagrammatic algorithms for theoretical physics.
The ZX-calculus, and the variant we consider in this paper (ZXH-calculus), are formal diagrammatic languages for qubit quantum computing. We show that it can also be used to describe SU(2) representation theory. To achieve this, we first recall the definition of Yutsis diagrams, a standard graphical calculus used in quantum chemistry and quantum gravity, which captures the main features of SU(2) representation theory. Second, we show precisely how it embed within Penrose's binor calculus. Third, we subsume both calculus into ZXH-diagrams. In the process we show how the SU(2) invariance of Wigner symbols is trivially provable in the ZXH-calculus. Additionally, we show how we can explicitly diagrammatically calculate 3jm, 4jm and 6j symbols. It has long been thought that quantum gravity should be closely aligned to quantum information theory. In this paper, we present a way in which this connection can be made exact, by writing the spin-networks of loop quantum gravity (LQG) in the ZX-diagrammatic language of quantum computation.
This paper has two tightly intertwined aims: (i) to introduce an intuitive and universal graphical calculus for multi-qubit systems, the ZX -calculus, which greatly simplifies derivations in the area of quantum computation and information. (ii) To axiomatize complementarity of quantum observables within a general framework for physical theories in terms of dagger symmetric monoidal categories. We also axiomatize phase shifts within this framework. Using the well-studied canonical correspondence between graphical calculi and dagger symmetric monoidal categories, our results provide a purely graphical formalisation of complementarity for quantum observables. Each individual observable, represented by a commutative special dagger Frobenius algebra, gives rise to an Abelian group of phase shifts, which we call the phase group. We also identify a strong form of complementarity, satisfied by the Z - and X -spin observables, which yields a scaled variant of a bialgebra.
While the quantum circuit model remains the dominant framework for representing and executing quantum algorithms, it is not always the most optimal framework for representing the quantum circuit’s deeper structure. In a standard circuit diagram, computation is shown mainly as a time ordered sequence of gates acting on qubit wires. ZX calculus, by contrast, represents the same process using two families of generators, the Z spiders and the X spiders, together with wires and Hadamard structure, so that algebraic identities, entanglement structure, and rewrite based simplifications become visible directly in the diagram itself. ZX calculus is not another visual representation for the same quantum circuit we have been using, but a graphical calculus in which diagrams carry both computational meaning and transformation rules. This is also the reason for the name ZX ; the formalism is built around the interaction between Z -type and X -type observables, phases, and basis changes. By turning matrix manipulations into diagrammatic rewrites, ZX calculus reduces much of the redundancies and abstract nature usually associated with quantum computation and provides a more structural language for understanding, optimizing, and verifying quantum processes.
ZX-calculus is graphical language for quantum computing which usually focuses on qubits. In this paper, we generalise qubit ZX-calculus to qudit ZX-calculus in any finite dimension by introducing suitable generators, especially a carefully chosen triangle node. As a consequence we obtain a set of rewriting rules which can be seen as a direct generalisation of qubit rules, and a normal form for any qudit vectors. Based on the qudit ZX-calculi, we propose a graphical formalism called qufinite ZX-calculus as a unified framework for all qudit ZX-calculi, which is universal for finite quantum theory due to a normal form for matrix of any finite size. As a result, it would be interesting to give a fine-grained version of the diagrammatic reconstruction of finite quantum theory [Selby2021reconstructing] within the framework of qufinite ZX-calculus.