Dataset 1

Spin Glass Dataset

Description of Ising Model Type

A spin glass Ising model is characterized by frustrated interactions, meaning some neighboring spins prefer to align while others prefer to oppose. These conflicting orientations prevent the system from reaching a uniformly ordered state. The energy Hamiltonian of this model can be given by

\[H = -J \sum_{i<j}s_i s_j\]

Here, \(J_{<i,j>}\) can be either positive or negative. If \(J_{<i,j>}\) is positive, \(s_i\) and \(s_j\) prefer to align and, conversely, they will anti-align. This is the precise phenomenon that defines a spin glass.

Instances

The instances we are providing for the spin glass Ising model type consist of weighted edges. Note that there are both positive and negative edge weights. The complexity of an instance is determined not only by the size, but also the pattern of interactions and frustration.

1D

Instance	# of Instances	# of Spins	Complexity	Graph Geometry	Reference
Spin Glass	6	300	Complex	Chain	[11]
Spin Glass	6	250	Complex	Chain	[11]
Spin Glass	6	200	Complex	Chain	[11]
Spin Glass	6	150	Complex	Chain	[11]
Spin Glass	6	100	Complex	Chain	[11]
Spin Glass	500	10	Simple	Chain	[4]
Spin Glass	500	9	Simple	Chain	[4]
Spin Glass	500	8	Simple	Chain	[4]
Spin Glass	500	7	Simple	Chain	[4]
Spin Glass	500	6	Simple	Chain	[4]
Spin Glass	500	5	Simple	Chain	[4]
Spin Glass	500	4	Simple	Chain	[4]
Spin Glass	500	3	Simple	Chain	[4]

2D

Instance	# of Instances	# of Spins	Complexity	Graph Geometry	Reference
Spin Glass	25	1,600	Complex	EA	[1]
Spin Glass	50	1,225	Complex	EA	[2]
Spin Glass	3	20	Complex	Toroidal	[11]
Spin Glass	3	15	Complex	Toroidal	[11]
Spin Glass	3	10	Complex	Toroidal	[11]
Spin Glass	50	900	Intermediate	EA	[2]
Spin Glass	50	625	Intermediate	EA	[2]
Spin Glass	100	256	Intermediate	Random	[3]_
Spin Glass	50	400	Simple	EA	[2]
Spin Glass	50	225	Simple	EA	[2]
Spin Glass	100	196	Simple	Random	[3]_
Spin Glass	100	144	Simple	Random	[3]_
Spin Glass	1,000	100	Simple	EA	[2]
Spin Glass	100	100	Simple	Random	[3]_
Spin Glass	100	64	Simple	Random	[3]_
Spin Glass	20	64	Simple	Cylindrical	[8]
Spin Glass	20	49	Simple	Cylindrical	[8]
Spin Glass	100	36	Simple	Random	[3]_
Spin Glass	20	36	Simple	Cylindrical	[8]
Spin Glass	20	25	Simple	Cylindrical	[8]
Spin Glass	100	16	Simple	Random	[3]_
Spin Glass	20	16	Simple	Cylindrical	[8]

3D

Instance	# of Instances	# of Spins	Complexity	Graph Geometry	Reference
Spin Glass	100	8,000	Complex	EA	[2]
Spin Glass	1	5,730	Complex	EA	[12]
Spin Glass	1	4,644	Complex	EA	[12]
Spin Glass	1	4,312	Complex	EA	[12]
Spin Glass	1	4,101	Complex	EA	[12]
Spin Glass	1	3,920	Complex	EA	[12]
Spin Glass	1	3,558	Complex	EA	[12]
Spin Glass	100	3,375	Complex	EA	[2]
Spin Glass	100	1,000	Complex	EA	[2]
Spin Glass	3	7	Complex	Toroidal	[11]
Spin Glass	3	6	Complex	Toroidal	[11]
Spin Glass	3	5	Complex	Toroidal	[11]
Spin Glass	100	512	Intermediate	EA	[2]
Spin Glass	100	216	Simple	EA	[2]
Spin Glass	100	64	Simple	EA	[2]
Spin Glass	20	54	Simple	Cubic	[8]
Spin Glass	20	50	Simple	Diamond	[8]
Spin Glass	20	36	Simple	Cubic	[8]
Spin Glass	20	32	Simple	Diamond	[8]
Spin Glass	20	24	Simple	Cubic	[8]
Spin Glass	20	18	Simple	Diamond	[8]

4D

Instance	# of Instances	# of Spins	Complexity	Graph Geometry	Reference
Spin Glass	100	4,096	Complex	EA	[2]
Spin Glass	100	2,041	Complex	EA	[2]
Spin Glass	100	1,296	Complex	EA	[2]
Spin Glass	100	625	Intermediate	EA	[2]
Spin Glass	50	256	Intermediate	EA	[2]
Spin Glass	1,000	81	Simple	EA	[2]

