Adaptive protocols enable the construction of more efficient state preparation circuits in variational quantum algorithms (VQAs) by utilizing data obtained from the quantum processor during the execution of the algorithm. This idea originated with Adaptive Derivative-Assembled Problem-Tailored variational quantum eigensolver (ADAPT-VQE), an algorithm that iteratively grows the state preparation circuit operator by operator, with each new operator accompanied by a new variational parameter, and where all parameters acquired thus far are optimized in each iteration. In ADAPT-VQE and other adaptive VQAs that followed it, it has been shown that initializing parameters to their optimal values from the previous iteration speeds up convergence and avoids shallow local traps in the parameter landscape. However, no other data from the optimization performed at one iteration is carried over to the next. In this work, we propose an improved quasi-Newton optimization protocol specifically tailored to adaptive VQAs. The distinctive feature in our proposal is that approximate second derivatives of the cost function are recycled across iterations in addition to optimal parameter values. We implement a quasi-Newton optimizer where an approximation to the inverse Hessian matrix is continuously built and grown across the iterations of an adaptive VQA. The resulting algorithm has the flavor of a continuous optimization where the dimension of the search space is augmented when the gradient norm falls below a given threshold. We show that this inter-optimization exchange of second-order information leads the approximate Hessian in the state of the optimizer to be consistently closer to the exact Hessian. As a result, our method achieves a superlinear convergence rate even in situations where the typical implementation of a quasi-Newton optimizer converges only linearly. Our protocol decreases the measurement costs in implementing adaptive VQAs on quantum hardware as well as the runtime of their classical simulation.

中文翻译：

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通过在自适应变分量子算法中回收 Hessian 矩阵来降低测量成本


自适应协议通过在算法执行期间利用从量子处理器获得的数据，在变分量子算法 （VQA） 中构建更高效的状态准备电路。这个想法起源于自适应导数组装问题定制的变分量子特征求解器 （ADAPT-VQE），这是一种逐个运算符迭代增长状态准备电路运算符的算法，每个新运算符都伴随着一个新的变分参数，并且迄今为止获得的所有参数都在每次迭代中得到优化。在 ADAPT-VQE 和随后的其他自适应 VQA 中，已经表明将参数初始化为上一次迭代的最佳值可以加快收敛速度，并避免参数景观中的浅层局部陷阱。但是，在一次迭代中执行的优化中的其他数据不会转移到下一次迭代中。在这项工作中，我们提出了一种改进的准牛顿优化协议，专门为自适应 VQA 量身定制。我们提案的显着特点是，除了最佳参数值外，成本函数的近似二阶导数还可以在迭代中循环使用。我们实现了一个准牛顿优化器，其中逆 Hessian 矩阵的近似值在自适应 VQA 的迭代中不断构建和增长。生成的算法具有持续优化的风格，其中当梯度范数低于给定阈值时，搜索空间的维度会增强。我们表明，这种二阶信息的相互优化交换导致优化器状态下的近似 Hessian 矩阵始终更接近精确的 Hessian。 因此，即使在准牛顿优化器的典型实现仅线性收敛的情况下，我们的方法也能实现超线性收敛率。我们的协议降低了在量子硬件上实施自适应 VQA 的测量成本，以及其经典模拟的运行时间。