The best published ground-state energies of open problems

Hamiltonian

Heisenberg#

22 instances, energies as E/N (S.S)

lattice size

instance	energy to beat	method	source	record (estimate)	results	verifier
chain, 200 sites, open	-0.4422078	DMRG	run script, Wu et al. (2024)		3	not yet (chain lattice)
chain, 200 sites	-0.4431678	DMRG	run script, Wu et al. (2024)		4	not yet (chain lattice)
kagome 6x6 (108 sites)	no variational energy yet; the first verified state sets it			-0.4383(3) extrapolated	2	not yet (kagome-6x6 lattice)
kagome 8x8 (192 sites)	-0.42987(1)	VMC	run script, Wu et al. (2024)		4	not yet (kagome-8x8 lattice)
pyrochlore 2x2x2 (128 sites)	-0.49229(7)	mVMC + RBM	Pohle et al. (2023)	-0.49434(5) extrapolated	8	not yet (pyrochlore-2x2x2 lattice)
pyrochlore 3x3x3 (108 sites)	-0.4871(1)	mVMC	Astrakhantsev et al. (2021)		5	not yet (pyrochlore-3x3x3 lattice)
pyrochlore 3x3x3 (432 sites)	-0.48851(3)	mVMC	Pohle et al. (2023)	-0.4924(2) extrapolated	5	not yet (pyrochlore-3x3x3 lattice)
pyrochlore 4x4x4 (256 sites)	-0.485584(3)	VMC	Cheng & Li (2026)		3	not yet (pyrochlore-4x4x4 lattice)
pyrochlore 4x4x4 (1024 sites)	-0.48800(1)	mVMC	Pohle et al. (2023)	-0.4921(2) extrapolated	3	not yet (pyrochlore-4x4x4 lattice)
shuriken, 96 sites	-0.43826(1)	mVMC	Wu et al. (2024)		1	not yet (shuriken lattice)
shuriken, 216 sites	-0.4376(1)	mVMC	Wu et al. (2024)		1	not yet (shuriken lattice)
shuriken, 384 sites	-0.4371(1)	mVMC	Wu et al. (2024)		1	not yet (shuriken lattice)
triangular 10x10, open	-0.5147668	DMRG	run script, Wu et al. (2024)		7	not yet (triangular lattice)
triangular, 108 sites	-0.55315(3)	GCNN	Roth et al. (2022)		2	not yet (triangular lattice)
triangular 12x12, open	-0.5150648	DMRG	run script, Wu et al. (2024)		5	not yet (triangular lattice)
triangular 12x12	-0.5493(1)	2D RNN	Moss et al. (2025)		6	not yet (triangular lattice)
triangular 14x14, open	-0.5138(1)	2D RNN	Wu et al. (2024)		5	not yet (triangular lattice)
triangular 16x16, open	-0.51668(9)	2D RNN	Wu et al. (2024)		4	not yet (triangular lattice)
triangular 18x18	-0.551703(1)	ViT	Viteritti et al. (2026)	-0.55200(1) extrapolated	8	not yet (triangular lattice)
triangular 24x24	-0.551602(1)	ViT	Viteritti et al. (2026)	-0.55189(1) extrapolated	3	not yet (triangular lattice)
triangular 30x30	-0.551528(1)	ViT	Viteritti et al. (2026)	-0.55180(1) extrapolated	4	not yet (triangular lattice)
triangular 42x42	-0.551429(1)	ViT	Viteritti et al. (2026)	-0.55171(1) extrapolated	2	not yet (triangular lattice)

