TensorSuite

Real Tensor

Quantum Physics

This TensorSuite collection includes block-wise, block-sparse tensors from density matrix renormalization group (DMRG) workloads, where tensor contractions naturally expose structured sparsity patterns [1], [2].

The tensors are generated from a 2D Hubbard model with a 6 x 3 lattice, 18 sites, and bond dimension up to 1600 using ITensors.jl [3]. From the two-site DMRG update workflow, we include five representative contraction systems: E1, E2, S1, S2, and S3. The collection covers four symmetry regimes: kysznf, sznf, nf, and nfparity.

For each contraction, TensorSuite lists both the canonical tensor layout and an internal permuted layout. The canonical layout follows the DMRG-generated index order, while the permuted layout groups contracted indices to expose larger block operations, a common strategy for improving tensor contraction performance on modern hardware [3], [4], [5].

Tensor Pairs

Each contraction system contains a tensor A and tensor B pair. The pairs are grouped by layout so links can be attached separately for the canonical formulation and the internal permuted layout. The tensor contractions are expressed in einsum notation below.

Canonical Formulation

Einsum equations: E1: dea,fag->defg   E2: fbag,bhai->fghi   S1: dae,fga->defg   S2: dea,fagh->defgh   S3: fbghai,baj->fghij

Symmetry System Tensor A Tensor B
kysznf E1 canonical_formulation_kysznf_E1_A [Link]
order: 3, dim: 1600×20×1600, nnz: 836,286, nnz_block: 1,717		 canonical_formulation_kysznf_E1_B [Link]
order: 3, dim: 4×1600×1599, nnz: 183,811, nnz_block: 379
kysznf E2 canonical_formulation_kysznf_E2_A [Link]
order: 4, dim: 1600×20×4×1599, nnz: 3,133,605, nnz_block: 6,159		 canonical_formulation_kysznf_E2_B [Link]
order: 4, dim: 20×20×4×4, nnz: 119, nnz_block: 101
kysznf S1 canonical_formulation_kysznf_S1_A [Link]
order: 3, dim: 4×1600×1599, nnz: 188,454, nnz_block: 361		 canonical_formulation_kysznf_S1_B [Link]
order: 3, dim: 4×1598×1600, nnz: 189,096, nnz_block: 353
kysznf S2 canonical_formulation_kysznf_S2_A [Link]
order: 3, dim: 1599×20×1599, nnz: 859,824, nnz_block: 1,720		 canonical_formulation_kysznf_S2_B [Link]
order: 4, dim: 4×1599×4×1598, nnz: 692,685, nnz_block: 1,310
kysznf S3 canonical_formulation_kysznf_S3_A [Link]
order: 6, dim: 1599×1598×20×4×20×4, nnz: 13,847,461, nnz_block: 24,193		 canonical_formulation_kysznf_S3_B [Link]
order: 3, dim: 1598×20×1598, nnz: 872,892, nnz_block: 1,661
sznf E1 canonical_formulation_sznf_E1_A [Link]
order: 3, dim: 1597×20×1597, nnz: 2,494,370, nnz_block: 372		 canonical_formulation_sznf_E1_B [Link]
order: 3, dim: 4×1597×1600, nnz: 550,488, nnz_block: 138
sznf E2 canonical_formulation_sznf_E2_A [Link]
order: 4, dim: 1597×20×4×1600, nnz: 9,388,237, nnz_block: 1,326		 canonical_formulation_sznf_E2_B [Link]
order: 4, dim: 20×20×4×4, nnz: 327, nnz_block: 85
sznf S1 canonical_formulation_sznf_S1_A [Link]
order: 3, dim: 4×1600×1600, nnz: 566,262, nnz_block: 126		 canonical_formulation_sznf_S1_B [Link]
order: 3, dim: 4×1597×1600, nnz: 566,696, nnz_block: 126
sznf S2 canonical_formulation_sznf_S2_A [Link]
order: 3, dim: 1600×20×1600, nnz: 2,583,778, nnz_block: 361		 canonical_formulation_sznf_S2_B [Link]
order: 4, dim: 4×1600×4×1597, nnz: 2,077,091, nnz_block: 478
sznf S3 canonical_formulation_sznf_S3_A [Link]
order: 6, dim: 1600×1597×20×4×20×4, nnz: 127,625,198, nnz_block: 7,996		 canonical_formulation_sznf_S3_B [Link]
order: 3, dim: 1597×20×1597, nnz: 2,608,738, nnz_block: 361
nf E1 canonical_formulation_nf_E1_A [Link]
order: 3, dim: 1599×20×1599, nnz: 8,732,294, nnz_block: 57		 canonical_formulation_nf_E1_B [Link]
order: 3, dim: 4×1599×1599, nnz: 1,849,041, nnz_block: 25
nf E2 canonical_formulation_nf_E2_A [Link]
order: 4, dim: 1599×20×4×1599, nnz: 33,651,549, nnz_block: 161		 canonical_formulation_nf_E2_B [Link]
order: 4, dim: 20×20×4×4, nnz: 849, nnz_block: 39
nf S1 canonical_formulation_nf_S1_A [Link]
order: 3, dim: 4×1600×1599, nnz: 1,882,422, nnz_block: 25		 canonical_formulation_nf_S1_B [Link]
order: 3, dim: 4×1600×1600, nnz: 1,873,919, nnz_block: 25
nf S2 canonical_formulation_nf_S2_A [Link]
order: 3, dim: 1599×20×1599, nnz: 8,906,772, nnz_block: 57		 canonical_formulation_nf_S2_B [Link]
order: 4, dim: 4×1599×4×1600, nnz: 7,126,739, nnz_block: 71
nf S3 canonical_formulation_nf_S3_A [Link]
order: 6, dim: 1599×1600×20×4×20×4, nnz: 736,523,150, nnz_block: 794		 canonical_formulation_nf_S3_B [Link]
order: 3, dim: 1600×20×1600, nnz: 8,903,114, nnz_block: 57
nfparity E1 canonical_formulation_nfparity_E1_A [Link]
order: 3, dim: 1600×20×1600, nnz: 25,600,000, nnz_block: 8		 canonical_formulation_nfparity_E1_B [Link]
order: 3, dim: 4×1600×1599, nnz: 5,116,800, nnz_block: 6
nfparity E2 canonical_formulation_nfparity_E2_A [Link]
order: 4, dim: 1600×20×4×1599, nnz: 102,336,000, nnz_block: 24		 canonical_formulation_nfparity_E2_B [Link]
order: 4, dim: 20×20×4×4, nnz: 2,789, nnz_block: 36
nfparity S1 canonical_formulation_nfparity_S1_A [Link]
order: 3, dim: 4×1599×1599, nnz: 5,113,602, nnz_block: 6		 canonical_formulation_nfparity_S1_B [Link]
order: 3, dim: 4×1599×1599, nnz: 5,113,602, nnz_block: 6
nfparity S2 canonical_formulation_nfparity_S2_A [Link]
order: 3, dim: 1599×20×1599, nnz: 25,567,960, nnz_block: 8		 canonical_formulation_nfparity_S2_B [Link]
order: 4, dim: 4×1599×4×1599, nnz: 20,454,408, nnz_block: 18
nfparity S3 canonical_formulation_nfparity_S3_A [Link]
order: 6, dim: 1599×1599×20×4×20×4, nnz: 3,743,156,664, nnz_block: 144		 canonical_formulation_nfparity_S3_B [Link]
order: 3, dim: 1599×20×1599, nnz: 25,567,992, nnz_block: 8

