# Techniques: Boson sampling beyond the dilute regime: second moments and anti-concentration

## What the paper contributes (one paragraph)

The paper supplies a representation-theoretic toolkit for computing exact second moments of particle-number-preserving observables under Haar-random passive linear-optical (boson sampling) circuits. It shows the second moment of an expectation value over random interferometers decomposes into Hilbert-Schmidt norms of projections onto irreducible components of the operator space, and gives a recursive sl_2 ladder method that evaluates these projection norms in closed form without Clebsch-Gordan coefficients or the hiding property. Applying this, it derives a closed-form expression for the normalized average outcome collision probability P_2(m,n), valid for arbitrary mode and photon numbers, and uses its asymptotic scaling to prove anti-concentration of Fock-state boson sampling beyond the dilute regime, including the experimentally relevant saturated (linear) regime where photon collisions are common. This resolves an open anti-concentration conjecture and strengthens hardness evidence; the same collision probability also characterises the ideal score in linear cross-entropy benchmarking.

## Techniques offered

- **Second-moment characterisation of linear-optical observables (representation-theoretic twirl)** - closed-form second moments of particle-number-preserving bosonic observables over Haar-random interferometers, expressed as Hilbert-Schmidt norms of irrep projections, usable for variance and concentration analysis of boson sampling estimators without the hiding assumption.
  - guarantee: exact analytical second-moment identity (statistical second-moment bound) valid across all mode/photon regimes
  - quote: "In this work, we leverage  representation-theoretic tools to address this gap, obtaining closed-form expressions for second moments of generic particle-number-preserving bosonic observables."

- **Recursive sl_2 projection method for irrep norms** - a basis-independent recursive procedure using raising and lowering maps to isolate irreducible components and evaluate their Hilbert-Schmidt projection norms in closed form, avoiding explicit Clebsch-Gordan coefficients or projectors.
  - guarantee: static proof of a recursive closed-form expression for each irrep projection norm, efficiently computable for a broad class of observables including Fock states
  - quote: "To evaluate these quantities, we introduce in \autoref{th:iterative_removal_main} a recursive projection method based on the underlying $\mathfrak{sl}_2$ structure, which allows us to isolate irreducible contributions without constructing explicit projectors."

- **Anti-concentration proof via closed-form collision probability (linear cross-entropy benchmarking quantity)** - a closed-form normalized average outcome collision probability P_2(m,n) with proven asymptotic scaling that establishes anti-concentration beyond the dilute regime and supplies the ideal-distribution score for linear cross-entropy benchmarking of boson samplers.
  - guarantee: proves the anti-concentration property (lower bound on the fraction of non-small outcome probabilities via the Paley-Zygmund inequality) in the saturated regime, resolving the conjecture of Bouland et al.
  - quote: "Moreover, this quantity is also of independent interest, as it naturally arises in linear cross-entropy benchmarking, where it characterizes the score of the ideal distribution."

## Where it could apply

- Target primitive(s): sampling (boson sampling, Fock-state and extendable to Gaussian-input); random-circuit expectation-value estimation under passive linear optics
- Target application group(s): platforms_tooling (benchmarking, certification, randomised benchmarking, classical shadows for linear optics), quantum_advantage / complexity-theoretic hardness analysis
- Code family / hardware assumptions: photonic (passive linear-optical interferometers on m modes, U(m) Haar-random unitaries); no error-correcting code assumed

## Caveats

This is primarily a theoretical statistics paper. Its suppliable outputs are analytical characterisation, benchmarking, and anti-concentration tools rather than a runnable protocol, decoder, or code. Results assume Haar-random interferometers and (for the closed forms) fixed-photon-number states and observables; relaxing the fixed-particle-number assumption, handling photon loss and partial distinguishability, and extending to higher moments or Gaussian inputs are left as future work. The benchmarking relevance is via the ideal-distribution collision probability, not a turnkey verification protocol.