封闭和开放系统量子优化的最优控制
Optimal Control for Closed and Open System Quantum Optimization
摘要：
我们提供了一个严格的分析的量子最优控制问题的设置的线性组合$s(t)B+(1-s(t))C$的两个noncommuting哈密顿$B$和$C$。这包括量子退火（QA）和量子近似优化算法（QAOA）。目标是最小化最终"问题“哈密尔顿算子$C$的能量，对于时间依赖和有界控制调度$s(t)\in [0,1]$和$t\in \mc{I}:= [0,t_f]$。最近，在一个纯封闭系统的设置，这个问题的最佳解决方案是一个"bang-anneal-bang“的时间表，与量子退火的标准处方$s(0)=1$和$s(t_f)=0$相反，在$\mc{I}$的有限子区间，特别是$s(0)=0$和$s(t_f)=1$的$s(t)= 0$和$s(t)= 1$的刘海的特点。在这里，我们将这个结果扩展到开放系统的设置，其中系统是由密度矩阵，而不是一个纯态描述。这是QA和QAOA实验实现的自然环境。对于有限维环境，没有任何近似，我们确定充分的条件，确保无论是邦退火，退火邦，或邦退火邦计划是最佳的，并恢复$s(0)=0$和$s(t_f)=1$的最优性。然而，对于无限维的环境和绝热Redfield主方程描述的系统，我们不恢复Bang-type的最优解。事实上，我们只能确定条件下，$s(t_f)=1$，甚至这一结果是不能恢复的完全马尔可夫极限。分析，我们进行了完全的几何框架内庞特里亚金最大值原理，简化使用密度矩阵制剂相比，状态向量制剂。
英文摘要：
We provide a rigorous analysis of the quantum optimal control problem in the setting of a linear combination $s(t)B+(1-s(t))C$ of two noncommuting Hamiltonians $B$ and $C$. This includes both quantum annealing (QA) and the quantum approximate optimization algorithm (QAOA). The target is to minimize the energy of the final ``problem'' Hamiltonian $C$, for a time-dependent and bounded control schedule $s(t)\in [0,1]$ and $t\in \mc{I}:= [0,t_f]$. It was recently shown, in a purely closed system setting, that the optimal solution to this problem is a ``bang-anneal-bang'' schedule, with the bangs characterized by $s(t)= 0$ and $s(t)= 1$ in finite subintervals of $\mc{I}$, in particular $s(0)=0$ and $s(t_f)=1$, in contrast to the standard prescription $s(0)=1$ and $s(t_f)=0$ of quantum annealing. Here we extend this result to the open system setting, where the system is described by a density matrix rather than a pure state. This is the natural setting for experimental realizations of QA and QAOA. For finite-dimensional environments and without any approximations we identify sufficient conditions ensuring that either the bang-anneal, anneal-bang, or bang-anneal-bang schedules are optimal, and recover the optimality of $s(0)=0$ and $s(t_f)=1$. However, for infinite-dimensional environments and a system described by an adiabatic Redfield master equation we do not recover the bang-type optimal solution. In fact we can only identify conditions under which $s(t_f)=1$, and even this result is not recovered in the fully Markovian limit. The analysis, which we carry out entirely within the geometric framework of Pontryagin Maximum Principle, simplifies using the density matrix formulation compared to the state vector formulation.
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