Internal Permuted Layout

Einsum equations: E1 / E2 / S1: ade,afg->defg   S2: ade,afgh->defgh   S3: adefg,ah->defgh

Symmetry System Tensor A Tensor B
kysznf E1 block_permuted_internal_only_kysznf_E1_A [Link]
order: 3, dim: 1600×1600×20, nnz: 836,286, nnz_block: 1,717		 block_permuted_internal_only_kysznf_E1_B [Link]
order: 3, dim: 1600×4×1599, nnz: 183,811, nnz_block: 379
kysznf E2 block_permuted_internal_only_kysznf_E2_A [Link]
order: 3, dim: 80×1600×1599, nnz: 3,136,409, nnz_block: 3,225		 block_permuted_internal_only_kysznf_E2_B [Link]
order: 3, dim: 80×20×4, nnz: 244, nnz_block: 66
kysznf S1 block_permuted_internal_only_kysznf_S1_A [Link]
order: 3, dim: 1600×4×1599, nnz: 188,454, nnz_block: 361		 block_permuted_internal_only_kysznf_S1_B [Link]
order: 3, dim: 1600×4×1598, nnz: 189,096, nnz_block: 353
kysznf S2 block_permuted_internal_only_kysznf_S2_A [Link]
order: 3, dim: 1599×1599×20, nnz: 859,824, nnz_block: 1,720		 block_permuted_internal_only_kysznf_S2_B [Link]
order: 4, dim: 1599×4×4×1598, nnz: 692,685, nnz_block: 1,310
kysznf S3 block_permuted_internal_only_kysznf_S3_A [Link]
order: 5, dim: 31960×1599×20×4×4, nnz: 142,653,316, nnz_block: 15,797		 block_permuted_internal_only_kysznf_S3_B [Link]
order: 2, dim: 31960×1598, nnz: 872,892, nnz_block: 107
sznf E1 block_permuted_internal_only_sznf_E1_A [Link]
order: 3, dim: 1597×1597×20, nnz: 2,494,370, nnz_block: 372		 block_permuted_internal_only_sznf_E1_B [Link]
order: 3, dim: 1597×4×1600, nnz: 550,488, nnz_block: 138
sznf E2 block_permuted_internal_only_sznf_E2_A [Link]
order: 3, dim: 80×1597×1600, nnz: 9,391,335, nnz_block: 462		 block_permuted_internal_only_sznf_E2_B [Link]
order: 3, dim: 80×20×4, nnz: 614, nnz_block: 42
sznf S1 block_permuted_internal_only_sznf_S1_A [Link]
order: 3, dim: 1600×4×1600, nnz: 566,262, nnz_block: 126		 block_permuted_internal_only_sznf_S1_B [Link]
order: 3, dim: 1600×4×1597, nnz: 566,696, nnz_block: 126
sznf S2 block_permuted_internal_only_sznf_S2_A [Link]
order: 3, dim: 1600×1600×20, nnz: 2,583,778, nnz_block: 361		 block_permuted_internal_only_sznf_S2_B [Link]
order: 4, dim: 1600×4×4×1597, nnz: 2,077,091, nnz_block: 478
sznf S3 block_permuted_internal_only_sznf_S3_A [Link]
order: 5, dim: 31940×1600×20×4×4, nnz: 536,305,715, nnz_block: 4,000		 block_permuted_internal_only_sznf_S3_B [Link]
order: 2, dim: 31940×1597, nnz: 2,608,738, nnz_block: 39
nf E1 block_permuted_internal_only_nf_E1_A [Link]
order: 3, dim: 1599×1599×20, nnz: 8,732,294, nnz_block: 57		 block_permuted_internal_only_nf_E1_B [Link]
order: 3, dim: 1599×4×1599, nnz: 1,849,041, nnz_block: 25
nf E2 block_permuted_internal_only_nf_E2_A [Link]
order: 3, dim: 80×1599×1599, nnz: 33,653,602, nnz_block: 51		 block_permuted_internal_only_nf_E2_B [Link]
order: 3, dim: 80×20×4, nnz: 1,426, nnz_block: 19
