A learning theory for quantum photonic processors and beyond

TL;DR

The work establishes a statistical-learning theory for continuous-variable quantum circuits, showing that learning CV states, measurements, and channels implemented by Gaussian and non-Gaussian (GG) circuits is efficient in the number of modes $n$ and largely independent of circuit depth. By employing fat-shattering dimensions, pseudo-dimensions, and covering numbers, the authors derive explicit sample-complexity bounds for Gaussian, GP, and GG circuit classes across state, measurement, and channel learning, as well as for discrimination and synthesis tasks with polynomial-encoding inputs. A key insight is that Gaussian and GG circuits admit polynomial-in-$n$ sample complexity (with GG requiring energy-like constraints), while depth does not generally degrade learnability in CV settings; the GG case remains feasible under fixed non-Gaussian coefficients. The results provide a rigorous foundation for efficiently training photonic CV processors and guide future work on encodings, non-Gaussian resources, and circuit compilation in CV quantum information processing.

Abstract

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.

Figures (2)

• Figure 1: Depiction of the simplest learning problem studied for CV architectures. We aim to reproduce the statistics of an unknown state $\rho$with respect to random couples of CV channels and measurements $(\Phi_i, M_i)$. For each $i$, the state is passed through channel $\Phi_i$ and measured with a binary measurement $M_i$ obtaining outcome $b_{i}$. Then the learner produces a hypothesis $\sigma$ by tuning the parameters of a CV circuit, e.g., a photonic chip, and obtains its output statistics on the channels and measurements of the training set. A suitable empirical loss evaluates how well $\sigma$ approximates $\rho$ on the training set. This can be used to optimize $\sigma$ iteratively.
• Figure 2: Depiction of the discrimination task learning problem studied for CV architectures. We receive random samples $(\rho_x,M_x,b_x)$ from a class of input states $\{\rho_x\}$ to be discriminated with a fixed measurement device $\{M_x\}$. Then the learner produces a hypothesis channel $\Phi$ that processes the input $\rho_x$ before measurement, and obtains its output statistics. A suitable empirical loss evaluates how well $\Phi$ is able to discriminate the states in the training set. This can be used to optimize $\Phi$ iteratively.

Theorems & Definitions (30)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• Definition 3
• Theorem 3
• Corollary 1
• proof
• Theorem 4
• Definition 4