Quantum gates are realized as differential operators on holomorphic functions that preserve the qubit subspace and act as canonical transformations on a toroidal geometry.
For example: •SWAPacts as a permutation of toroidal coor- dinates: (ϕ aj , ϕbj , ϕak , ϕbk)7→(ϕ ak , ϕbk , ϕaj , ϕbj), which is a global isometry ofT 2N
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Geometry of Quantum Logic Gates
Quantum gates are realized as differential operators on holomorphic functions that preserve the qubit subspace and act as canonical transformations on a toroidal geometry.