Geometry, optimal control and quantum computing.

摘要

Quantum computation promises solution to problems that are hard to solve by classical computers. The efficient construction of quantum circuits that can solve interesting tasks is a fundamental challenge in the field. Such efficient construction also reduces decoherence losses in physical implementations of quantum algorithms by reducing interaction time with the environment. Therefore, finding time-optimal ways to synthesize unitary transformations from available physical resources is a problem of both fundamental and practical interest in quantum information processing. In this thesis, we study these problems in general mathematical frame as well as in some concrete real physical settings. We give a complete characterization of all the unitary transformations that can be synthesized in a given time for a two-qubit system in presence of general time varying coupling tensor, assuming that the local unitary transformation on two qubits can be performed arbitrarily fast (on a time scale governed by the strength of couplings). A generalization of this result on general Lie group is also presented. We then give the time optimal ways for coherence transfer on three linearly coupled spin chain, and an efficient way of constructing a CNOT gate between two indirectly coupled spins.