Mean-Field

Instance	# of Instances	# of Spins	Complexity	Graph Geometry	Reference
Spin Glass	100	900	Intermediate	Hopfield	[14]
Spin Glass	100	900	Intermediate	SK	[14]
Spin Glass	100	800	Intermediate	Hopfield	[14]
Spin Glass	100	800	Intermediate	SK	[14]
Spin Glass	100	700	Intermediate	Hopfield	[14]
Spin Glass	100	700	Intermediate	SK	[14]
Spin Glass	100	600	Intermediate	Hopfield	[14]
Spin Glass	100	600	Intermediate	SK	[14]
Spin Glass	100	500	Intermediate	Hopfield	[14]
Spin Glass	100	500	Intermediate	SK	[14]
Spin Glass	100	400	Intermediate	Hopfield	[14]
Spin Glass	100	400	Intermediate	SK	[14]
Spin Glass	100	300	Intermediate	Hopfield	[14]
Spin Glass	100	300	Intermediate	SK	[14]
Spin Glass	100	200	Simple	Hopfield	[14]
Spin Glass	100	200	Simple	SK	[14]
Spin Glass	100	100	Simple	Hopfield	[14]
Spin Glass	100	100	Simple	SK	[14]
Spin Glass	40	100	Simple	Hopfield	[14]
Spin Glass	25	32	Simple	WPE	[1]
Spin Glass	500	10	Simple	FC	[4]
Spin Glass	500	9	Simple	FC	[4]
Spin Glass	500	8	Simple	FC	[4]
Spin Glass	500	7	Simple	FC	[4]
Spin Glass	500	6	Simple	FC	[4]
Spin Glass	500	5	Simple	FC	[4]
Spin Glass	500	4	Simple	FC	[4]
Spin Glass	500	3	Simple	FC	[4]
Spin Glass	10	72	Simple	SK	[9]
Spin Glass	10	56	Simple	SK	[9]
Spin Glass	10	40	Simple	SK	[9]
Spin Glass	10	24	Simple	SK	[9]
Spin Glass	10	8	Simple	SK	[9]
Spin Glass	1	16	Simple	SK	[5]

Dataset References

Below contain the references to the datasets we gathered on this website.

[1] (1,2)

Variational Neural Annealing

VNA proposes a parameterized annealing model to stochastically search for ground states of the Ising model. It features fully connected spin glass instances, as well as Edwards-Anderson (EA) and Wishart Planted Ensemble (WPE) graph instances of the Ising model.

[2] (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)

Searching for Spin Glass Ground States Through Deep Reinforcement Learning

This paper introduces DIRAC, a deep reinforcement learning framework that can be trained on small-scale spin glass instances and applied to arbitrarily large ones. DIRAC has found success in scalability compared to other methods and has been tested on 2D, 3D, and 4D EA spin-glass instances.

[3]

Finding the Ground State of Spin Hamiltonians with Reinforcement Learning

This paper demonstrates that an RL agent is able to surpass the performance of standard heuristic temperature schedule for two classes of Hamiltonians. They show the performance of their implementation by training on weak-strong clusters (bipartite graph with two fully connected clusters) and nearest-neighbor square spin glasses.

[4] (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)

Quantum-enhanced Markov Chain Monte Carlo

This paper provides an algorithm that leverages MCMC to sample from the Boltzmann distribution of classical Ising models. It performs testing analysis on spin-glass Ising models ranging from 3 to 10 spins, and each set of spins featuring both a fully connected and line connected configuration.

[5]

Wavelength-Division Multiplexing Optical Ising Simulator

This paper uses a spatial photonic Ising machine to compute Ising Hamiltonians of programmable Ising models. To demonstrate the programming capabilities, testing is performed on $pm$ J models, Sherrington-Kirkpatrick (SK) models, and locally connected $J_1-J_2$ models

[8] (1,2,3,4,5,6,7,8,9,10,11)

Beyond-classical computation in quantum simulation

This paper uses seeks to establish quantum computing’s capability of handling complex computational problems, within the field of quantum simulation. With the problem scope focusing on the Ising model, they compare their QPU’s results against classical algorithms (Schrodinger equation dependent) on various spin-glass models. Specifically, testing was performed on 2D cylindrical spin-glass, 3D cubic spin-glass, 3D diamond spin-glass, and bi-clique spin-glass instances.

[9] (1,2,3,4,5)

Quantum-enhanced greedy combinatorial optimization solver

This paper introduces a quantum heuristic optimization algorithm for combinatorial optimization problems. Attempting to overcome noise constraints by reducing the problem to a classical greedy problem. Specifically, this heuristic algorithm is tested on Sherrington-Kirkpatrick spin-glass problems.

