Quantum Control In A Noisy Environment

Precise quantum control of quantum systems is considered as a prerequisite for many important quantum technology applications,such as quantum computing,quantum logic gate design,and quantum information processing.These advanced quantum technologies impose almost harsh requirements on precise quantum control.Generally,the quantum computer require a fidelity of 99.99%,which poses a huge challenge to current quantum control research.The difficulty lies in the fact that the real quantum system is an open quantum system,which will be affected by noise and the coupling between the system and the environment.This causes quantum decoherence and reduces the fidelity of manipulating qubits.Therefore,quantum error correction and the study of quantum dynamics in noisy environments are very meaningful,which are crucial to the large-scale application of quantum technologies represented by quantum computing.This paper studies quantum control problems in noisy environments and how to improve the fidelity of quantum control operations.First,based on the control theory of a single qubit and starting from the classical Hamiltonian of two-level system,the standard Bloch equation describing the motion of a qubit is obtained.The influence of environmental noise is introduced by the Kubo-Einstein fluctuation dissipation theorem,and the equation of motion of the complete quantum control problem in classical form is obtained.Next,based on full quantum-mechanical consideration,starting from the Hamiltonian of Caldeira-Leggett model,the quantum equation of motion is re-derived by the coherent-state path integral method.The Feymann-Vernon influence functional method is used to include the impact of the environment.An crucial effective-field reduction scheme and a pure state truncation approximation method are proposed.The quantum Langevin equation for the reduced density matrix is obtained by the effectivefield reduction scheme.In particular,we outlines how to eliminate the inevitable adiabatic trapping due to the environmental coupling.The results in the classical form are finally verified.Based on that a classical path on a two-dimensional sphere can completely describe the motion of a single qubit,this paper presents a variational optimization scheme of gradient descent algorithm that can be used to improve the fidelity of quantum control operations.Taking the ohmic bath as an example,the operation steps and specific numerical results of the optimization scheme are shown in detail in this paper.It is found that the main factor affecting the fidelity of quantum control at the high temperature limit is thermal fluctuations,while at the zero temperature limit,nonMarkovian quantum dissipation also has a significant contribution.The result of numerical simulations demonstrate the usefulness of the optimization scheme.In conclusion,this paper demonstrates a novel quantum control theoretical framework that can be used to analyze and minimize the effects of environmental coupling,providing theoretical directions for implementing quantum gates with highfidelity.The framework is universal and is expected to be experimentally verified.