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 show 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：本文研究连续变量量子电路生成的量子态、测量和信道的学习任务。这类电路适用于描述光学量子技术，特别包含了能够展现量子优势的先进光子处理器。我们定义了一系列函数类别，这些函数将编码到连续变量电路参数中的经典变量映射至电路输出概率。通过计算这些函数类的伪维度或覆盖数界限，我们建立了相应类别的高效可学习性保证，证明连续变量量子电路的学习样本复杂度随电路规模（即模式数量）呈多项式增长。我们的结果表明，连续变量电路能够以与电路深度无关的训练样本量实现高效训练，这一特性与有限维对应系统形成鲜明对比。