nf S1 block_permuted_internal_only_nf_S1_A [Link]
order: 3, dim: 1600×4×1599, nnz: 1,882,422, nnz_block: 25		 block_permuted_internal_only_nf_S1_B [Link]
order: 3, dim: 1600×4×1600, nnz: 1,873,919, nnz_block: 25
nf S2 block_permuted_internal_only_nf_S2_A [Link]
order: 3, dim: 1599×1599×20, nnz: 8,906,772, nnz_block: 57		 block_permuted_internal_only_nf_S2_B [Link]
order: 4, dim: 1599×4×4×1600, nnz: 7,126,739, nnz_block: 71
nf S3 block_permuted_internal_only_nf_S3_A [Link]
order: 5, dim: 32000×1599×20×4×4, nnz: 2,172,410,244, nnz_block: 394		 block_permuted_internal_only_nf_S3_B [Link]
order: 2, dim: 32000×1600, nnz: 8,903,114, nnz_block: 9
nfparity E1 block_permuted_internal_only_nfparity_E1_A [Link]
order: 3, dim: 1600×1600×20, nnz: 25,600,000, nnz_block: 8		 block_permuted_internal_only_nfparity_E1_B [Link]
order: 3, dim: 1600×4×1599, nnz: 5,116,800, nnz_block: 6
nfparity E2 block_permuted_internal_only_nfparity_E2_A [Link]
order: 3, dim: 80×1600×1599, nnz: 102,336,000, nnz_block: 4		 block_permuted_internal_only_nfparity_E2_B [Link]
order: 3, dim: 80×20×4, nnz: 3,200, nnz_block: 12
nfparity S1 block_permuted_internal_only_nfparity_S1_A [Link]
order: 3, dim: 1599×4×1599, nnz: 5,113,602, nnz_block: 6		 block_permuted_internal_only_nfparity_S1_B [Link]
order: 3, dim: 1599×4×1599, nnz: 5,113,602, nnz_block: 6
nfparity S2 block_permuted_internal_only_nfparity_S2_A [Link]
order: 3, dim: 1599×1599×20, nnz: 25,567,960, nnz_block: 8		 block_permuted_internal_only_nfparity_S2_B [Link]
order: 4, dim: 1599×4×4×1599, nnz: 20,454,408, nnz_block: 18
nfparity S3 block_permuted_internal_only_nfparity_S3_A [Link]
order: 5, dim: 31980×1599×20×4×4, nnz: 8,181,763,200, nnz_block: 72		 block_permuted_internal_only_nfparity_S3_B [Link]
order: 2, dim: 31980×1599, nnz: 25,567,992, nnz_block: 2

References

[1] U. Schollwock, "The density-matrix renormalization group in the age of matrix product states," Annals of Physics, 2011. DOI

[2] R. Levy, E. Solomonik, and B. K. Clark, "Distributed-memory DMRG via sparse and dense parallel tensor contractions," SC20, 2020. DOI

[3] M. Fishman, S. White, and E. Stoudenmire, "The ITensor software library for tensor network calculations," SciPost Physics Codebases, 2022. DOI

[4] P. Springer and P. Bientinesi, "Design of a high-performance GEMM-like tensor-tensor multiplication," ACM Transactions on Mathematical Software, 2018. DOI

[5] I. A. Kaliman and A. I. Krylov, "New algorithm for tensor contractions on multi-core CPUs, GPUs, and accelerators enables CCSD and EOM-CCSD calculations with over 1000 basis functions on a single compute node," Journal of Computational Chemistry, 2017. DOI

Acknowledgments

We thank Karl Pierce and Miles Stoudenmire for their help in gathering the data.