# Techniques: Shedding light on classical shadows: learning photonic quantum states

## What the paper contributes (one paragraph)

The paper introduces a classical shadow protocol for learning properties of unknown photonic quantum states using only passive linear optics and photon-number measurement. Random Haar linear-optical interferometers are applied to the input state, followed by photon-number-resolving detection; the resulting list of unitary-outcome pairs is post-processed to estimate expectation values of observables. Using representation theory of the unitary group, the authors derive a closed form for the measurement channel and a photonic shadow-norm, giving rigorous sample-complexity and time-complexity bounds that are efficient for low-degree observables. They contribute an exact post-processing algorithm based on matrix permanents to keep classical processing tractable, and a pseudo-PNR scheme that emulates photon-number resolution with threshold detectors plus a debiasing step. The protocol is demonstrated on twelve-mode and twenty-four-mode integrated photonic processors across five applications, including correlation functions, Lie-algebraic invariants, Bose-Hubbard ground energy, and learning Boson Sampling states.

## Techniques offered

- **Photonic classical shadows (randomized-measurement shadow tomography for linear optics)** - estimates physical properties of an unknown photonic state from randomized passive linear-optical evolutions and photon-number measurement, supplying a sample-efficient characterisation primitive for fixed-photon-number photonic platforms.
  - guarantee: statistical bound; unbiased expectation-value estimation in the visible space with a sample-complexity bound controlled by the photonic shadow-norm.
  - quote: "Here, we introduce a practical classical shadow protocol for learning photonic quantum states via randomized passive linear optical transformations and photon-number measurement."

- **Photonic shadow-norm sample-complexity guarantee** - bounds the number of samples needed to estimate many observables to additive precision, supplying a rigorous performance certificate that scales with observable degree.
  - guarantee: statistical bound; a collection of T linear functions estimated to additive precision epsilon with shadow size O(max shadow-norm squared times log T over epsilon squared) at constant success probability.
  - quote: "We provide rigorous theoretical guarantees showing that our scheme is sample- and time-efficient for measuring physical observables of interest."

- **Exact permanent-based post-processing algorithm for expectation values** - computes expectation values of degree-d observables from each shadow pair via a polynomial number of small matrix permanents (Ryser's algorithm), supplying time-efficient classical post-processing that avoids full-permanent and channel-matrix costs.
  - guarantee: static computational bound; exact expectation values of constant-degree observables computed in polynomial time, matching the sample-efficient regime.
  - quote: "Rather, we introduce in the \arxiv{\autoref{app:exactTechnique}}{Supplementary Information} a novel technique for computing $\expval{O}_{\varphi_m(U)\ket{\bm s}}$ from the pair $(U, \bm s)$ in time $\smash{\bigo{m^{\deg(O)}}}$, allowing for efficient post-processing."

- **Pseudo-PNR measurement with bias mitigation** - emulates photon-number-resolving detection using threshold detectors in a larger Fourier interferometer, with a debiasing step that removes the resulting distribution bias, supplying photon-number resolution and an error-mitigation correction on hardware without true PNR detectors.
  - guarantee: statistical bound; the mitigated total variation distance to the true PNR distribution converges to zero given enough samples, at a sample overhead.
  - quote: "In the experiments, we achieve photon-number resolving measurements via pseudo-PNR data processing (see \arxiv{\autoref{sec:ppnr}}{Supplementary Information})."

## Where it could apply

- Target primitive(s): sampling (boson sampling state learning), Trotterised/Hamiltonian simulation (Bose-Hubbard ground-energy estimation), expectation-value estimation, state characterisation and certification.
- Target application group(s): platforms_tooling, chemistry_materials (Hubbard-type Hamiltonians), quantum_ml.
- Code family / hardware assumptions: photonic (passive linear optics, integrated Mach-Zehnder interferometers, Fock-state encoding, PNR or threshold/pseudo-PNR detection); fixed-photon-number subspaces; applies to dual-rail qubit encodings.

## Caveats

PNR detection restricts tomography to a single fixed photon-number subspace, so states that are superpositions of different photon numbers are only partially characterised (only the maximal-photon-number block). The sample and time bounds are efficient only for constant-degree observables and are pessimistic worst-case bounds. Accuracy in experiments is limited by partial photon distinguishability, chip characterisation, multi-photon emission, and input-state generation rate. The pseudo-PNR debiasing requires a sample overhead to reach zero bias.