Realizable solutions of the Thouless-Anderson-Palmer equations

Aspelmeier, Timo; Moore, M. A.

doi:10.1103/physreve.100.032127

“…Thus, extending RSB to glassy systems on sparse networks, i.e., random graphs [17] of finite average or fixed degree ("Bethe lattices", BL), constituted another major breakthrough [18]. More recently, the one-dimensional long-range model [19] has gained popularity in numerical studies [20][21][22][23] for the ability to interpolate between SK and the EA (but on a 1d -ring geometry) based on the range of interactions. That model has effective upper and lower dimensions, but all results obtained are numerical.…”

mentioning

confidence: 99%

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

Boettcher

¹

2020

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We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond-diluted to an overall density p, such that the average degree diverges as ∼ pN with the system size N . Ground-state energies are obtained with high accuracy for random instances for given p over a wide range of densities p. Since this is a NP-hard combinatorial problem, we employ the Extremal Optimization heuristic to that end. We find that the exponent describing the finite-size corrections, ω, varies continuously with p, a somewhat surprising result, as one would not expect that gradual bond-dilution would change the universality class of a statistical model. For p → 1, the familiar result of ω(p = 1) ≈ 2 3 for SK is obtained. In the limit of small p, ω(p) appears to diverge hyperbolically. arXiv:1912.04974v1 [cond-mat.dis-nn]

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“…The lowest value of the Gibbs potential g m 0 consistent with c 1 = 0, c 2 0 and Σ 0 is again determined analogue to procedure of subsection 3.2. As before a vanishing complexity and two temperature regions are found with a sticking below a different critical temperature [7], to [12] and to [11], respectively.…”

Section: Postulated Marginalitymentioning

confidence: 58%

“…Early numerical investigations [7,[9][10][11] claim marginal metastable states based on c 1 → 0 in the thermodynamic limit. The recent work of Aspelmeier and Moore [12] have numerically studied the N-dependence of the two lowest eigenvalues of the Hessian. They found that both eigenvalues tend to zero in the thermodynamic limit.…”

Section: Postulated Marginalitymentioning

confidence: 99%

“…This value is in remarkable agreement with the value of ĝm 0 = −0.076 19 found by the dynamical investigation [11] with c 1 0 and with a vanishing complexity. The result ĝm 0 is added to figure 8 as point C together with the points A and B found by further investigations [7,12]. Even though c 1 0 is satisfied no results for the complexity are available for the points A and B.…”

Section: Postulated Marginalitymentioning

confidence: 99%

“…An alternative method to work out the characteristic properties of the metastable TAP states are numerical investigations [7,[9][10][11][12] based on iteration techniques or phenomenological dynamics for systems of finite size. The numerical determined fix-points are interpreted as metastable TAP states.…”

Section: Introductionmentioning

confidence: 99%

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The marginal stability of the metastable TAP states

Plefka

¹

2020

J. Phys. A: Math. Theor.

1

0

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The existing investigations on the complexity are extended. In addition to the Edward–Anderson parameter q 2 the fourth moment q 4 = 1 / N ∑ i m i 4 of the magnetizations m i is included to the set of constrained variables and the constrained complexity Σ(T; q 2, q 4) is numerical determined. The maximum of Σ(T; q 2, q 4) (representing the total complexity) sticks at the boundary for temperatures at and below a new critical temperature. This implies marginal stability for the nearly all metastable states. The temperature dependence of the lowest value of the Gibbs potential consistent with various physical requirements is presented.

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Scaling advantage of chaotic amplitude control for high-performance combinatorial optimization

Leleu

¹

,

Khoyratee

²

,

Levi

³

et al. 2021

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The development of physical simulators, called Ising machines, that sample from low energy states of the Ising Hamiltonian has the potential to transform our ability to understand and control complex systems. However, most of the physical implementations of such machines have been based on a similar concept that is closely related to relaxational dynamics such as in simulated, mean-field, chaotic, and quantum annealing. Here we show that dynamics that includes a nonrelaxational component and is associated with a finite positive Gibbs entropy production rate can accelerate the sampling of low energy states compared to that of conventional methods. By implementing such dynamics on field programmable gate array, we show that the addition of nonrelaxational dynamics that we propose, called chaotic amplitude control, exhibits exponents of the scaling with problem size of the time to find optimal solutions and its variance that are smaller than those of relaxational schemes recently implemented on Ising machines.

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Realizable solutions of the Thouless-Anderson-Palmer equations

Cited by 7 publications

References 26 publications

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

The marginal stability of the metastable TAP states

Scaling advantage of chaotic amplitude control for high-performance combinatorial optimization

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