36,309 research outputs found
Counting, Fanout, and the Complexity of Quantum ACC
We propose definitions of \QAC^0, the quantum analog of the classical class
\AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in,
and \QACC[q], the analog of the class \ACC[q] where \Mod_q gates are also
allowed. We prove that parity or fanout allows us to construct quantum \MOD_q
gates in constant depth for any , so \QACC[2] = \QACC. More generally, we
show that for any , \MOD_q is equivalent to \MOD_p (up to constant
depth). This implies that \QAC^0 with unbounded fanout gates, denoted
\QACwf^0, is the same as \QACC[q] and \QACC for all . Since \ACC[p]
\ne \ACC[q] whenever and are distinct primes, \QACC[q] is strictly
more powerful than its classical counterpart, as is \QAC^0 when fanout is
allowed. This adds to the growing list of quantum complexity classes which are
provably more powerful than their classical counterparts.
We also develop techniques for proving upper bounds for \QACC^0 in terms of
related language classes. We define classes of languages \EQACC, \NQACC and
\BQACC_{\rats}. We define a notion of -planar \QACC operators and
show the appropriately restricted versions of \EQACC and \NQACC are
contained in \P/\poly. We also define a notion of -gate restricted
\QACC operators and show the appropriately restricted versions of \EQACC
and \NQACC are contained in \TC^0
On the Complexity of Quantum ACC
For any , let \MOD_q be a quantum gate that determines if the number
of 1's in the input is divisible by . We show that for any ,
\MOD_q is equivalent to \MOD_t (up to constant depth). Based on the case
, Moore \cite{moore99} has shown that quantum analogs of AC,
ACC, and ACC, denoted QAC, QACC, QACC respectively,
define the same class of operators, leaving as an open question. Our
result resolves this question, proving that QAC QACC
QACC for all . We also develop techniques for proving upper bounds for QACC
in terms of related language classes. We define classes of languages EQACC,
NQACC and BQACC_{\rats}. We define a notion -planar QACC operators and
show the appropriately restricted versions of EQACC and NQACC are contained in
P/poly. We also define a notion of -gate restricted QACC operators and
show the appropriately restricted versions of EQACC and NQACC are contained in
TC. To do this last proof, we show that TC can perform iterated
addition and multiplication in certain field extensions. We also introduce the
notion of a polynomial-size tensor graph and show that families of such graphs
can encode the amplitudes resulting from apply an arbitrary QACC operator to an
initial state.Comment: 22 pages, 4 figures This version will appear in the July 2000
Computational Complexity conference. Section 4 has been significantly revised
and many typos correcte
A scalable multi-core architecture with heterogeneous memory structures for Dynamic Neuromorphic Asynchronous Processors (DYNAPs)
Neuromorphic computing systems comprise networks of neurons that use
asynchronous events for both computation and communication. This type of
representation offers several advantages in terms of bandwidth and power
consumption in neuromorphic electronic systems. However, managing the traffic
of asynchronous events in large scale systems is a daunting task, both in terms
of circuit complexity and memory requirements. Here we present a novel routing
methodology that employs both hierarchical and mesh routing strategies and
combines heterogeneous memory structures for minimizing both memory
requirements and latency, while maximizing programming flexibility to support a
wide range of event-based neural network architectures, through parameter
configuration. We validated the proposed scheme in a prototype multi-core
neuromorphic processor chip that employs hybrid analog/digital circuits for
emulating synapse and neuron dynamics together with asynchronous digital
circuits for managing the address-event traffic. We present a theoretical
analysis of the proposed connectivity scheme, describe the methods and circuits
used to implement such scheme, and characterize the prototype chip. Finally, we
demonstrate the use of the neuromorphic processor with a convolutional neural
network for the real-time classification of visual symbols being flashed to a
dynamic vision sensor (DVS) at high speed.Comment: 17 pages, 14 figure
A 100-MIPS GaAs asynchronous microprocessor
The authors describe how they ported an asynchronous microprocessor previously implemented in CMOS to gallium arsenide, using a technology-independent asynchronous design technique. They introduce new circuits including a sense-amplifier, a completion detection circuit, and a general circuit structure for operators specified by production rules. The authors used and tested these circuits in a variety of designs
An average-case depth hierarchy theorem for Boolean circuits
We prove an average-case depth hierarchy theorem for Boolean circuits over
the standard basis of , , and gates.
Our hierarchy theorem says that for every , there is an explicit
-variable Boolean function , computed by a linear-size depth- formula,
which is such that any depth- circuit that agrees with on fraction of all inputs must have size This
answers an open question posed by H{\aa}stad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial
hierarchy is infinite relative to a random oracle with probability 1,
confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result
to show that there is no "approximate converse" to the results of Linial,
Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus
answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which
generalize random restrictions
The Computational Complexity of Generating Random Fractals
In this paper we examine a number of models that generate random fractals.
The models are studied using the tools of computational complexity theory from
the perspective of parallel computation. Diffusion limited aggregation and
several widely used algorithms for equilibrating the Ising model are shown to
be highly sequential; it is unlikely they can be simulated efficiently in
parallel. This is in contrast to Mandelbrot percolation that can be simulated
in constant parallel time. Our research helps shed light on the intrinsic
complexity of these models relative to each other and to different growth
processes that have been recently studied using complexity theory. In addition,
the results may serve as a guide to simulation physics.Comment: 28 pages, LATEX, 8 Postscript figures available from
[email protected]
Particle Computation: Complexity, Algorithms, and Logic
We investigate algorithmic control of a large swarm of mobile particles (such
as robots, sensors, or building material) that move in a 2D workspace using a
global input signal (such as gravity or a magnetic field). We show that a maze
of obstacles to the environment can be used to create complex systems. We
provide a wide range of results for a wide range of questions. These can be
subdivided into external algorithmic problems, in which particle configurations
serve as input for computations that are performed elsewhere, and internal
logic problems, in which the particle configurations themselves are used for
carrying out computations. For external algorithms, we give both negative and
positive results. If we are given a set of stationary obstacles, we prove that
it is NP-hard to decide whether a given initial configuration of unit-sized
particles can be transformed into a desired target configuration. Moreover, we
show that finding a control sequence of minimum length is PSPACE-complete. We
also work on the inverse problem, providing constructive algorithms to design
workspaces that efficiently implement arbitrary permutations between different
configurations. For internal logic, we investigate how arbitrary computations
can be implemented. We demonstrate how to encode dual-rail logic to build a
universal logic gate that concurrently evaluates and, nand, nor, and or
operations. Using many of these gates and appropriate interconnects, we can
evaluate any logical expression. However, we establish that simulating the full
range of complex interactions present in arbitrary digital circuits encounters
a fundamental difficulty: a fan-out gate cannot be generated. We resolve this
missing component with the help of 2x1 particles, which can create fan-out
gates that produce multiple copies of the inputs. Using these gates we provide
rules for replicating arbitrary digital circuits.Comment: 27 pages, 19 figures, full version that combines three previous
conference article
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