Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation

Details Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation

2004-06-07

arxiv.org/abs/quant-ph/0406038v1

Journal of Mathematical Physics 46 (2005) 022101, 1-15

Physics

Quantum Physics

19 pages, no figures, LaTeX2e

Scientific paper

10.1063/1.1835545

The isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a $ k $-dimensional unitary gate which operates on an $ N $-dimensional Hilbert space with $ N \geq 2k $. Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the two-qubit discrete Fourier transformation gate. Controllers for these gates are explicitly constructed.

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