10,858 research outputs found
The Classical Complexity of Boson Sampling
We study the classical complexity of the exact Boson Sampling problem where
the objective is to produce provably correct random samples from a particular
quantum mechanical distribution. The computational framework was proposed by
Aaronson and Arkhipov in 2011 as an attainable demonstration of `quantum
supremacy', that is a practical quantum computing experiment able to produce
output at a speed beyond the reach of classical (that is non-quantum) computer
hardware. Since its introduction Boson Sampling has been the subject of intense
international research in the world of quantum computing. On the face of it,
the problem is challenging for classical computation. Aaronson and Arkhipov
show that exact Boson Sampling is not efficiently solvable by a classical
computer unless and the polynomial hierarchy collapses to
the third level.
The fastest known exact classical algorithm for the standard Boson Sampling
problem takes time to produce samples for a
system with input size and output modes, making it infeasible for
anything but the smallest values of and . We give an algorithm that is
much faster, running in time and
additional space. The algorithm is simple to implement and has low constant
factor overheads. As a consequence our classical algorithm is able to solve the
exact Boson Sampling problem for system sizes far beyond current photonic
quantum computing experimentation, thereby significantly reducing the
likelihood of achieving near-term quantum supremacy in the context of Boson
Sampling.Comment: 15 pages. To appear in SODA '1
No imminent quantum supremacy by boson sampling
It is predicted that quantum computers will dramatically outperform their
conventional counterparts. However, large-scale universal quantum computers are
yet to be built. Boson sampling is a rudimentary quantum algorithm tailored to
the platform of photons in linear optics, which has sparked interest as a rapid
way to demonstrate this quantum supremacy. Photon statistics are governed by
intractable matrix functions known as permanents, which suggests that sampling
from the distribution obtained by injecting photons into a linear-optical
network could be solved more quickly by a photonic experiment than by a
classical computer. The contrast between the apparently awesome challenge faced
by any classical sampling algorithm and the apparently near-term experimental
resources required for a large boson sampling experiment has raised
expectations that quantum supremacy by boson sampling is on the horizon. Here
we present classical boson sampling algorithms and theoretical analyses of
prospects for scaling boson sampling experiments, showing that near-term
quantum supremacy via boson sampling is unlikely. While the largest boson
sampling experiments reported so far are with 5 photons, our classical
algorithm, based on Metropolised independence sampling (MIS), allowed the boson
sampling problem to be solved for 30 photons with standard computing hardware.
We argue that the impact of experimental photon losses means that demonstrating
quantum supremacy by boson sampling would require a step change in technology.Comment: 25 pages, 9 figures. Comments welcom
On permanents of Sylvester Hadamard matrices
It is well-known that the evaluation of the permanent of an arbitrary
-matrix is a formidable problem. Ryser's formula is one of the fastest
known general algorithms for computing permanents. In this paper, Ryser's
formula has been rewritten for the special case of Sylvester Hadamard matrices
by using their cocyclic construction. The rewritten formula presents an
important reduction in the number of sets of distinct rows of the matrix to
be considered. However, the algorithm needs a preprocessing part which remains
time-consuming in general
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