The Computational Complexity of Linear Optics

We give new evidence that quantum computers—moreover, rudimentary quantum computers built entirely out of linear-optical elements—cannot be efficiently simulated by classical comput-ers. 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 com-putation, 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 = BPPNP, 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 al-ready imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures