Summary: The Computational Complexity of Linear Optics
Scott Aaronson # Alex Arkhipov +
We give new evidence that quantum computers---moreover, rudimentary quantum computers
built entirely out of linearoptical elements---cannot be e#ciently simulated by classical comput
ers. In particular, we define a model of computation in which identical photons are generated,
sent through a linearoptical 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 polynomialtime classical algorithm that samples
from the same probability distribution as a linearoptical 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 al
ready imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures:
the PermanentofGaussians Conjecture, which says that it is #Phard to approximate the per
manent of a matrix A of independent N (0, 1) Gaussian entries, with high probability over A;
and the Permanent AntiConcentration Conjecture, which says that |Per (A)| # # n!/ poly (n)