ZH-calculus extends ZX-calculus with a third generator, the H-box, which represents a Hadamard-like node whose entries can be arbitrary complex numbers rather than just $\pm 1$. This lets ZH-calculus directly express non-linear boolean functions (AND, Toffoli-style multi-controlled gates) as a single diagram element, something ZX-calculus can only build out of several spiders.

An H-box with all-1 entries except a single $-1$ recovers the ordinary Hadamard gate, but more general H-boxes can represent an AND gate on any number of inputs in one node. This is useful because many practical circuits, especially arithmetic and oracle circuits used in Grover-style algorithms, are built almost entirely out of multi-controlled gates.

  H-box (n-input AND-like node):
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      [H]  --- arbitrary tensor entries
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ZH-calculus keeps ZX-calculus's spider fusion rules and adds H-box-specific rules: two H-boxes connected in sequence multiply their tensors together, and an H-box with a single leg of value 1 is the identity. Like ZX-calculus, ZH-calculus is complete for all of quantum mechanics, not just the Clifford fragment, once general H-boxes are allowed.

ZH-calculus is the natural graphical language for reasoning about Toffoli-based circuits and other multi-controlled gate constructions common in arithmetic circuits and Grover's algorithm oracles. It's typically used alongside ZX-calculus and ZW-calculus, with a given diagram translated into whichever of the three calculi makes the structure being studied easiest to manipulate.