We consider the tasks of learning quantum states, measurements and channels generated by continuous-variable (CV) quantum circuits. This family of circuits is suited to describe optical quantum technologies and in particular it includes state-of-the-art photonic processors capable of showing quantum advantage. We define classes of functions that map classical variables, encoded into the CV circuit parameters, to outcome probabilities evaluated on those circuits. We then establish efficient learnability guarantees for such classes, by computing bounds on their pseudo-dimension or covering numbers, showing that CV quantum circuits can be learned with a sample complexity that scales polynomially with the circuit's size, i.e., the number of modes. Our results establish that CV circuits can be trained efficiently using a number of training samples that, unlike their finite-dimensional counterpart, does not scale with the circuit depth.

翻译：我们考虑的是学习由连续可变量子电路产生的量子状态、测量和信道的任务。这种电路组适合于描述光量技术,特别是它包括能够显示量子优势的最先进的光子处理器。我们定义了将古典变量编码为CV电路参数的功能类别,以及这些电路上评估的结果概率。然后,我们通过计算其伪分解或覆盖数字的界限,为这些类别建立高效的学习保障,表明CV量子电路可以以样本复杂性来学习,这种复杂性与电路的大小(即模式数量)成倍地成比例。我们的结果证明,CV电路可以有效地使用一些培训样品来培训,这些样品不同于其多维对等,不与电路深度成比例。