60,359 research outputs found
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 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.Comment: 94 pages, 4 figure
Spatio-angular Minimum-variance Tomographic Controller for Multi-Object Adaptive Optics systems
Multi-object astronomical adaptive-optics (MOAO) is now a mature wide-field
observation mode to enlarge the adaptive-optics-corrected field in a few
specific locations over tens of arc-minutes.
The work-scope provided by open-loop tomography and pupil conjugation is
amenable to a spatio-angular Linear-Quadratic Gaussian (SA-LQG) formulation
aiming to provide enhanced correction across the field with improved
performance over static reconstruction methods and less stringent computational
complexity scaling laws.
Starting from our previous work [1], we use stochastic time-progression
models coupled to approximate sparse measurement operators to outline a
suitable SA-LQG formulation capable of delivering near optimal correction.
Under the spatio-angular framework the wave-fronts are never explicitly
estimated in the volume,providing considerable computational savings on
10m-class telescopes and beyond.
We find that for Raven, a 10m-class MOAO system with two science channels,
the SA-LQG improves the limiting magnitude by two stellar magnitudes when both
Strehl-ratio and Ensquared-energy are used as figures of merit. The
sky-coverage is therefore improved by a factor of 5.Comment: 30 pages, 7 figures, submitted to Applied Optic
Inefficiency of classically simulating linear optical quantum computing with Fock-state inputs
Aaronson and Arkhipov recently used computational complexity theory to argue
that classical computers very likely cannot efficiently simulate linear,
multimode, quantum-optical interferometers with arbitrary Fock-state inputs
[Aaronson and Arkhipov, Theory Comput. 9, 143 (2013)]. Here we present an
elementary argument that utilizes only techniques from quantum optics. We
explicitly construct the Hilbert space for such an interferometer and show that
its dimension scales exponentially with all the physical resources. We also
show in a simple example just how the Schr\"odinger and Heisenberg pictures of
quantum theory, while mathematically equivalent, are not in general
computationally equivalent. Finally, we conclude our argument by comparing the
symmetry requirements of multiparticle bosonic to fermionic interferometers
and, using simple physical reasoning, connect the nonsimulatability of the
bosonic device to the complexity of computing the permanent of a large matrix.Comment: 7 pages, 1 figure Published in PRA Phys. Rev. A 89, 022328 (2014
The Equivalence of Sampling and Searching
In a sampling problem, we are given an input x, and asked to sample
approximately from a probability distribution D_x. In a search problem, we are
given an input x, and asked to find a member of a nonempty set A_x with high
probability. (An example is finding a Nash equilibrium.) In this paper, we use
tools from Kolmogorov complexity and algorithmic information theory to show
that sampling and search problems are essentially equivalent. More precisely,
for any sampling problem S, there exists a search problem R_S such that, if C
is any "reasonable" complexity class, then R_S is in the search version of C if
and only if S is in the sampling version. As one application, we show that
SampP=SampBQP if and only if FBPP=FBQP: in other words, classical computers can
efficiently sample the output distribution of every quantum circuit, if and
only if they can efficiently solve every search problem that quantum computers
can solve. A second application is that, assuming a plausible conjecture, there
exists a search problem R that can be solved using a simple linear-optics
experiment, but that cannot be solved efficiently by a classical computer
unless the polynomial hierarchy collapses. That application will be described
in a forthcoming paper with Alex Arkhipov on the computational complexity of
linear optics.Comment: 16 page
- …