J1-J2#

66 instances, energies as E/N (S.S)

lattice size

instance	energy to beat	method	source	record (estimate)	results	verifier
square 8x8, J2 = 0.1	-0.633093	ViT	Golubev et al. (2026)		1	yes
square 8x8, J2 = 0.2	-0.594508	ViT	Golubev et al. (2026)		1	yes
square 8x8, J2 = 0.3	-0.558295	ViT	Golubev et al. (2026)		1	yes
square 8x8, J2 = 0.4	-0.525653(1)	RBM + PP	Nomura & Imada (2021)	-0.5262(1) extrapolated	10	yes
square 8x8, J2 = 0.45	-0.511331(1)	RBM + PP	Nomura & Imada (2021)	-0.5116(1) extrapolated	11	yes
square 8x8, J2 = 0.48	no variational energy yet; the first verified state sets it			-0.5034(4) extrapolated	1	yes
square 8x8, J2 = 0.5	-0.498963(2)	RBM + PP	Wu et al. (2024)	-0.4992(1) extrapolated	27	yes
square 8x8, J2 = 0.52	no variational energy yet; the first verified state sets it			-0.4941(1) extrapolated	1	yes
square 8x8, J2 = 0.55	-0.488820(2)	RBM + PP	Nomura & Imada (2021)	-0.4891(1) extrapolated	13	yes
square 8x8, J2 = 0.6	-0.481606	ViT	Golubev et al. (2026)		2	yes
square 8x8, J2 = 0.65	-0.498312	ViT	Golubev et al. (2026)		1	yes
square 8x8, J2 = 0.7	-0.522091	ViT	Golubev et al. (2026)		1	yes
square 8x8, J2 = 0.8	-0.57731	ViT	Golubev et al. (2026)		1	yes
square 8x8, J2 = 0.9	-0.637518	ViT	Golubev et al. (2026)		1	yes
square 8x8, J2 = 1	-0.700508	ViT	Golubev et al. (2026)		1	yes
square 10x10, J2 = 0.1	-0.631365	ViT	Golubev et al. (2026)		1	yes
square 10x10, J2 = 0.2	-0.592982	ViT	Golubev et al. (2026)		3	yes
square 10x10, J2 = 0.3	-0.556912	ViT	Golubev et al. (2026)		5	yes
square 10x10, J2 = 0.4	-0.524361	ViT	Golubev et al. (2026)	-0.5253 extrapolated	16	yes
square 10x10, J2 = 0.44	-0.51273	CNN	10.1109/TPDS.2022.3145163,		1	yes
square 10x10, J2 = 0.45	-0.51001(1)	VMC	Hu et al. (2013)	-0.511 extrapolated	13	yes
square 10x10, J2 = 0.46	-0.50744	CNN	Liang et al. (2023)		2	yes
square 10x10, J2 = 0.48	-0.50231	CNN	Liang et al. (2023)		2	yes
square 10x10, J2 = 0.5	-0.4976939(2)	CNN-MPS	Fan et al. (2026)	-0.4988 extrapolated	64	yes
square 10x10, J2 = 0.55	-0.48693(3)	VMC	Hu et al. (2013)	-0.488 extrapolated	21	yes
square 10x10, J2 = 0.6	-0.47839	CNN	Liang et al. (2023)		9	yes
square 10x10, J2 = 0.65	-0.494672	ViT	Golubev et al. (2026)		1	yes
square 10x10, J2 = 0.7	-0.519035	ViT	Golubev et al. (2026)		5	yes
square 10x10, J2 = 0.8	-0.57413	ViT	Golubev et al. (2026)		7	yes
square 10x10, J2 = 0.9	-0.63417	ViT	Golubev et al. (2026)		4	yes
square 10x10, J2 = 1	-0.69670(2)	RBM	Chen et al. (2022)		7	yes
square 12x12, J2 = 0.44	-0.51225	CNN	10.1109/TPDS.2022.3145163,		1	yes
square 12x12, J2 = 0.45	no variational energy yet; the first verified state sets it			-0.5088(1) extrapolated	1	yes
