Shedding light on classical shadows: learning photonic quantum states

TL;DR

This work addresses the challenge of efficiently learning properties of unknown quantum states beyond full tomography by extending classical shadows to photonic systems. It introduces a photonic shadow protocol that uses randomized passive linear-optical transformations and photon-number resolving measurements, rendering tomography within fixed photon-number sectors and enabling scalable prediction of linear and some nonlinear properties. The authors formalize a photonic shadow-norm to bound estimator variance and show favorable sample complexity for low-degree observables, then demonstrate the method experimentally on a $12$-mode, $6$-photon photonic processor, achieving accurate predictions for correlators and Lie-algebraic invariants from a single shadow of size $N$ on the order of $10^3$. This work broadens the practical applicability of shadow tomography to photonic platforms, with potential impacts on benchmarking, certification, and photonic machine learning, while highlighting computational considerations tied to permanents and future error-mitigation strategies.

Abstract

Efficient learning of quantum state properties is both a fundamental and practical problem in quantum information theory. Classical shadows have emerged as an efficient method for estimating properties of unknown quantum states, with rigorous statistical guarantees, by performing randomized measurement on a few number of copies. With the advent of photonic technologies, formulating efficient learning algorithms for such platforms comes out as a natural problem. Here, we introduce a classical shadow protocol for learning photonic quantum states via randomized passive linear optical transformations and photon-number measurement. We show that this scheme is efficient for a large class of observables of interest. We experimentally demonstrate our findings on a twelve-mode photonic integrated quantum processing unit. Our protocol allows for scalable learning of a wide range of photonic state properties and paves the way to applying the already rich variety of applications of classical shadows to photonic platforms.

Figures (4)

• Figure 1: Implementation pipeline of the classical shadow experimental protocol in linear optics. The quantum processing unit (QPU) Ascella maring_versatile_2024 is controlled with the Perceval Python library heurtel_perceval_2023 and the experimental results are post-processed with a Julia package github_loshadows. In the experimental setting, the input state $\rho$ is known. The state preparation involves evolving the $4$-mode $3$-photon input state $\ket{1,1,1,0}$ through $U_{\text{prep}}$. During the data-collection phase, the evolution is given by a randomly chosen $U_i$. At the end, the outcome $\boldsymbol{s}_i$ is obtained via on-chip pseudo-PNR measurement perfectly emulating PNR measurement. The inset QPU Ascella illustrates the configuration of the chip of the experiment reported in \ref{['fig:experiments']}. The Pseudo-PNR phase consists in plugging each output mode of the evolution step to the top mode of a 3-mode Fourier interferometer ($\mathcal{F}_3$)---maximising the scattering probability of the input state into different modes---whose output is measured with thresholds SNSPD (see SI for details).
• Figure 2: Experimental property estimation via classical shadows on Ascella. Experimental results of \ref{['exp:loc', 'exp:inv', 'exp:bpb']} performed on Ascella are documented in \ref{['fig:expTPC', 'fig:expINV', 'fig:expBPB']}, respectively. The classical shadow of the input is the same for all three experiments. It consists of $N=1100$ randomly sampled $U_i$; for each of which an average of 19 3-photon samples were obtained. \ref{['fig:expTPC']}: Comparison of the two-mode correlation matrices of the true one and that obtained from the shadow. (right) true correlation matrix (center) estimate obtained from the shadows (right) entry-wise absolute estimation error. \ref{['fig:expINV']}: Numerical values of the Lie-algebraic invariants. (left) evolution of the estimation via bootstrapping, (center and right) estimated spectrum of $\rho_T$ and $\Gamma(\rho)$. \ref{['fig:expBPB']}: Binned-distribution for all possible bipartitions of the modes. This way, each bipartition is defined by a subset of the modes and its complement. For instance, $\mathcal{K} = [1]$ represents the partition $(\{1\}, \{2, 3,4\})$. Certain bins are omitted due to symmetry. The horizontal axis corresponds to the possible occupations of the labelled bin.
• Figure 3: Effect of the Pseudo-PNR mitigation on the total variation distance (TVD) to the distribution obtained with perfect PNR measurement for the output distribution of $U_{\text{prep}}$. The TVD of non-mitigated distribution reaches a plateau at around $0.12$. On the contrary, the TVD of the mitigated distribution eventually attains zero provided enough samples are available.
• Figure 4: Pseudo photon-number resolving measurement. Illustration of the pseudo-PNR circuit and the effect of mitigating the output distribution on the total variation distance.

Theorems & Definitions (16)

• Theorem 1: Sample complexity of photonic classical shadows
• Theorem 2: Computational complexity of photonic classical shadows
• Theorem 3: Fock state classical shadows
• proof
• Proposition 1
• proof
• Lemma 1: Bound on estimators variance
• proof
• Lemma 2
• proof