Fock State BosonSampling

• Fock State BosonSampling is a quantum model leveraging interference of indistinguishable bosons in fixed Fock states to generate complex, nonuniform output distributions.
• It employs efficient row-norm estimators to certify outputs, maintaining statistical structure even with photon loss and nonideal, mixed inputs.
• Contrasted with FermionSampling, its reliance on the #P-hard matrix permanent underpins the model's potential for demonstrating quantum advantage.

Fock State BosonSampling is a computational model that leverages the quantum interference of noninteracting, indistinguishable bosons prepared in definite-number (Fock) states and injected into a passive linear-optical network. The resulting output distribution in the photon number basis exhibits complexity deeply rooted in the combinatorial structure of matrix permanents, yielding sampling problems conjectured to be classically intractable under plausible complexity-theoretic assumptions. The structure and properties of the distribution, certification strategies, robustness to experimental imperfections, and contrast to alternative models such as FermionSampling or sampling with non-Fock initial states, have been the focus of significant research interest, particularly as experimental techniques push towards demonstrating “quantum advantage” via BosonSampling devices.

1. Structural Properties and Distinguishability of BosonSampling Distributions

A defining feature of Fock state BosonSampling is that, for an $n$-photon input traversing an %%%%1%%%%-mode Haar-random linear network, the output photon number distribution $\mathcal{D}_A$ is far from uniform in total variation distance. The paper proves that, with high probability over a Haar-random matrix $A$, the total variation distance between the BosonSampling distribution and the uniform distribution is bounded below by a constant: $$\|\mathcal{D}_A - \mathcal{U}\| = \frac{1}{2} \sum_{S} \left| p_S - \frac{1}{|\Lambda_{m, n}|} \right| \geq \frac{1}{9}$$ for sufficiently large $n$ and $m \gg n^{5.1}$, where $|\Lambda_{m, n}|$ denotes the size of the output configuration space (Aaronson et al., 2013). This demonstrates persistent statistical “structure” in the BosonSampling distribution which can be exploited for device certification.

Contrary to certain claims [e.g., (Gogolin et al., 2013)], this nonuniformity persists: the output is not “operationally indistinguishable” from uniform random distributions, even in the absence of detailed a priori knowledge. A key innovation of the paper is the construction of a linear-time computable row-norm estimator $R^*$: $$R^*(X) = \frac{\prod_{i=1}^n \left( \sum_{j=1}^n |x_{ij}|^2 \right)}{n^n}$$ for an $n \times n$ matrix $X$, such that

$$\Pr_{S\sim\mathcal{D}_A}[R^*(A_S)\geq 1] - \Pr_{S \sim \mathcal{U}}[R^*(A_S)\geq 1] \geq \frac{1}{9}$$

This enables efficient discrimination between the BosonSampling distribution and uniform, achieving constant bias in polynomial time. Thus, there exist experimentally practical statistics indicating that the device is sampling from the intended quantum distribution and not producing trivial or uniform randomness.

2. Robustness Against Loss, Non-Fock Inputs, and Experimental Imperfections

One crucial concern is the stability of BosonSampling features under realistic experimental imperfections, particularly photon loss and deviations from ideal single-photon Fock states. The row-norm estimator $R^*$ retains its effectiveness for distinguishing the true distribution from uniform even in the presence of:

• Photon loss: If some photons are lost (i.e., only a subset of the $n$ input photons are detected), the structure in the output persists, since only the occupied modes’ rows contribute significantly to the estimator.
• General input states: Even with more general (possibly mixed) input states – such as coherent or Gaussian states – the output can still be described in terms of the evolution of Fock coefficients, and the same row-norm–based tests apply.

Thus, the nonuniformity and distinguishability of BosonSampling distributions are robust features, persisting under nonideal conditions and making the same classical polynomial-time diagnosis strategies widely applicable.

