A note on diagonal gates in SZX-calculus

SZX-calculus: Scalable Graphical Quantum Reasoning

We introduce the Scalable ZX-calculus (SZX-calculus for short), a formal and compact graphical language for the design and verification of quantum computations. The SZX-calculus is an extension of the ZX-calculus, a powerful framework that captures graphically the fundamental properties of quantum mechanics through its complete set of rewrite rules. The ZX-calculus is, however, a low level language, with each wire representing a single qubit. This limits its ability to handle large and elaborate quantum evolutions. We extend the ZX-calculus to registers of qubits and allow compact representati…

A graphical calculus for integration over random diagonal unitary matrices

We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially ordered set of uniform block permutations. A similar calculus is developed for random vectors consisting of independent uniform signs, based on the combinatorics of the partially ordered set of even partitions. We employ our method to extend some of the results by Johnston and MacLean on the family of local diagonal unitary invariant matrices. Furthermore, our …

ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity

We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximately (real-valued) universal family of Hadamard+CCZ circuits. The diagrammatic language is generated by two kinds of nodes: the so-called 'spider' associated with the computational basis, as well as a new arity-N generalisation of the Hadamard gate, which satisfies a variation of the spider fusion law. Unlike previous graphical calculi, this admits compact encodings of non-linear classical functions. For exampl…

构建来自相位小工具的 qutrit 对角门

Building Qutrit Diagonal Gates from Phase Gadgets

本文研究 qutrit 相位小工具，提出 flexsymmetric ZX 演算，推广多重控制概念，实现无辅助量子比特的 qutrit 相位门，并给出仅需三个非 Clifford 门的 CCZ 门模拟，优于 qubit 方案。

可扩展蜘蛛巢（……或如何以图形化方式参悟横向非Clifford门）

Scalable Spider Nests (...Or How to Graphically Grok Transversal Non-Clifford Gates)

作为图形化参悟系列第二篇，本文用可扩展ZX-演算简洁表述蜘蛛巢恒等式并证明其可由单条新规则归纳导出，结合CSS码ZX图景刻画了任意CSS码中Clifford层级第三级的全部横向对角门。

Techniques to Reduce $π/4$-Parity-Phase Circuits, Motivated by the ZX Calculus

To approximate arbitrary unitary transformations on one or more qubits, one must perform transformations which are outside of the Clifford group. The gate most commonly considered for this purpose is the T = diag(1, exp(i π/4)) gate. As T gates are computationally expensive to perform fault-tolerantly in the most promising error-correction technologies, minimising the "T-count" (the number of T gates) required to realise a given unitary in a Clifford+T circuit is of great interest. We describe techniques to find circuits with reduced T-count in unitary circuits, which develop on the ideas of H…