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

Abstract: Boson sampling is a leading candidate for demonstrating quantum advantage in photonic systems. Despite significant experimental and theoretical progress, a characterization of its output statistics remains incomplete. This is especially true beyond the dilute regime, where photon collisions and bunching become significant. The associated saturated regime, characterized by mode number growing linearly with photon number, or more generally sub-quadratically, is precisely the regime of greatest experimental interest. As a consequence, anti-concentration of the output distribution--a key ingredient in hardness arguments--remains poorly understood in boson sampling. 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. We express these quantities in terms of Hilbert-Schmidt norms of projections onto irreducible components of the operator space and show that these projection norms admit compact analytical expressions by exploiting the underlying symmetry structure. Focusing on Fock state output probabilities, we further establish anti-concentration beyond the dilute regime. Together with recent complexity-theoretic results, our findings strengthen hardness guarantees for boson sampling in experimentally interesting settings.

• The paper introduces a representation-theoretic framework to compute second moments for boson sampling in both dilute and saturated regimes.
• It provides exact closed-form expressions for collision probabilities and anti-concentration using group-theoretic decompositions.
• The study reinforces boson sampling's complexity-theoretic hardness by demonstrating robust anti-concentration even with high photon collisions.

Boson Sampling Beyond the Dilute Regime: Second Moments and Anti-Concentration

Introduction and Motivation

Boson sampling is a paradigmatic model for quantum advantage demonstrations in photonic systems, relying on the computational intractability of simulating the output distributions of multi-photon interference in large random linear optical networks. Traditional complexity-theoretic arguments have focused on the so-called dilute regime, where the number of modes $m$ scales at least quadratically with the number of photons $n$, yielding negligible photon collisions. However, experimental advances are pushing into the saturated regime ($m=O(n)$), where high rates of photon collisions and bunching fundamentally alter the output statistics. Prior analyses, especially those dealing with anti-concentration—a property crucial for complexity-theoretic hardness arguments—are largely restricted to the dilute, collision-free regime due to their reliance on random matrix theory and the hiding property. This work addresses this gap by developing a closed-form, representation-theoretic treatment of second moment statistics and anti-concentration for boson sampling beyond the dilute regime (2604.14323).

Representation-Theoretic Framework for Linear Optics

A central technical contribution is the construction of a general group-theoretic decomposition of the operator space associated with $n$-photon states on $m$ modes under passive linear optics (i.e., transformations from $U(m)$ preserving photon number). The Hilbert space, $\mathcal{F}_n$, admits a decomposition into occupation-number (Fock) states indexed by photon configurations. Observables preserving particle number form a space $\mathcal{W}$, which is shown to decompose cleanly into irreducible representations (irreps) of $U(m)$ in a way that is compatible with a ladder structure induced by a dual $\mathfrak{sl}_2(\mathbb{C})$ action—an instance of Howe duality.

The explicit construction organizes the operator space into a direct sum of irreps labeled by $n$0, each appearing as a ladder spanning different particle-number sectors. This mathematical structure is leveraged to express second moments and projection norms in fully general, efficiently computable forms, sidestepping the limitations of Clebsch-Gordan coefficients and recasting the analysis without the hiding property.

Analytical Second Moment Calculations

For an observable $n$1 acting on the $n$2-photon sector, consider the Haar-averaged second moment of expectation values under evolution by random linear optical unitaries: $n$3 where $n$4 is an input state. Proposition 1 gives the explicit result: $n$5 where $n$6 projects onto the $n$7-th irrep component and $n$8 is its dimension. The core technical challenge is the efficient, basis-independent calculation of these projection norms.

This is resolved using a recursive procedure exploiting the $n$9 ladder structure: projection onto each irrep is constructed using iterative application of raising and lowering maps, allowing the explicit calculation of the Hilbert-Schmidt norm of each irrep contribution without requiring explicit Clebsch-Gordan decomposition. This provides the machinery for computing second moments of generic number-preserving observables in all regimes of $m=O(n)$0 and $m=O(n)$1.

Exact Results for Anti-Concentration Beyond the Dilute Regime

A principal statistic of interest is the normalized outcome collision probability: $m=O(n)$2 which quantifies the probability of two independent samples yielding the same outcome $m=O(n)$3 and directly governs anti-concentration via Paley–Zygmund inequalities.

Using the developed formalism, the authors derive an exact closed-form expression for $m=O(n)$4 in all parameter regimes: $m=O(n)$5 where $m=O(n)$6 is the hypergeometric function and $m=O(n)$7 is the Pochhammer symbol. This formula generalizes previous results and, crucially, remains valid beyond the dilute regime.

Scaling Behavior and Anti-Concentration

By analyzing the asymptotics of $m=O(n)$8 for various scalings of $m=O(n)$9 with $n$0, the authors establish:

• Dilute regime, $n$1: $n$2. This reproduces known results.
• Intermediate/saturated regime, $n$3 with $n$4: $n$5.
• Saturated (linear) regime, $n$6: $n$7, i.e., asymptotically constant.

Through the Paley–Zygmund inequality, these results imply anti-concentration persists in the saturated regime, with the fraction of outputs of above-mean probability scaling as $n$8. Thus, even for experimentally feasible values $n$9, the output distribution remains sufficiently broad to preclude efficient classical simulation based on sparsity assumptions.

Implications and Connections to Complexity Theory

This work rigorously establishes that anti-concentration—previously known analytically only in the dilute regime—also holds in the regime of greatest experimental relevance, where photon collisions cannot be neglected. This removes a key obstruction for direct application of worst-to-average-case reductions and Stockmeyer-based hardness amplification arguments in the linear regime, thus tightening the theoretical foundations for boson sampling as a candidate for quantum computational advantage.

The developed group-theoretic techniques supersede the need for the hiding property and Gaussian approximations, enabling analytical results where previous work relied on numerics or indirect arguments (2604.14323).

On a practical level, closed-form expressions for second moments support exact diagnostics for certification (e.g., via linear cross-entropy benchmarking) and improved error analysis for quantum advantage experiments. The formalism can be further extended to Gaussian boson sampling, classical shadows, randomized benchmarking, and potentially analyses of noise-robustness in variational quantum algorithms.

Future Directions

The paper suggests several promising directions for further research:

• Higher moment analysis: Extending the representation-theoretic framework to compute all higher moments of output probabilities, to access even finer spectral properties and potentially prove full concentration results.
• Generalized measurements and mixed states: Incorporating non–particle-number-preserving observables and states, including Gaussian states, to analyze broader classes of photonic platforms.
• Noisy and imperfect devices: Adapting the theory to include experimental imperfections such as loss, partial distinguishability, or realistic noise channels.
• Resource theories: Employing projection norms and irrep purities as monotones or invariants in resource-theoretic settings for linear optics.

Conclusion

This work develops a comprehensive representation-theoretic approach for evaluating second moments of bosonic observables under Haar-random linear optics, and fully resolves the status of anti-concentration for boson sampling in all experimentally relevant regimes, including the linear mode regime where collisions dominate. By providing exact, constructive formulae for key statistical diagnostics without reliance on asymptotic or dilute approximations, the results directly reinforce the complexity-theoretic soundness of boson sampling as a candidate for scalable quantum advantage (2604.14323).