3. Comparison to FermionSampling and Computational Complexity

The contrast with FermionSampling – where identical fermions traverse a linear network – is instructive. For fermions, the transition amplitudes involve determinants rather than permanents: $$\Pr_{\mathcal{F}_A}[S] = |\operatorname{Det}(A_S)|^2$$ The determinant is efficiently computable, and the paper provides a classical $O(m n^2)$ algorithm for sampling from the FermionSampling distribution (Aaronson et al., 2013). Additionally, the statistical structure (e.g., the distribution of $|\operatorname{Det}(X)|^2$ for Gaussian $X$) is substantially different; it converges to a lognormal distribution at rate $O(1/\log^{3/2} n)$.

FermionSampling thus serves as a tractable “control” model, further emphasizing the exceptional computational hardness of Fock state BosonSampling, which is rooted in the #P-hardness of the matrix permanent.

4. Mockup Distributions and Computational Indistinguishability

A notable result, due to Fernando Brandão and included as a “bonus,” concerns the possibility of efficiently sampling a distribution $\mathcal{D}'$ that is—by construction—indistinguishable from $\mathcal{D}_A$ by any fixed polynomial-size classical circuit, yet has high min-entropy and is efficiently samplable classically. Formally, for every fixed circuit size $T$ and any $\varepsilon > 0$, there exists a distribution on collision-free outcomes that is $\varepsilon$–indistinguishable from BosonSampling as long as testing is limited to polynomial-size circuits. This means that no efficient certification test, by itself, can completely rule out “mockup” distributions that mimic the certified properties of genuine BosonSampling outputs—unless the test inspects fine-grained or computationally hard features of the distribution.

This insight delineates the boundary between efficient, operational certification (e.g., using row-norms and other efficiently computable statistics) and the full complexity-theoretic difficulty of distinguishing arbitrary quantum-generated vs. efficiently-samplable classical distributions.

5. Mathematical Framework and Core Formulas

The statistics and complexity of Fock state BosonSampling are encapsulated in the combinatorial structure of the output amplitudes and probabilities:

• The probability for output configuration $S = (s_1, \ldots, s_m)$:

$$\Pr_{\mathcal{D}_A}[S] = \frac{|\operatorname{Per}(A_S)|^2}{s_1! \cdots s_m!}$$

where $A_S$ is the $n \times n$ matrix formed by selecting $s_i$ rows for each occupied mode $i$.

• The classical-hardness stems from the fact that $\operatorname{Per}(A_S)$ is #P-hard to compute, even approximately.
• Alternative estimators (such as $R^*(X)$) provide efficiently computable proxies, weakly correlated with the permanent, yet exhibiting sufficient bias to distinguish BosonSampling output.

The total variation distance, row-norm estimators, and the manner in which Fock coefficients transform under the network’s unitary evolution form the critical mathematical backbone for both analysis and practical certification strategies.

6. Experimental and Theoretical Implications

The theoretical insights yield clear prescriptions for both verifying and leveraging Fock state BosonSampling experimentally:

• Certification: Efficient row-norm–based protocols enable experimenters to distinguish output distributions from trivial or uniform noise using polynomial resources—a critical consideration in scaling experiments and validating claims of quantum advantage.
• Robustness: These certification techniques generalize to imperfect settings, including partial photon loss and non-ideal initial states, ensuring the verifiability of experimental outcomes even in nonideal operating regimes.
• Limitations: While operationally accessible tests are effective against “trivial” classical distractors (such as the uniform distribution), complexity-theoretic arguments show that cryptographically-embedded mockup distributions can evade efficient detection, highlighting fundamental limitations of classical verification unless one seeks features conjectured to be classically intractable.

In summary, the statistical structure of Fock state BosonSampling distributions is both highly nontrivial and robust, and can be effectively characterized and certified with polynomial-time statistical tests even under experimental imperfections. This property distinguishes it sharply from trivial or classically simulatable scenarios, and clarifies the requirements and limitations for certifying and exploiting genuine quantum sampling advantages in both theoretical and experimental settings (Aaronson et al., 2013).