Quantum state tomography (QST), the task of estimating an unknown quantum state given measurement outcomes, is essential to building reliable quantum computing devices. Whereas computing the maximum-likelihood (ML) estimate corresponds to solving a finite-sum convex optimization problem, the objective function is not smooth nor Lipschitz, so most existing convex optimization methods lack sample complexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithm should be highly scalable with respect to the dimension and sample size, just like stochastic gradient descent. In this paper, we propose a stochastic first-order algorithm that computes an $\varepsilon$-approximate ML estimate in $O( ( D \log D ) / \varepsilon ^ 2 )$ iterations with $O( D^3 )$ per-iteration time complexity, where $D$ denotes the dimension of the unknown quantum state and $\varepsilon$ denotes the optimization error. Our algorithm is an extension of Soft-Bayes to the quantum setup.
翻译:量子状态断层(QST)是估算一个未知量状态的测量结果的任务,对于建立可靠的量子计算设备至关重要。计算最大似值(ML)的估算相当于解决一个有限和共浮优化问题,但目标功能并不平滑,而利普西茨则不平滑,因此,大多数现有的convex优化方法都缺乏样本复杂性保障;此外,样本大小和尺寸随着QST实验中的量子数量而成倍增长,因此,理想的算法在尺寸和样本大小方面应具有高度可伸缩性,就像随机梯度梯度下降一样。在本文中,我们提议一种随机一级算法,在$O(D\log D) / \ varepsilon = 2) 中计算一个近于ML的估算值;此外,样本大小和尺寸随着QST实验中的量位数(D3) 美元(美元) / 美元(美元) / 时间复杂性而成倍增长,因此,理想的算法在尺寸和 $\ vareplon 表示未知梯位值的高度误差。 我们的算算算算法是SQ- basqasion- 的扩展的延伸。