[11] (1,2,3,4,5,6,7,8,9,10,11)

Biq Mac Library

A dataset for QUBO and Max-Cut instances curated for the development of a QUBO and Max-Cut solver (Biq Mac). Within this dataset contains Ising model instances, specifically, Toroidal grid graphs and Ising chain instances.

[12] (1,2,3,4,5,6)

On Computational Complexity of 3D Ising Spin Glass: Lessons from D-Wave Annealer

This paper searches for the ground state of 3D spin-glass instances using annealing methods on the D-Wave quantum annealer. To demostrate the NP-hard complexity of 3D spin-glass instances, they performed testing on 3D instances ranging from 3558 to 5627 spins.

[14] (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)

Solving Statistical Mechanics Using Variational Autoregressive Networks

Ferromagnetic Dataset

Description of Ising Model Type

In a ferromagnetic Ising model, neighboring spins want to align. In other words, they want to point in the same direction to minimize the system’s energy. Uniformity, whether it be in the form of all spins pointing up or down, is a key trait of the ferromagnetic instance. The energy Hamiltonian of this model can be given by

\[E = -J \sum_{i<j}s_i s_j\]

If the neighboring spins point in the same direction, the energy of the system will decrease and, conversely arranged, will increase. Essential behaviors of the ferromagnetic model are the randomness of spins at high temperatures, and the aligned spins at lower temperatures.

Instances

The instances we are providing for the ferromagnetic Ising model type consist of weighted edges. Note that there are only positive weights present for these instances. The complexity of an instance is determined by its size (# of nodes).

2D

Instance	# of Instances	# of Spins	Complexity	Graph Geometry	Reference
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_

Dataset References

Below contain the references to the datasets we gathered on this website.

[13]

16. Mehta, M. Bukov, C.-H. Wang, A.G. Day, C. Richardson, C.K. Fisher, D.J. Schwab. A high-bias, low-variance introduction to machine learning for physicists. Physics Reports, 810 (2019), pp. 1-124, 10.1016/j.physrep.2019.03.001

This textbook introduces Machine Learning and its applications towards physics problems and research. Within the textbook, contains an Ising model dataset of locally-connected 2D Ising models.

Anti-Ferromagnetic Dataset

Description of Ising Model Type

In an anti-ferromagnetic Ising model, neighboring spins want to anti-align. In other words, they want to point in the opposite directions to minimize the system’s energy. The energy Hamiltonian of this model can be given by

\[E = -J \sum_{i<j}s_i s_j\]

If the neighboring spins point in the same direction, the energy of the system will increase and, conversely arranged, will decrease.

Instances

The instances we are providing for the anti-ferromagnetic Ising model type consist of weighted edges. Note that there are only negative weights present for these instances. The complexity of an instance is determined by its size (# of nodes).

2D

Instance	# of Instances	# of Spins	Complexity	Graph Geometry	Reference
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	10,000	1,600	Complex	Metropolis	[13]_
Anti-Ferromagnetic	1	256	Intermediate	…	[3]_
Anti-Ferromagnetic	1	196	Intermediate	…	[3]_
Anti-Ferromagnetic	1	144	Intermediate	…	[3]_
Anti-Ferromagnetic	1	100	Intermediate	…	[3]_
Anti-Ferromagnetic	1	64	Intermediate	…	[3]_
Anti-Ferromagnetic	1	36	Intermediate	…	[3]_
Anti-Ferromagnetic	1	16	Intermediate	…	[3]_

Dataset References

Below contain the references to the datasets we gathered on this website.

[3]

Finding the Ground State of Spin Hamiltonians with Reinforcement Learning

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Citation: Mills, K., Ronagh, P. & Tamblyn, I. Finding the ground state of spin Hamiltonians with reinforcement learning. Nat Mach Intell 2, 509–517 (2020). https://doi.org/10.1038/s42256-020-0226-x

This paper demonstrates that an RL agent is able to surpass the performance of standard heuristic temperature schedule for two classes of Hamiltonians. They show the performance of their implementation by training on weak-strong clusters (bipartite graph with two fully connected clusters) and nearest-neighbor square spin glasses.

[13]

A high-bias, low-variance introduction to Machine Learning for physicists

This textbook introduces Machine Learning and its applications towards physics problems and research. Within the textbook, contains an Ising model dataset of locally-connected 2D Ising models.

Spin-Ice Dataset

Description

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Instances

Instance	# of Instances	# of Spins	Complexity	Graph Geometry	Reference

Dataset References

Below contain the references to the datasets we gathered on this website.