  There are various gate sets used for describing quantum computation. A
particularly popular one consists of Clifford gates and arbitrary single-qubit
phase gates. Computations in this gate set can be elegantly described by the
ZX-calculus, a graphical language for a class of string diagrams describing
linear maps between qubits. The ZX-calculus has proven useful in a variety of
areas of quantum information, but is less suitable for reasoning about
operations outside its natural gate set such as multi-linear Boolean operations
like the Toffoli gate. In this paper we study the ZH-calculus, an alternative
graphical language of string diagrams that does allow straightforward encoding
of Toffoli gates and other more complicated Boolean logic circuits. We find a
set of simple rewrite rules for this calculus and show it is complete with
respect to matrices over $\mathbb{Z}[\frac12]$, which correspond to the
approximately universal Toffoli+Hadamard gateset. Furthermore, we construct an
extended version of the ZH-calculus that is complete with respect to matrices
over any ring $R$ where $1+1$ is not a zero-divisor.
