The ground state energy of the ±J spin glass from the genetic algorithm

1994 ◽  
Vol 4 (9) ◽  
pp. 1281-1285 ◽  
Author(s):  
P. Sutton ◽  
D. L. Hunter ◽  
N. Jan
Keyword(s):  
Genetic Algorithm ◽  
Spin Glass ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy

"INHERENT STRUCTURES" OF THE ±J ISING SPIN GLASS IN FOUR DIMENSIONS

1999 ◽  
Vol 10 (07) ◽  
pp. 1327-1333
Author(s):  
COLIN CHISHOLM ◽  
MARK LUKEMAN ◽  
N. JAN ◽  
D. L. HUNTER
Keyword(s):  
Transition Temperature ◽  
Spin Glass ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Precise Estimate ◽  
Ising Spin Glass ◽  
Four Dimensions ◽  
Ising Spin ◽  
2D And 3D

We measure the "inherent structures" of the ±J Ising spin glass in four dimensions (4D) and find a behavior similar to that seen for the 2D and 3D systems. We are able to determine the transition temperature from the overlap between the quenched states and the equilibrium states. We find that the transition temperature Tsg is 2.07±0.05 which agrees well with the recently reported value of 2.03±0.03 by Maranari and Zuliani. We also find that the ground state energy for the 4D spin glass is near -2.087±0.005, a more precise estimate than the value of -1.83 reported earlier.

Ground-state energy of the ±J spin glass with dimension greater than three

1996 ◽  
Vol 99 (4) ◽  
pp. 247-248 ◽  
Author(s):  
T. Wanschura ◽  
D.A. Coley ◽  
S. Migowsky
Keyword(s):  
Spin Glass ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy

On ℓp-Gaussian–Grothendieck Problem

2021 ◽  
Author(s):  
Wei-Kuo Chen ◽  
Arnab Sen
Keyword(s):  
Spin Glass ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Mean Field ◽  
Stability Estimates ◽  
Spin Glass Model ◽  
Maximum Value ◽  
Glass Model ◽  
Gaussian Orthogonal Ensemble

Abstract For $p\geq 1$ and $(g_{ij})_{1\leq i,j\leq n}$ being a matrix of i.i.d. standard Gaussian entries, we study the $n$-limit of the $\ell _p$-Gaussian–Grothendieck problem defined as $$\begin{align*} & \max\Bigl\{\sum_{i,j=1}^n g_{ij}x_ix_j: x\in \mathbb{R}^n,\sum_{i=1}^n |x_i|^p=1\Bigr\}. \end{align*}$$The case $p=2$ corresponds to the top eigenvalue of the Gaussian orthogonal ensemble; when $p=\infty $, the maximum value is essentially the ground state energy of the Sherrington–Kirkpatrick mean-field spin glass model and its limit can be expressed by the famous Parisi formula. In the present work, we focus on the cases $1\leq p<2$ and $2<p<\infty .$ For the former, we compute the limit of the $\ell _p$-Gaussian–Grothendieck problem and investigate the structure of the set of all near optimizers along with stability estimates. In the latter case, we show that this problem admits a Parisi-type variational representation and the corresponding optimizer is weakly delocalized in the sense that its entries vanish uniformly in a polynomial order of $n^{-1}$.

Classical approximation schemes for the ground-state energy of quantum and classical Ising spin Hamiltonians on planar graphs

2009 ◽  
Vol 9 (7&8) ◽  
pp. 701-720
Author(s):  
N. Bansal ◽  
S. Bravyi ◽  
B.M. Terhal
Keyword(s):  
Approximation Algorithm ◽  
Spin Glass ◽  
Planar Graph ◽  
Ground State Energy ◽  
Ground State ◽  
State Energy ◽  
Approximation Schemes ◽  
Ising Spin Glass ◽  
Ising Spin ◽  
Classical Approximation

We describe a classical approximation algorithm for evaluating the ground state energy of the classical Ising Hamiltonian with linear terms on an arbitrary planar graph. The running time of the algorithm grows linearly with the number of spins and exponentially with $1/\epsilon$, where $\epsilon$ is the worst-case relative error. This result contrasts the well known fact that exact computation of the ground state energy for the two-dimensional Ising spin glass model is NP-hard. We also present a classical approximation algorithm for the quantum Local Hamiltonian Problem or Quantum Ising Spin Glass problem on a planar graph {\em with bounded degree} which is known to be a QMA-complete problem. Using a different technique we find a classical approximation algorithm for the quantum Ising spin glass problem on the simplest planar graph with unbounded degree, the star graph.

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