  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.
