Matrix Calculus for Classical and Quantum Circuits

Quantum computation on <i>w</i> qubits is represented by the infinite unitary group U(2<sup><i>w</i></sup>); classical reversible computation on <i>w</i> bits is represented by the finite symmetric group <b>S</b><sub>2</sub><sup><i>w</i></sup>. In order to establish the relationship between classical reversible computing and quantum computing, we introduce two Lie subgroups XU(<i>n</i>) and ZU(<i>n</i>) of the unitary group U(<i>n</i>). The former consists of all unitary <i>n</i> × <i>n</i> matrices with all line sums equal to 1; the latter consists of all unitary diagonal <i>n</i> × <i>n</i> 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.

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