Technical Summary: Collision-aware Bayesian Mitigation of Photon Loss in Linear Optical Quantum Circuits

1. Problem Statement

Photon loss is the dominant noise source in linear optical quantum circuits (LOQC), particularly in Boson Sampling experiments. Unlike dephasing, loss causes photons to disappear entirely, transforming ideal $n$-photon output patterns into outcomes with fewer photons. This distortion renders the ideal lossless distribution exponentially rare as the photon number grows, making standard postselection (discarding lossy events) inefficient at realistic loss rates.

Existing classical mitigation strategies, specifically the "recycling" framework proposed by Mills and Mezher (implemented in Quandela's photon_recycling), attempt to recover lossless probabilities by reweighting lossy samples. However, this approach suffers from two critical limitations:

1. Statistical Bias: The recycling estimators are inherently biased due to approximations in their reconstruction (e.g., replacing interference terms with expectations and heuristic exponential extrapolation). This bias does not vanish even as the sample count $N \to \infty$.
2. Structural Restriction: The recycling kernel assumes the "no-collision" (NC) sector, where each output mode contains at most one photon. Real Boson Sampling outputs frequently exhibit "collision" states (multiple photons in a single mode) due to Hong–Ou–Mandel bunching. Recycling cannot process these samples; discarding them sacrifices significant data and information, particularly in regimes where collision patterns constitute a large fraction of the lossless output mass (e.g., 71% at $n=4, m=10$).

2. Methodology

The authors propose a collision-aware Bayesian framework that treats raw lossy detector counts as Multinomial observations under a per-photon Bernoulli loss model.

Forward Model

The method models photon loss as an independent per-photon Bernoulli channel with survival probability $1-\eta$.

• No-Collision Regime: The model recovers the recycling kernel as a boundary case where input and output vectors are binary ($s, y \in \{0, 1\}^m$).
• Collision-Aware Regime: The framework extends to the full space of weak compositions ($s \in \mathbb{Z}_{\ge 0}^m$ with $\sum s_i = n$). It incorporates a per-mode binomial loss factor, $\binom{s_i}{y_i}$, which accounts for the exchangeability of photons within a mode. The likelihood for observing outcome $y$ given ideal distribution $p$ and loss rate $\eta$ is:
  $$\pi_y(p, \eta) = \eta^k (1-\eta)^{n-k} \sum_{s \in G_{\text{coll}}(y)} \left[ \prod_{i=1}^m \binom{s_i}{y_i} \right] p(s)$$
  where $k$ is the number of lost photons and $G_{\text{coll}}(y)$ is the set of valid lossless states that could produce $y$.

Inference

Instead of deriving point estimators via reweighting, the authors infer the joint posterior of the ideal distribution $p$ and the loss rate $\eta$ using Bayes' theorem:
$$\Pr(p, \eta \mid c) \propto \prod_{y \in Y} \pi_y(p, \eta)^{c(y)} \cdot \Pr(p) \cdot \Pr(\eta)$$

• Priors: Symmetric Dirichlet prior for $p$ ($\alpha=1$) and Beta prior for $\eta$ ($\beta=1$).
• Sampling: The posterior is sampled using the No-U-Turn Sampler (NUTS) on a sparse incidence representation of the kernel. This representation reduces the computational cost of the inner loop from $O(K \cdot |Y|)$ to $O(E + K)$, making inference tractable for pattern spaces up to $K \approx 8855$.

3. Key Contributions

The paper makes three primary contributions:

1. Reduction of Reconstruction Error: By fitting the forward model directly via Multinomial likelihood, the Bayesian approach avoids the specific reweighting approximations and interference-term substitutions inherent in recycling estimators. This results in lower Kullback-Leibler (KL) divergence and Mean Squared Error (MSE) compared to both recycling and postselection.
2. Collision-Regime Support: The framework naturally handles collision states, which recycling methods cannot process. In collision-rich regimes, the Bayesian posterior recovers loss rates and collision-pattern probabilities that recycling cannot estimate, while maintaining lower reconstruction error.
3. Actionable Uncertainty Quantification: Unlike point-estimator baselines, the joint posterior provides credible intervals for both the reconstructed distribution $p$ and the loss rate $\eta$. This allows for retrospective coverage analysis, revealing how well the intervals capture the true distribution under various noise conditions.

4. Experimental Results

Experiments were conducted using data from the independent Perceval simulator (SLOS engine) at scales matching prior work: $n=4$ photons in $m=20$ modes (no-collision headline) and $n=4$ in $m=10$ modes (collision-aware).

• No-Collision Benchmark ($n=4, m=20$):

  • The Bayesian posterior achieved the lowest KL divergence at every tested loss rate ($\eta \in \{0.3, 0.7, 0.9\}$) and indistinguishability level ($V \in \{1.0, 0.8\}$).
  • Recycling estimators exhibited a "reconstruction-error plateau" (e.g., KL $\approx 0.31-0.33$ nats at $\eta=0.3$) that did not decrease with increasing sample count $N$, confirming their inherent bias.
  • The Bayesian error continued to decrease as $N$ increased (e.g., from 0.348 at $N=10^4$ to 0.068 at $N=2\times 10^5$), widening the gap over recycling.
  • Postselection performed poorly at high loss due to the collapse of the effective sample size.

• Collision-Aware Benchmark ($n=4, m=10$):

  • Recycling methods were structurally inapplicable (requiring the discarding of 71% of lossless output mass).
  • The Bayesian posterior successfully recovered the loss rate exactly ($\hat{\eta} = \eta$) and achieved significantly lower KL divergence than postselection at high loss.
  • At low loss, the Bayesian method was comparable to postselection, with slight variations depending on indistinguishability.

• Uncertainty Quantification:

  • The 95% credible intervals showed empirical coverage of 0.88–0.96 on the no-collision headline.
  • In regimes with partial photon distinguishability ($V < 1$), the intervals correctly covered the partially distinguishable distribution $p_V$ (the actual data-generating distribution) but showed undercoverage for the ideal lossless distribution $p^*$, reflecting the shift in the target rather than a failure of the estimator.

5. Significance and Claims

The paper claims that this Bayesian framework offers a finite-sample advantage over existing classical mitigation techniques for linear optical quantum circuits.

• Superiority over Recycling: It eliminates the structural bias caused by interference-term approximations and the data loss caused by discarding collision events.
• Scalability: While exact NUTS sampling faces computational limits at very large $N$ or $K$ (requiring future approximate inference methods like variational inference), the current implementation is tractable for the scales of current experimental demonstrations ($n=4$).
• Robustness: The method degrades gracefully under multiphoton emission and correctly identifies the target distribution shift under partial distinguishability.
• Limitations: The authors explicitly state that the results demonstrate finite-sample improvements, not asymptotic unbiasedness (which would require a population-limit analysis). The method currently assumes uniform per-photon loss and does not model mode-dependent loss or partial distinguishability within the likelihood itself (though it handles the resulting data shifts).

The work establishes a new baseline for photon-loss mitigation that is valid in the collision regime and provides rigorous uncertainty quantification, addressing key gaps in the current state-of-the-art recycling methods.