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The isoholonomic problem in a homogeneous bundle is formulated and solvedexactly. The problem takes a form of a boundary value problem of a variationalequation. The solution is applied to the optimal control problem in holonomicquantum computer. We provide a prescription to construct an optimal controllerfor an arbitrary unitary gate and apply it to a $ k $-dimensional unitary gatewhich operates on an $ N $-dimensional Hilbert space with $ N \geq 2k $. Ourconstruction is applied to several important unitary gates such as the Hadamardgate, the CNOT gate, and the two-qubit discrete Fourier transformation gate.Controllers for these gates are explicitly constructed.Comment: 19 pages, no figures, LaTeX2

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