Laplace expansions and tree decompositions: A faster polytime algorithm for shallow nearest-neighbour Boson Sampling

In a Boson Sampling quantum optical experiment we send $n$ individual photons into an $m$-mode interferometer and we measure the occupation pattern on the output. The statistics of this process depending on the permanent of a matrix representing the experiment, a #P-hard problem to compute, is the reason behind ideal and fully general Boson Sampling being hard to simulate on a classical computer. We exploit the fact that for a nearest-neighbour shallow circuit, i.e. depth $D = \mathcal{O}(\log m)$, one can adapt the algorithm by Clifford & Clifford (2018) to exploit the sparsity of the shallow interferometer using an algorithm by Cifuentes & Parrilo (2016) that can efficiently compute a permanent of a structured matrix from a tree decomposition. Our algorithm generates a sample from a shallow circuit in time $\mathcal{O}(n^2 2^ωω^2) + \mathcal{O}(ωn^3)$, where $ω$ is the treewidth of the decomposition which satisfies $ω\le 2D$ for nearest-neighbour shallow circuits. The key difference in our work with respect to previous work using similar methods is the reuse of the structure of the tree decomposition, allowing us to adapt the Laplace expansion used by Clifford & Clifford which removes a significant factor of $m$ from the running time, especially as $m>n^2$ is a requirement of the original Boson Sampling proposal.

💡 Research Summary

The paper presents a novel classical algorithm for sampling from Boson Sampling experiments when the underlying linear‑optical circuit is a shallow nearest‑neighbour architecture with depth (D = O(\log m)). The key observation is that in this regime the transition matrix (V) (the sub‑matrix of the interferometer unitary that governs the amplitudes of the (n) photons) is highly sparse, and its sparsity can be captured by a tree decomposition whose treewidth (\omega) satisfies (\omega \le 2D).

Cifuentes and Parrilo (2016) showed that the permanent of a matrix whose associated bipartite graph has treewidth (\omega) can be computed in (\mathcal{O}(2^{\omega},\omega^{2})) time using a dynamic‑programming scheme over the tree. Clifford and Clifford (2018) introduced a sampling strategy that builds a sample photon‑by‑photon and uses the Laplace expansion of the permanent to reuse many sub‑permanent calculations, achieving a runtime of (\mathcal{O}(n^{2}2^{n}) + \mathcal{O}(mn^{2})). However, the factor (\mathcal{O}(mn^{2})) remains problematic when the number of modes (m) is much larger than the number of photons (n).

The authors combine these two ideas. They construct a single global tree decomposition of the bipartite graph representing the current photon configuration, arranging the tree as a linear path so that each node corresponds to exactly one input photon. When the Laplace expansion removes the last row (the photon being sampled), they “temporarily delete” the corresponding node and replace it with a dummy node that carries minimal information. Because the tree structure is preserved, the dynamic‑programming tables built for the original tree can be updated locally rather than recomputed from scratch. Consequently, each term of the Laplace expansion can be evaluated in (\mathcal{O}(2^{\omega},\omega^{2})) time, and the whole sampling procedure runs in

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