17,393 research outputs found
Polynomial-time Solvable #CSP Problems via Algebraic Models and Pfaffian Circuits
A Pfaffian circuit is a tensor contraction network where the edges are
labeled with changes of bases in such a way that a very specific set of
combinatorial properties are satisfied. By modeling the permissible changes of
bases as systems of polynomial equations, and then solving via computation, we
are able to identify classes of 0/1 planar #CSP problems solvable in
polynomial-time via the Pfaffian circuit evaluation theorem (a variant of L.
Valiant's Holant Theorem). We present two different models of 0/1 variables,
one that is possible under a homogeneous change of basis, and one that is
possible under a heterogeneous change of basis only. We enumerate a series of
1,2,3, and 4-arity gates/cogates that represent constraints, and define a class
of constraints that is possible under the assumption of a ``bridge" between two
particular changes of bases. We discuss the issue of planarity of Pfaffian
circuits, and demonstrate possible directions in algebraic computation for
designing a Pfaffian tensor contraction network fragment that can simulate a
swap gate/cogate. We conclude by developing the notion of a decomposable
gate/cogate, and discuss the computational benefits of this definition
Sublogarithmic uniform Boolean proof nets
Using a proofs-as-programs correspondence, Terui was able to compare two
models of parallel computation: Boolean circuits and proof nets for
multiplicative linear logic. Mogbil et. al. gave a logspace translation
allowing us to compare their computational power as uniform complexity classes.
This paper presents a novel translation in AC0 and focuses on a simpler
restricted notion of uniform Boolean proof nets. We can then encode
constant-depth circuits and compare complexity classes below logspace, which
were out of reach with the previous translations.Comment: In Proceedings DICE 2011, arXiv:1201.034
Variational Quantum Time Evolution without the Quantum Geometric Tensor
The real- and imaginary-time evolution of quantum states are powerful tools
in physics and chemistry to investigate quantum dynamics, prepare ground states
or calculate thermodynamic observables. They also find applications in wider
fields such as quantum machine learning or optimization. On near-term devices,
variational quantum time evolution is a promising candidate for these tasks, as
the required circuit model can be tailored to trade off available device
capabilities and approximation accuracy. However, even if the circuits can be
reliably executed, variational quantum time evolution algorithms quickly become
infeasible for relevant system sizes. They require the calculation of the
Quantum Geometric Tensor and its complexity scales quadratically with the
number of parameters in the circuit. In this work, we propose a solution to
this scaling problem by leveraging a dual formulation that circumvents the
explicit evaluation of the Quantum Geometric Tensor. We demonstrate our
algorithm for the time evolution of the Heisenberg Hamiltonian and show that it
accurately reproduces the system dynamics at a fraction of the cost of standard
variational quantum time evolution algorithms. As an application, we calculate
thermodynamic observables with the QMETTS algorithm
Enabling High-Dimensional Hierarchical Uncertainty Quantification by ANOVA and Tensor-Train Decomposition
Hierarchical uncertainty quantification can reduce the computational cost of
stochastic circuit simulation by employing spectral methods at different
levels. This paper presents an efficient framework to simulate hierarchically
some challenging stochastic circuits/systems that include high-dimensional
subsystems. Due to the high parameter dimensionality, it is challenging to both
extract surrogate models at the low level of the design hierarchy and to handle
them in the high-level simulation. In this paper, we develop an efficient
ANOVA-based stochastic circuit/MEMS simulator to extract efficiently the
surrogate models at the low level. In order to avoid the curse of
dimensionality, we employ tensor-train decomposition at the high level to
construct the basis functions and Gauss quadrature points. As a demonstration,
we verify our algorithm on a stochastic oscillator with four MEMS capacitors
and 184 random parameters. This challenging example is simulated efficiently by
our simulator at the cost of only 10 minutes in MATLAB on a regular personal
computer.Comment: 14 pages (IEEE double column), 11 figure, accepted by IEEE Trans CAD
of Integrated Circuits and System
Tensor Computation: A New Framework for High-Dimensional Problems in EDA
Many critical EDA problems suffer from the curse of dimensionality, i.e. the
very fast-scaling computational burden produced by large number of parameters
and/or unknown variables. This phenomenon may be caused by multiple spatial or
temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit
simulation), nonlinearity of devices and circuits, large number of design or
optimization parameters (e.g. full-chip routing/placement and circuit sizing),
or extensive process variations (e.g. variability/reliability analysis and
design for manufacturability). The computational challenges generated by such
high dimensional problems are generally hard to handle efficiently with
traditional EDA core algorithms that are based on matrix and vector
computation. This paper presents "tensor computation" as an alternative general
framework for the development of efficient EDA algorithms and tools. A tensor
is a high-dimensional generalization of a matrix and a vector, and is a natural
choice for both storing and solving efficiently high-dimensional EDA problems.
This paper gives a basic tutorial on tensors, demonstrates some recent examples
of EDA applications (e.g., nonlinear circuit modeling and high-dimensional
uncertainty quantification), and suggests further open EDA problems where the
use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and
System
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