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：我们研究了由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，尤其包含了能够展示量子优势的先进光子处理器。我们定义了函数类别，这些函数将编码到CV电路参数中的经典变量映射至这些电路上评估的结果概率。随后，通过计算其伪维度或覆盖数的界限，我们为此类函数建立了高效可学习性保证，表明CV量子电路可在样本复杂度随电路规模（即模式数量）多项式增长的条件下被学习。我们的结果表明，CV电路能够使用与电路深度无关的训练样本数量进行高效训练，这一点与其有限维对应物不同。