Volume 9 (2013) Article 4 pp. 143-252
The Computational Complexity of Linear Optics
by
Revised: December 30, 2012
Published: February 9, 2013
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 ${\mathsf P}^{\mathsf{\#P}}=\mathsf{BPP}^{\mathsf{NP}}$, 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 $\mathsf{\#P}$-hard to approximate the permanent of a matrix $A$ of independent $\mathcal{N}\left( 0,1\right)$ Gaussian entries, with high probability over $A$; and the Permanent Anti-Concentration Conjecture, which says that $\left\vert \operatorname*{Per}\left( A\right) \right\vert \geq\sqrt{n!}/\operatorname*{poly}\left( n\right)$ 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.