square 12x12, J2 = 0.46	-0.50697	CNN	Liang et al. (2023)		2	yes
square 12x12, J2 = 0.48	-0.50183	CNN	Liang et al. (2023)		3	yes
square 12x12, J2 = 0.5	-0.496989	CNN	Liang et al. (2023)		12	yes
square 12x12, J2 = 0.52	no variational energy yet; the first verified state sets it			-0.4920(1) extrapolated	1	yes
square 12x12, J2 = 0.55	-0.48607(1)	GCNN	Roth et al. (2022)	-0.4862(2) extrapolated	5	yes
square 12x12, J2 = 0.6	-0.47715	CNN	Liang et al. (2023)	-0.4784(3) extrapolated	3	yes
square 14x14, J2 = 0.4	-0.52287(1)	VMC	Hu et al. (2013)	-0.52351(2) extrapolated	3	yes
square 14x14, J2 = 0.44	-0.512	CNN	10.1109/TPDS.2022.3145163,		1	yes
square 14x14, J2 = 0.45	-0.50899(1)	VMC	Hu et al. (2013)	-0.50953(1) extrapolated	3	yes
square 14x14, J2 = 0.46	-0.50675	CNN	Liang et al. (2023)		2	yes
square 14x14, J2 = 0.48	-0.50159	CNN	Liang et al. (2023)		2	yes
square 14x14, J2 = 0.5	-0.49674	CNN	Liang et al. (2023)	-0.49722(2) extrapolated	13	yes
square 14x14, J2 = 0.55	-0.48581	CNN	Liang et al. (2023)	-0.48696(5) extrapolated	5	yes
square 14x14, J2 = 0.6	-0.47678	CNN	Liang et al. (2023)		2	yes
square 16x16, J2 = 0.44	-0.51188	CNN	10.1109/TPDS.2022.3145163,		1	yes
square 16x16, J2 = 0.45	no variational energy yet; the first verified state sets it			-0.5087(1) extrapolated	1	yes
square 16x16, J2 = 0.46	-0.50662	CNN	Liang et al. (2023)		2	yes
square 16x16, J2 = 0.48	-0.50147	CNN	Liang et al. (2023)		3	yes
square 16x16, J2 = 0.5	-0.4969140(5)	CNN-MPS	Fan et al. (2026)		11	yes
square 16x16, J2 = 0.52	no variational energy yet; the first verified state sets it			-0.4920(1) extrapolated	1	yes
square 16x16, J2 = 0.55	-0.48565	CNN	Liang et al. (2023)	-0.4859(4) extrapolated	5	yes
square 16x16, J2 = 0.6	-0.47656	CNN	Liang et al. (2023)	-0.4777(3) extrapolated	3	yes
square 18x18, J2 = 0.4	-0.52259(1)	VMC	Hu et al. (2013)	-0.52333(1) extrapolated	3	yes
square 18x18, J2 = 0.44	-0.51179	CNN	10.1109/TPDS.2022.3145163,		1	yes
square 18x18, J2 = 0.45	-0.50874(1)	VMC	Hu et al. (2013)	-0.50940(1) extrapolated	3	yes
square 18x18, J2 = 0.46	-0.50651	CNN	Liang et al. (2023)		2	yes
square 18x18, J2 = 0.48	-0.50139	CNN	Liang et al. (2023)		2	yes
square 18x18, J2 = 0.5	-0.496513	CNN	Liang et al. (2023)	-0.49717(2) extrapolated	6	yes
square 18x18, J2 = 0.55	-0.48557	CNN	Liang et al. (2023)	-0.48698(5) extrapolated	5	yes
square 18x18, J2 = 0.6	-0.47652	CNN	Liang et al. (2023)		2	yes
square 20x20, J2 = 0.5	-0.4967987(6)	CNN-MPS	Fan et al. (2026)	-0.49684(1) extrapolated	3	yes
triangular, 108 sites, J2 = 0.125	-0.51268(9)	GCNN	Roth et al. (2022)		3	not yet (triangular lattice)
triangular 12x12, J2 = 0.125	-0.51218(9)	GCNN	Roth et al. (2022)		3	not yet (triangular lattice)

