We introduce the qudit ZH-calculus and show how to generalise all the phase-free qubit rules to qudits. We prove that for prime dimensions d, the phase-free qudit ZH-calculus is universal for matrices over the ring Z[e^2(pi)i/d]. For qubits, there is a strong connection between phase-free ZH-diagrams and Toffoli+Hadamard circuits, a computationally universal fragment of quantum circuits. We generalise this connection to qudits, by finding that the two-qudit |0>-controlled X gate can be used to construct all classical reversible qudit logic circuits in any odd qudit dimension, which for qubits requires the three-qubit Toffoli gate. We prove that our construction is asymptotically optimal up to a logarithmic term. Twenty years after the celebrated result by Shi proving universality of Toffoli+Hadamard for qubits, we prove that circuits of |0>-controlled X and Hadamard gates are approximately universal for qudit quantum computing for any odd prime d, and moreover that phase-free ZH-diagrams correspond precisely to such circuits allowing post-selections.

翻译：我们引入量子d维ZH-演算，并展示如何将所有无相位量子比特规则推广至量子d维情形。对于素数维度d，我们证明了无相位量子d维ZH-演算在环Z[e^(2πi/d)]上关于矩阵具有普适性。针对量子比特，无相位ZH-图与Toffoli+Hadamard电路（量子电路中可计算普适的子片段）之间存在强关联。我们将此关联推广至量子d维，发现在任意奇数量子d维中，两量子d维|0>控制的X门可用于构造所有经典可逆量子d维逻辑电路（而对量子比特而言，这需要三量子比特Toffoli门）。我们证明该构造在渐进意义上达到对数因子范围内的最优性。在Shi关于量子比特Toffoli+Hadamard普适性的著名结果发表二十年后，我们证明了由|0>控制的X门与Hadamard门构成的电路对任意奇数素数d的量子d维计算近似普适，且无相位ZH-图恰好对应允许后选择的此类电路。