Continuous-variable (CV) quantum information processing is a promising candidate for large-scale fault-tolerant quantum computation. However, analysis of CV quantum process relies mostly on direct computation of the evolution of operators in the Heisenberg picture, and the features of CV space has yet to be thoroughly investigated in an intuitive manner. One key ingredient for further exploration of CV quantum computing is the construction of a computational model that brings visual intuition and new tools for the analysis of quantum protocols, especially non-Gaussian processes. In this paper, we delve into a graphical computational model, inspired by a similar model for qubit-based systems called the ZX calculus, which enables the representation of arbitrary CV quantum process as a simple directed graph. We demonstrate the utility of our model as a graphical tool to comprehend CV processes intuitively by showing how equivalences between two distinct quantum processes can be proven as a sequence of diagrammatic transformations for various quantum protocols. We also examine possible applications of our model, such as measurement-based quantum computing, characterization of Gaussian and non-Gaussian processes, and circuit optimization.
Continuous-variable (CV) quantum information processing is a promising candidate for large-scale fault-tolerant quantum computation. However, analysis of CV quantum process relies mostly on direct computation of the evolution of operators in the Heisenberg picture, and the features of CV space has yet to be thoroughly investigated in an intuitive manner. One key ingredient for further exploration of CV quantum computing is the construction of a computational model that brings visual intuition and new tools for the analysis of quantum protocols, especially non-Gaussian processes. In this paper, we delve into a graphical computational model, inspired by a similar model for qubit-based systems called the ZX calculus, which enables the representation of arbitrary CV quantum process as a simple directed graph. We demonstrate the utility of our model as a graphical tool to comprehend CV processes intuitively by showing how equivalences between two distinct quantum processes can be proven as a sequence of diagrammatic transformations for various quantum protocols. We also examine possible applications of our model, such as measurement-based quantum computing, characterization of Gaussian and non-Gaussian processes, and circuit optimization.
While the ZX and ZW calculi have been effective as graphical reasoning tools for finite-dimensional quantum computation, the possibilities for continuous-variable quantum computation (CVQC) in infinite-dimensional Hilbert space are only beginning to be explored. In this work, we formulate a graphical language for CVQC. Each diagram is an undirected graph made of two types of spiders: the Z spider from the ZX calculus defined on the reals, and the newly introduced Fock spider defined on the natural numbers. The Z and X spiders represent functions in position and momentum space respectively, while the Fock spider represents functions in the discrete Fock basis. In addition to the Fourier transform between Z and X, and the Hermite transform between Z and Fock, we present exciting new graphical rules capturing heftier CVQC interactions. We ensure this calculus is complete for all of Gaussian CVQC interpreted in infinite-dimensional Hilbert space, by translating the completeness in affine Lagrangian relations by Booth, Carette, and Comfort. Applying our calculus for quantum error correction, we derive graphical representations of the Gottesman-Kitaev-Preskill (GKP) code encoder, syndrome measurement, and magic state distillation of Hadamard eigenstates. Finally, we elucidate Gaussian boson sampling by providing a fully graphical proof that its circuit samples submatrix hafnians.
We provide a unified graphical calculus for all Gaussian pure states, including graph transformation rules for all local and semilocal Gaussian unitary operations, as well as local quadrature measurements. We then use this graphical calculus to analyze continuous-variable (CV) cluster states, the essential resource for one-way quantum computing with CV systems. Current graphical approaches to CV cluster states are only valid in the unphysical limit of infinite squeezing, and the associated graph transformation rules only apply when the initial and final states are of this form. Our formalism applies to all Gaussian pure states and subsumes these rules in a natural way. In addition, the term ``CV graph state'' currently has several inequivalent definitions in use. Using this formalism we provide a single unifying definition that encompasses all of them. We provide many examples of how the formalism may be used in the context of CV cluster states: defining the ``closest'' CV cluster state to a given Gaussian pure state and quantifying the error in the approximation due to finite squeezing; analyzing the optimality of certain methods of generating CV cluster states; drawing connections between this graphical formalism and bosonic Hamiltonians with Gaussian ground states, including those useful for CV one-way quantum computing; and deriving a graphical measure of bipartite entanglement for certain classes of CV cluster states. We mention other possible applications of this formalism and conclude with a brief note on fault tolerance in CV one-way quantum computing.
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.
Optimizing quantum circuits is a key challenge for quantum computing. The PyZX compiler broke new ground by optimizing circuits via the ZX calculus, a powerful graphical alternative to the quantum circuit model. Still, it carries no guarantee of its correctness. To address this, we developed VyZX, a verified ZX-calculus in the Coq proof assistant. VyZX provides two distinct representations of ZX diagrams for ease of programming and proof: A graph-based representation for writing high-level functions on diagrams and a block-based representation for proving ZX diagrams equivalent. Through these two different views, VyZX provides the tools necessary to verify properties and transformations of ZX diagrams. This paper explores the proofs and design choices underlying VyZX and its application and the challenges of verifying a graphical programming language.
Quantum computing is a fast-growing research area. A quantum computer can solve problems which are out of reach of the classical computers. It is essential to understand the fundamental structures of quantum information processing in order to develop specific and effective tools. In this habilitation thesis we consider several examples of graphical approaches for quantum computing : first of all the graph state formalism, the measurement-based model of quantum computing and the quantum secret sharing protocol. We then explore in depth the ZX-calculus, a category theory based diagrammatic language for quantum reasoning. This language is equipped with an equational theory to transform any diagram into an equivalent diagram. The language is said to be complete if, for any pair of diagrams representing the same quantum evolution,there exists a transformation from one to the other using the rules of the language. We show the completeness of ZX-calculation for several fragments of quantum mechanics.
Graphical calculi are vital tools for representing and reasoning about quantum circuits and processes. Some are not only graphically intuitive but also logically complete. The best known of these is the ZX-calculus, which is an industry candidate for an Intermediate Representation; a language that sits between the algorithm designer's intent and the quantum hardware's gate instructions. The ZX calculus, built from generalised Z and X rotations, has difficulty reasoning about arbitrary rotations. This contrasts with the cross-hardware compiler TriQ which uses these arbitrary rotations to exploit hardware efficiencies. In this paper we introduce the graphical calculus ZQ, which uses quaternions to represent these arbitrary rotations, similar to TriQ, and the phase-free Z spider to represent entanglement, similar to ZX. We show that this calculus is sound and complete for qubit quantum computing, while also showing that a fully spider-based representation would have been impossible. This new calculus extends the zoo of qubit graphical calculi, each with different strengths, and we hope it will provide a common language for the optimisation procedures of both ZX and TriQ.