Hubbard#

62 instances, energies as E/site

lattice size

instance	energy to beat	method	source	record (estimate)	results	verifier
rectangular 14x16, U = 8, n = 0.875	-0.75830(4)	VAFQMC	Wu et al. (2024)		1	yes
rectangular 4x12, U = 8, n = 0.875	-0.76608(1)	HFPS	Chen et al. (2026)		1	yes
rectangular 4x16, U = 8, n = 0.875	-0.76623(1)	ACE	Gu et al. (2026)	-0.76748 extrapolated	22	yes
rectangular 4x16, periodic/open, U = 2, n = 0.8125	-1.2852656	DMRG	run script, Wu et al. (2024)		1	yes
rectangular 4x16, periodic/open, U = 8, n = 0.8125	-0.853	DMRG	run script, Wu et al. (2024)		1	yes
rectangular 4x16, periodic/open, U = 2, n = 0.875	-1.2611875	DMRG	run script, Wu et al. (2024)		1	yes
rectangular 4x16, periodic/open, U = 8, n = 0.875	-0.7508328	DMRG	run script, Wu et al. (2024)	-0.75114(6) extrapolated	4	yes
rectangular 4x16, periodic/open, U = 2, n = 0.9375	-1.2199062	DMRG	run script, Wu et al. (2024)		1	yes
rectangular 4x16, periodic/open, U = 8, n = 0.9375	-0.637875	DMRG	run script, Wu et al. (2024)		1	yes
rectangular 4x16, periodic/open, U = 2, n = 1	-1.1710781	DMRG	run script, Wu et al. (2024)		1	yes
rectangular 4x16, periodic/open, U = 8, n = 1	-0.5144219	DMRG	run script, Wu et al. (2024)		1	yes
rectangular 4x8, U = 8, n = 0.875	-0.768337(2)	NNBF	Loehr & Clark (2025)	-0.768416 extrapolated	12	yes
rectangular 4x8, periodic/open, U = 8, n = 0.875	-0.7363906	DMRG	run script, Wu et al. (2024)	-0.736431(9) extrapolated	7	yes
rectangular 6x16, U = 8, n = 0.875	-0.7486	VMC	Marino et al. (2022)		2	yes
rectangular 6x32, U = 8, n = 0.875	-0.7483	VMC	Marino et al. (2022)		2	yes
rectangular 6x48, U = 8, n ≈ 0.8333	-0.82075(3)	VMC	run script, Wu et al. (2024)	-0.82575(2) projected	4	yes
rectangular 6x48, U = 8, n = 0.875	-0.74834(3)	VMC	run script, Wu et al. (2024)	-0.75398(2) projected	6	yes
rectangular 6x48, U = 12, n = 0.875	-0.62556(3)	VMC	run script, Wu et al. (2024)	-0.63217(2) projected	6	yes
rectangular 6x48, U = 8, n ≈ 0.9167	-0.67269(3)	VMC	run script, Wu et al. (2024)	-0.6862(2) projected	7	yes
rectangular 6x64, U = 8, n = 0.875	-0.7482	VMC	Marino et al. (2022)		2	yes
rectangular 8x16, U = 8, n = 0.875	-0.7568	Tensor backflow	Liang (2025)		2	yes
rectangular 8x24, U = 8, n = 0.875	-0.7546	Tensor backflow	Liang (2025)		2	yes
rectangular 8x32, U = 8, n = 0.875	-0.7547	Tensor backflow	Liang (2025)		2	yes
rectangular 8x48, U = 8, n ≈ 0.9167	no variational energy yet; the first verified state sets it			-0.6854(2) projected	3	yes
square 6x6, U = 4, n ≈ 0.6667	-1.1834(1)	VAFQMC	Levy et al. (2023)	-1.18525(4) projected	2	yes
square 6x6, U = 4, n ≈ 1.3333	no variational energy yet; the first verified state sets it			0.14809(4) projected	1	yes
square, 50 sites, U = 8, n = 0.84	-0.82328(3)	VAFQMC	Wu et al. (2024)		1	yes
square 8x8, U = 4, n = 0.4375	no variational energy yet; the first verified state sets it			-1.07180(2) projected	1	yes
square 8x8, U = 4, n = 0.6875	-1.1846	Tensor backflow	Liang (2025)	-1.1858(2) projected	4	yes
