Matrix Calculus for Classical and Quantum Circuits

Quantum computation on w qubits is represented by the infinite unitary group U(2(w)); classical reversible computation on w bits is represented by the finite symmetric group S-2w. In order to establish the relationship between classical reversible computing and quantum computing, we introduce two Lie subgroups XU(n) and ZU(n) of the unitary group U(n). The former consists of all unitary n x n matrices with all line sums equal to 1; the latter consists of all unitary diagonal n x n matrices with first entry equal to 1. Such a group structure also reveals the relationship between matrix calculus and diagrammatic zx-calculus of quantum circuits.