The Computational Complexity of Linear Optics

TL;DR

This work introduces a noninteracting-boson model implemented via linear optics and analyzes its computational power. By tying output amplitudes to matrix permanents, it provides evidence that exact BosonSampling cannot be classically simulated without collapsing the polynomial hierarchy, and it strengthens this claim for approximate simulations under plausible conjectures about Gaussian permanents. The authors develop a robust technical framework linking Haar-random unitary truncations to Gaussian permanents, and they outline a concrete experimental path using linear optics to test these ideas. Overall, the paper raises strong theoretical and practical implications for the Extended Church-Turing Thesis and the potential quantum advantage of near-term optical experiments.

Abstract

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

Figures (4)

• Figure 1: Galton's board, a simple “ computer” to output samples from the binomial distribution. From MathWorld, http://mathworld.wolfram.com/GaltonBoard.html
• Figure 2: Summary of our hardness argument (modulo conjectures). If there exists a polynomial-time classical algorithm for approximate BosonSampling, then Theorem \ref{['mainresult']} says that $\left\vert \text{GPE}\right\vert _{\pm}^{2}\in\mathsf{BPP}^{\mathsf{NP}}$. Assuming Conjecture \ref{['pacc']} (the PACC), Theorem \ref{['decompthm']} says that this is equivalent to GPE$_{\times}\in\mathsf{BPP}^{\mathsf{NP}}$. Assuming Conjecture \ref{['pgc']} (the PGC), this is in turn equivalent to $\mathsf{P}^{\mathsf{\#P}}=\mathsf{BPP}^{\mathsf{NP}}$, which collapses the polynomial hierarchy by Toda's Theorem toda.
• Figure 3: The Hong-Ou-Mandel dip.
• Figure 4: Probability density functions of the random variables $D_{n}=\left\vert \operatorname*{Det}\left( X\right) \right\vert ^{2}/n!$ and $P_{n}=\left\vert \operatorname*{Per}\left( X\right) \right\vert ^{2}/n!$, where $X\sim\mathcal{G}^{n\times n}$ is a complex Gaussian random matrix, in the case $n=6$. Note that $\operatorname*{E}\left[ D_{n}\right] =\operatorname*{E}\left[ P_{n}\right] =1$. As $n$ increases, the bends on the left become steeper. We do not know whether the pdfs diverge at the origin.

Theorems & Definitions (70)

• Theorem 1
• Theorem 3: Main Result
• Conjecture 5: Permanent-of-Gaussians Conjecture or PGC
• Conjecture 6: Permanent Anti-Concentration Conjecture
• Theorem 7
• Definition 8: $\mathsf{PostBPP}$ and $\mathsf{PostBQP}$
• Theorem 9: Aaronson aar:pp
• Definition 10: $\mathsf{SampP}$ and $\mathsf{SampBQP}$
• Definition 11: $\mathsf{FBPP}$ and $\mathsf{FBQP}$
• Theorem 12: Sampling/Searching Equivalence Theorem aar:samp