square 8x8, U = 4, n ≈ 0.7813	-1.13048(8)	VAFQMC	Levy et al. (2023)	-1.13253(3) projected	5	yes
square 8x8, U = 8, n ≈ 0.7813	-0.91483(2)	mVMC + RBM	run script, Wu et al. (2024)	-0.9252(1) projected	5	yes
square 8x8, U = 4, n = 0.875	-1.01391(8)	VAFQMC	Levy et al. (2023)	-1.01923(6) projected	4	yes
square 8x8, U = 8, n = 0.875	-0.7469(2)	VAFQMC	Levy et al. (2023)	-0.7616(1) projected	10	yes
square 8x8, U = 8, n = 0.875, t12 (t2 = -0.2)	-0.75058(1)	Transformer backflow	Viteritti et al. (2026)	-0.7540(4) extrapolated	19	yes
square 10x10, U = 4, n = 0.8	-1.11097(6)	VAFQMC	Levy et al. (2023)	-1.1135(2) projected	4	yes
square 12x12, U = 4, n ≈ 0.8056	-1.1061	Tensor backflow	Liang (2025)	-1.1089(2) projected	3	yes
square 12x12, U = 8, n = 0.875, t12 (t2 = -0.2)	-0.74425(1)	SBP	Rende et al. (2026)		1	yes
square 16x16, U = 8, n ≈ 0.8281	-0.83788(6)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, U = 8, n ≈ 0.8438	-0.81198(6)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, U = 8, n ≈ 0.8516	-0.79896(5)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, U = 8, n ≈ 0.8594	-0.78530(6)	VAFQMC	Wu et al. (2024)		2	yes
square 16x16, U = 8, n ≈ 0.8672	-0.77196(6)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, U = 4, n = 0.875	-1.02916(2)	VAFQMC	Wu et al. (2024)		2	yes
square 16x16, U = 6, n = 0.875	-0.86592(2)	VAFQMC	Wu et al. (2024)		2	yes
square 16x16, U = 8, n = 0.875	-0.75865(3)	VAFQMC	Wu et al. (2024)		19	yes
square 16x16, U = 8, n = 0.875, t12 (t2 = -0.1)	-0.7370(1)	HFPS	Roth et al. (2025)		2	yes
square 16x16, U = 8, n = 0.875, t12 (t2 = -0.2)	-0.74411(1)	SBP	Rende et al. (2026)		15	yes
square 16x16, U = 8, n = 0.875, t12 (t2 = -0.3)	-0.7375(1)	HFPS	Roth et al. (2025)		2	yes
square 16x16, U = 8, n ≈ 0.8828	-0.74455(3)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, U = 4, n ≈ 0.8906	-1.00849(2)	VAFQMC	Wu et al. (2024)		2	yes
square 16x16, U = 6, n ≈ 0.8906	-0.84058(2)	VAFQMC	Wu et al. (2024)		2	yes
square 16x16, U = 8, n ≈ 0.8906	-0.73038(5)	VAFQMC	Wu et al. (2024)		2	yes
square 16x16, U = 8, n ≈ 0.8984	-0.71584(5)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, U = 8, n ≈ 0.9063	-0.70135(3)	VAFQMC	Wu et al. (2024)		2	yes
square 16x16, U = 8, n ≈ 0.9141	-0.68678(5)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, U = 8, n ≈ 0.9219	-0.67215(5)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, U = 8, n = 0.9375	-0.64053(5)	VAFQMC	Wu et al. (2024)		1	yes
square 16x16, periodic/antiperiodic, U = 8, n ≈ 0.8359	-0.82333(3)	VAFQMC	Wu et al. (2024)		2	yes
square 16x16, periodic/antiperiodic, U = 8, n ≈ 0.8359, t12 (t2 = -0.25)	-0.79937(4)	mVMC + RBM	Wu et al. (2024)		1	yes
square 16x16, periodic/antiperiodic, U = 8, n ≈ 0.8359, t12 (t2 = -0.25), UV1V2	1.25918(3)	mVMC + RBM	Wu et al. (2024)		1	yes
square 20x20, U = 8, n = 0.875, t12 (t2 = -0.2)	-0.74382(1)	SBP	Rende et al. (2026)		1	yes
square 24x24, U = 8, n = 0.875, t12 (t2 = -0.2)	-0.74397(1)	SBP	Rende et al. (2026)		1	yes

