线性系统和最小二乘的量子梯度下降
Quantum gradient descent for linear systems and least squares
摘要：
量子机器学习和优化是令人兴奋的新领域，由Harrow，Hassidim和Lloyd的突破性量子算法提出，用于求解线性方程组。{经典}线性系统求解器的效用扩展到线性代数之外，因为它们可以利用迭代方法（如梯度下降）来解决优化问题。在这项工作中，我们提供了第一个量子方法进行梯度下降时，梯度是一个仿射函数。执行梯度下降的$τ$步骤需要用于加权最小二乘问题的时间$O(τC_S)$，其中$C_S$是量子地执行梯度下降的一个步骤的成本，其有时可以比经典成本小得多。我们说明我们的方法提供两个应用程序：第一，解决半正定线性系统，第二，用于执行随机梯度下降的加权最小二乘问题，减少量子内存的要求。我们还在QRAM数据结构模型中提供了一个量子线性系统求解器，该模型为大型矩阵家族提供了显着的成本节省。
英文摘要：
Quantum machine learning and optimization are exciting new areas that have been brought forward by the breakthrough quantum algorithm of Harrow, Hassidim and Lloyd for solving systems of linear equations. The utility of {classical} linear system solvers extends beyond linear algebra as they can be leveraged to solve optimization problems using iterative methods like gradient descent. In this work, we provide the first quantum method for performing gradient descent when the gradient is an affine function. Performing $τ$ steps of the gradient descent requires time $O(τC_S)$ for weighted least squares problems, where $C_S$ is the cost of performing one step of the gradient descent quantumly, which at times can be considerably smaller than the classical cost. We illustrate our method by providing two applications: first, for solving positive semidefinite linear systems, and, second, for performing stochastic gradient descent for the weighted least squares problem with reduced quantum memory requirements. We also provide a quantum linear system solver in the QRAM data structure model that provides significant savings in cost for large families of matrices.
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