精确数值优化的离散伴随及其在量子控制中的应用
Discrete Adjoints for Accurate Numerical Optimization with Application to Quantum Control
摘要：
本文研究了在封闭量子系统中实现逻辑门的最优控制问题。量子态由薛定谔方程控制，我们将其表示为一个与时间相关的哈密顿系统，其中包含态矢量的实部和虚部。该系统离散的Stormer-Verlet计划，这是一个辛分区龙格库塔方法。我们的主要理论贡献是推导出一个兼容的时间离散化的伴随状态方程，这样的离散目标函数的梯度可以精确计算，在计算成本解决两个薛定谔系统，独立的控制函数中的参数的数量。还介绍了基于B样条与内置载波的控制函数的参数化。载波用于指定控制函数的频谱，而B样条函数指定其包络和相位。这种方法允许的控制参数的数量是独立的，并显着小于，积分薛定谔方程的时间步长的数量。我们认为哈密顿模型的超导多能级qudit的动力学和目前的数值例子，所提出的技术可以如何结合内点L-BFGS算法从IPOPT包实现量子门。在一组测试用例中，所提出的算法与QuTiP/pulse_optim和Grape-Tensorflow相比毫不逊色。
英文摘要：
This paper considers the optimal control problem for realizing logical gates in a closed quantum system. The quantum state is governed by Schrodinger's equation, which we formulate as a time-dependent Hamiltonian system in terms of the real and imaginary parts of the state vector. The system is discretized with the Stormer-Verlet scheme, which is a symplectic partitioned Runge-Kutta method. Our main theoretical contribution is the derivation of a compatible time-discretization of the adjoint state equation, such that the gradient of the discrete objective function can be calculated exactly, at a computational cost of solving two Schrödinger systems, independently of the number of parameters in the control functions. A parameterization of the control functions based on B-splines with built-in carrier waves is also introduced. The carrier waves are used to specify the frequency spectra of the control functions, while the B-spline functions specify their envelope and phase. This approach allows the number of control parameters to be independent of, and significantly smaller than, the number of time steps for integrating Schrodinger's equation. We consider Hamiltonians that model the dynamics of a superconducting multi-level qudit and present numerical examples of how the proposed technique can be combined with the interior point L-BFGS algorithm from the IPOPT package for realizing quantum gates. In a set of test cases, the proposed algorithm is shown to compare favorably with QuTiP/pulse_optim and Grape-Tensorflow.
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