Complexity-theoretic foundations of BosonSampling with a linear number of modes

TL;DR

This work establishes a complexity-theoretic hardness foundation for BosonSampling in the saturated regime where the number of modes scales linearly with the number of photons. It introduces a new worst-to-average-case reduction for computing the Permanent that remains robust under row repetitions and correlated entries, and extends the framework to Gaussian BosonSampling, supported by approximate invariance and anticoncentration evidence. By linking typical saturated outputs to permanents of structured submatrices and leveraging Lipton-style interpolation with Robust Berlekamp-Welch, the authors argue that efficient classical simulation would collapse PH under plausible average-case hardness assumptions. The results provide a unified hardness narrative for saturated BosonSampling and its Gaussian variant, with practical implications for experimental benchmarks and the potential for universal quantum computation using saturated linear optics.

Abstract

BosonSampling is the leading candidate for demonstrating quantum computational advantage in photonic systems. While we have recently seen many impressive experimental demonstrations, there is still a formidable distance between the complexity-theoretic hardness arguments and current experiments. One of the largest gaps involves the ratio of {particles} to modes -- all current hardness evidence assumes a dilute regime in which the number of linear optical modes scales at least quadratically in the number of particles. By contrast, current experiments operate in a saturated regime with a linear number of modes. In this paper we bridge this gap, bringing the hardness evidence for experiments in the saturated regime to the same level as had been previously established for the dilute regime. This involves proving a new worst-to-average-case reduction for computing the Permanent which is robust to both large numbers of row repetitions and also to distributions over matrices with correlated entries. We also apply similar arguments to give evidence for hardness of Gaussian BosonSampling in the saturated regime.

Figures (5)

• Figure 1: Two regimes of BosonSampling. (a) The dilute limit, where $m=\Omega(n^2)$. No-collision outcomes dominate, and the submatrix whose permanent we must compute (i.e. selecting the appropriate rows and columns) is sufficiently smaller than the Haar-random interferometer $U$ that its entries look i.i.d. Gaussian. (b) The saturated limit, where $m=\Theta(n)$. No-collision outcomes are rare, and complexity is determined by the number of detector clicks (purple) rather than output photons (blue). The entries of the submatrix of $U$ are very correlated, and its rows will be repeated according to collisions, as described in the main text.
• Figure 2: The figure shows how to extend the $c \times c$ matrix $X$ to handle $k$ row repetitions. The first step is to create the matrix $A=(X|Y)$ by appending $k$ extra columns, as described in the text. The matrix $Z$ just corresponds to all extra rows that are added according to the repetition pattern.
• Figure 3: Box plots for the distribution of random probabilities in the $m = 2n$ regime. For each $n$, we calculated $\ln(|\operatorname{Per}(U_S)|^2/\prod_i s_i!)$ for 20 Haar random matrices $U \in \mathbb C^{2n \times 2n}$ and 20 uniformly random outcomes $S$. Left: The blue box plots depict this distribution and the red dashed line show the anticoncentration bound (i.e., $- \ln \vert\mathcal{S}_{m,n}\vert$). Right: The blue box plots depict the same distribution shifted by the anticoncentration bound, hence why the red dashed line appears at $0$. Each box plot has the same format: min, 1st quartile, median, 3rd quartile, max.
• Figure 4: Gaussian Boson Sampling. Vertical dashed lines correspond to preparation of two-mode squeezed states with identical squeezing parameters $r$. The overall transition matrix is given by $C = (\tanh{r}) U W^\dag$, and transition probabilities are given by permanents of submatrices of $C$, as described in the main text. Each photon observed in one of the top (bottom) $m$ modes selects one of the rows (columns) of $C$.
• Figure 5: The spherical cap in a $\delta$ radius around $w_0$. The shaded region is a spherical cap, whose boundary is a circle (in general, $d-2$-dimensional sphere). The area of the cap is strictly larger than the area of the flat disk of radius $\sin\theta$ with the same boundary. The value of $\theta$ is determined by the equation $(1-\cos(\theta))^2 + (\sin\theta)^2 = \delta$, which can be seen by inspecting the right triangle drawn with vertex at $w_0$ and hypotenuse of length $\delta$.

Theorems & Definitions (39)

• Remark
• Theorem 1: Informal
• Conjecture 1
• Theorem 2: No classical sampler
• Lemma 1: Combinatorics of $c$ clicks
• proof
• Lemma 2: Concentration inequality for number of occupied modes for random
• Theorem 3: $\#\mathsf{P}$-hard permanents of worst-case submatrices
• Lemma 3
• proof : Proof of Theorem \ref{['thm:worst']}