Quantum tomography involves obtaining a full classical description of a prepared quantum state from experimental results. We propose a Langevin sampler for quantum tomography, that relies on a new formulation of Bayesian quantum tomography exploiting the Burer-Monteiro factorization of Hermitian positive-semidefinite matrices. If the rank of the target density matrix is known, this formulation allows us to define a posterior distribution that is only supported on matrices whose rank is upper-bounded by the rank of the target density matrix. Conversely, if the target rank is unknown, any upper bound on the rank can be used by our algorithm, and the rank of the resulting posterior mean estimator is further reduced by the use of a low-rank promoting prior density. This prior density is a complex extension of the one proposed in (Annales de l'Institut Henri Poincare Probability and Statistics, 56(2):1465-1483, 2020). We derive a PAC-Bayesian bound on our proposed estimator that matches the best bounds available in the literature, and we show numerically that it leads to strong scalability improvements compared to existing techniques when the rank of the density matrix is known to be small.

翻译：量子层析成像旨在通过实验结果获取已制备量子态的完整经典描述。我们提出了一种用于量子层析成像的朗之万采样器，其基于一种新的贝叶斯量子层析公式，该公式利用了埃尔米特半正定矩阵的布勒-蒙特罗分解。若目标密度矩阵的秩已知，该公式允许我们定义一个后验分布，其仅支持秩不超过目标密度矩阵秩的矩阵。反之，若目标秩未知，我们的算法可采用任意秩上界，并且通过使用一种促进低秩的先验密度，所得后验均值估计量的秩会进一步降低。该先验密度是（Annales de l'Institut Henri Poincare Probability and Statistics, 56(2):1465-1483, 2020）中所提出先验的复数推广。我们推导了所提估计量的一个PAC-贝叶斯界，该界与文献中可获得的最佳界相匹配，并且数值实验表明，当已知密度矩阵的秩较小时，与现有技术相比，该方法能带来显著的可扩展性提升。