We propose a distributed Quantum State Tomography (QST) protocol, named Local Stochastic Factored Gradient Descent (Local SFGD), to learn the low-rank factor of a density matrix over a set of local machines. QST is the canonical procedure to characterize the state of a quantum system, which we formulate as a stochastic nonconvex smooth optimization problem. Physically, the estimation of a low-rank density matrix helps characterizing the amount of noise introduced by quantum computation. Theoretically, we prove the local convergence of Local SFGD for a general class of restricted strongly convex/smooth loss functions, i.e., Local SFGD converges locally to a small neighborhood of the global optimum at a linear rate with a constant step size, while it locally converges exactly at a sub-linear rate with diminishing step sizes. With a proper initialization, local convergence results imply global convergence. We validate our theoretical findings with numerical simulations of QST on the Greenberger-Horne-Zeilinger (GHZ) state.

翻译：我们提出一个分布式量子国家地形学(QST)协议,名为本地软质因子渐变基因组(SFGD),以了解一组本地机器中密度矩阵的低位因子。QST是量子系统状态特征的典型化程序,我们把它发展成一个随机非孔状平稳优化问题。从物理上看,对低位密度矩阵的估计有助于量化计算带来的噪音量的特征化。理论上,我们证明本地软质质分数组(SFGD)对于一个限制型强 convex/smooth损失功能的一般类别(即当地软质分数组)具有本地趋同性,即:本地软质分解组合会以不变的步数速度聚集到一个全球最优化的小区域,同时以不变的步数速尺大小,而本地则完全在亚线速率上趋同。随着适当的初始化,本地趋同结果就意味着全球趋同。我们用格林伯格-霍恩-泽林格(GHGHGZ)州QST的数字模拟来验证我们的理论结论结论。