662 research outputs found
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
We consider quantum computations comprising only commuting gates, known as
IQP computations, and provide compelling evidence that the task of sampling
their output probability distributions is unlikely to be achievable by any
efficient classical means. More specifically we introduce the class post-IQP of
languages decided with bounded error by uniform families of IQP circuits with
post-selection, and prove first that post-IQP equals the classical class PP.
Using this result we show that if the output distributions of uniform IQP
circuit families could be classically efficiently sampled, even up to 41%
multiplicative error in the probabilities, then the infinite tower of classical
complexity classes known as the polynomial hierarchy, would collapse to its
third level. We mention some further results on the classical simulation
properties of IQP circuit families, in particular showing that if the output
distribution results from measurements on only O(log n) lines then it may in
fact be classically efficiently sampled.Comment: 13 page
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
On the Power of Many One-Bit Provers
We study the class of languages, denoted by \MIP[k, 1-\epsilon, s], which
have -prover games where each prover just sends a \emph{single} bit, with
completeness and soundness error . For the case that
(i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson
({\em Computational Complexity'02}) demonstrate that \SZK exactly
characterizes languages having 1-bit proof systems with"non-trivial" soundness
(i.e., ). We demonstrate that for the case that
, 1-bit -prover games exhibit a significantly richer structure:
+ (Folklore) When , \MIP[k, 1-\epsilon, s]
= \BPP;
+ When , \MIP[k,
1-\epsilon, s] = \SZK;
+ When , \AM \subseteq \MIP[k, 1-\epsilon,
s];
+ For and sufficiently large , \MIP[k, 1-\epsilon, s]
\subseteq \EXP;
+ For , \MIP[k, 1, 1-\epsilon, s] = \NEXP.
As such, 1-bit -prover games yield a natural "quantitative" approach to
relating complexity classes such as \BPP,\SZK,\AM, \EXP, and \NEXP.
We leave open the question of whether a more fine-grained hierarchy (between
\AM and \NEXP) can be established for the case when
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