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

Samo Novák, Raúl García-Patrón

TL;DR

The key difference in this 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.

Abstract

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.

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

TL;DR

The key difference in this 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 from the running time, especially as is a requirement of the original Boson Sampling proposal.

Abstract

In a Boson Sampling quantum optical experiment we send individual photons into an -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 , 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 , where is the treewidth of the decomposition which satisfies 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 from the running time, especially as is a requirement of the original Boson Sampling proposal.

Paper Structure

This paper contains 61 sections, 9 theorems, 65 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Informal summary
  3. Structure of the paper
  4. Boson Sampling and Clifford & Clifford algorithm
  5. Short introduction to single-photon Boson Sampling
  6. Outcome probabilities
  7. Graphical representation of $\mathcal{V}^{\underline{n}', \underline{n}}$
  8. Clifford and Clifford algorithms
  9. Algorithms CC-A and CC-B
  10. Laplace expansion: algorithm CC-C
  11. Cifuentes and Parillo Algorithm
  12. Tree Decompositions
  13. Treewidth
  14. Permanent algorithm
  15. Dynamic programming tables
  16. ...and 46 more sections

Key Result

Lemma B.1

The marginal pmf of a partial sample of $k \in [n]$ photons is

Figures (10)

  • Figure 1: Overview of our work using an example: (a) The shallow brickwork circuit of nearest-neighbour beamsplitters with inputs on right and outputs on the left, showing one measurement outcome with previously sampled photons "$\bullet$" and a new photon "$\bullet^*$" added in this step of the Clifford and Clifford algorithm to expand the sample. (b) A weighted bipartite graph represents the outcome: vertices are input and output photons, and non-zero transition amplitudes between them give weighted edges. The outcome amplitude is the sum of weighted perfect matchings in this graph, i.e. the permanent. (c) To exploit the sparsity structure, we construct a tree decomposition: a tree whose nodes contain sets of edges of the graph in (b), subject to simple rules. We use a linear decomposition (no branching) where each node corresponds uniquely to an input photon vertex. We then modify the tree decomposition in several steps, each time temporarily removing a different input photon $=$ column $=$ node of the tree, replacing the latter by a dummy with no new information, seen in (d). Permanents of these modified trees form the Laplace expansion, our main technical contribution. When done in the right order, conceptualized as a "machine head" walking the tree from one end to the other, we can reuse most of the dynamic programming tables required by the Cifuentes and Parrilo permanent algorithm. As seen in the middle term, we note that the dummy may need to be nonempty.
  • Figure 2: Graphical representation: (a) Graphs $G(\mathcal{U})$ and $G(\mathcal{V}^{\underline{n}', \underline{n}})$ and their relationship. Left (right) partition corresponds to outputs (inputs). Note that $\mathcal{U}$ is the biadjacency matrix of $G(\mathcal{U})$, and likewise $\mathcal{V}^{\underline{n}', \underline{n}}$ of $G(\mathcal{V}^{\underline{n}', \underline{n}})$. The output partition of the graph $G(\mathcal{V}^{\underline{n}', \underline{n}})$ consists of $n'_i$ copies $|i\rangle_1, \dots, |i\rangle_{n'_i}$ of output vertex $\left|i\right\rangle$ of $G(\mathcal{U})$; inputs are similarly copied. If there is an edge $e=(|i\rangle, \langle j|)$ in $G(\mathcal{U})$, then ${G(\mathcal{V}^{\underline{n}', \underline{n}})}$ has edges $(\left|i\right\rangle_a, \left\langle j\right|_b)$ for all $a,b$, and these have the same weight as $e$: We show in red the edge $(\left|1\right\rangle, \left\langle2\right|)$ of $G(\mathcal{U})$ and its copies in $G(\mathcal{V}^{\underline{n}', \underline{n}})$, all with weights $\mathcal{U}_{1,2}$. Some edges are omitted for clarity. (b) Example: $n = n' = 3$, $m = 5$, input $\left|\underline{n}\right\rangle = \left|1,1,1,0,0\right\rangle$ and output $\left|\underline{n}'\right\rangle = \left|2,0,0,1,0\right\rangle$.
  • Figure 3: Relation between $G(M)$ and $\mathop{\mathrm{per}}\nolimits M$ for an example binary matrix $M$: $\mathop{\mathrm{per}}\nolimits M$ counts the perfect matchings of $G(M)$.
  • Figure 4: Example tree decomposition $\mathrm{T}$ of some graph of matrix $G(M)$. This tree is rooted at $t_1$.
  • Figure 5: Bipartite permanent
  • ...and 5 more figures

Theorems & Definitions (23)

  • Definition 3.1: tree decomposition of a bipartite graph
  • Definition 3.2: treewidth Cifuentes2015Voigt2016
  • Lemma B.1: partial sample
  • proof
  • Lemma C.2
  • proof
  • Lemma C.3
  • proof
  • Theorem C.4: running time of CP
  • proof
  • ...and 13 more