Tutorial: Graphical Calculus for Quantum Circuits

Bob Coecke, Duncan, Ross

We explain the graphical zx-calculus for reasoning about qubits without any reference to the underlying categorical semantics, and illustrate its use on quantum circuits. © 2013 Springer-Verlag Berlin Heidelberg. © Springer-Verlag Berlin Heidelberg 2013.

Ayan Datta, P Prabhavathi

Vladimir P. Gerdt, Alexander N. Prokopenya

Larisse D. Voufo, Gerardo Ortíz, Amr Sabry

We present elements of quantum circuits trans- lations from the (standard) network or circuit model to the one-way one. We present a translation scheme, give an account of currently existing tools to apply the scheme, and propose an extension of those tools into a complete translation calculus. We analyse the set of difficulties incurred from such work, and show an engendered opening to new sets of problems and ideas. Among others, this paper introduces the notions of graphical concatenation, graph state reduction (GSR) and graph state extension (GSE) passes, as well as an algorithm for running the (extended) measurement calculus with acceptable efficiency.

Boldizsár Poór, Quanlong Wang, Razin Abdulrauf Shaikh +3 more

The ZX-calculus is a universal graphical language for qubit quantum computation, meaning that every linear map between qubits can be expressed in the ZX-calculus. Furthermore, it is a complete graphical rewrite system: any equation involving linear maps that is derivable in the Hilbert space formalism for quantum theory can also be derived in the calculus by rewriting. It has widespread usage within quantum industry and academia for a variety of tasks such as quantum circuit optimisation, error-correction, and education.The ZW-calculus is an alternative universal graphical language that is also complete for qubit quantum computing. In fact, its completeness was used to prove that the ZX-calculus is universally complete. This calculus has advanced how quantum circuits are compiled into photonic hardware architectures in the industry.Recently, by combining these two calculi, a new calculus has emerged for qubit quantum computation, the ZXW-calculus. Using this calculus, graphical-differentiation, -integration, and -exponentiation were made possible, thus enabling the development of novel techniques in the domains of quantum machine learning and quantum chemistry.Here, we generalise the ZXW-calculus to arbitrary finite dimensions, that is, to qudits. Moreover, we prove that this graphical rewrite system is complete for any finite dimension. This is the first completeness result for any universal graphical language beyond qubits.

Helio V. Fagundes