Matrix Calculus for Classical and Quantum Circuits
Alexis De Vos, Stijn De Baerdemacker · ACM Journal on Emerging Technologies in Computing Systems · 2014
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 2 w . 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 × n matrices with all line sums equal to 1; the latter consists of all unitary diagonal n × 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.