Transverse-field Ising#

6 instances, energies as E/N (Pauli)

lattice size

instance	energy to beat	method	source	results	verifier
square 8x8, open, h = 3	-3.1641828	DMRG	Dao (2026)	9	not yet (TFIsing model)
square 10x10, open, h = 3	-3.16977	DMRG	Dao (2026)	9	not yet (TFIsing model)
square 12x12, open, h = 2	-2.4096028	DMRG	Dao (2026)	4	not yet (TFIsing model)
square 12x12, open, h = 3	-3.1739018(2)	2D RNN	Hibat-Allah et al. (2020)	15	not yet (TFIsing model)
square 12x12, open, h = 4	-4.1217979	DMRG	Dao (2026)	4	not yet (TFIsing model)
square 20x20, open, h = 3	-3.18197	DMRG	Ganahl et al. (2023)	7	not yet (TFIsing model)

Impurity#

8 instances, energies as E (total)

lattice size

instance	energy to beat	method	source	results	verifier
TB-DMFT, size 9	-112.9042852	DMRG	run script, Wu et al. (2024)	3	not yet (Impurity model)
TB-DMFT, size 19	-224.3360713	DMRG	run script, Wu et al. (2024)	3	not yet (Impurity model)
TB-DMFT, size 119	-1307.5645417	DMRG	run script, Wu et al. (2024)	3	not yet (Impurity model)
TB-DMFT, size 309	-3378.6762368	DMRG	run script, Wu et al. (2024)	3	not yet (Impurity model)
TB-DMFT-SOC, size 9	-112.8717703	DMRG	run script, Wu et al. (2024)	3	not yet (Impurity model)
TB-DMFT-SOC, size 19	-219.766911	DMRG	run script, Wu et al. (2024)	3	not yet (Impurity model)
TB-DMFT-SOC, size 119	-1305.3902308	DMRG	run script, Wu et al. (2024)	3	not yet (Impurity model)
TB-DMFT-SOC, size 309	-3351.7573592	DMRG	run script, Wu et al. (2024)	3	not yet (Impurity model)

Electron gas, 3D#

112 instances, energies as E/N (Ha)

lattice size

instance	energy to beat	method	source	record (estimate)	results	verifier
bcc cell, N = 27, rs = 0.2, polarized	44.18067(2)	FermiNet	10.1103/PhysRevLett.130.036401		5	not yet (HEG3D model)
bcc cell, N = 27, rs = 0.5, polarized	6.318704(2)	FermiNet	10.1103/PhysRevLett.130.036401		5	not yet (HEG3D model)
bcc cell, N = 27, rs = 1, polarized	1.261547(1)	FermiNet	10.1103/PhysRevLett.130.036401		5	not yet (HEG3D model)
bcc cell, N = 27, rs = 10, polarized	-0.050320(5)	FermiNet	10.1103/PhysRevLett.130.036401	-0.0503349(8) projected	5	not yet (HEG3D model)
bcc cell, N = 27, rs = 2, polarized	0.152086(1)	FermiNet	10.1103/PhysRevLett.130.036401	0.152077(4) projected	5	not yet (HEG3D model)
bcc cell, N = 27, rs = 3, polarized	-0.006961(2)	FermiNet	10.1103/PhysRevLett.130.036401	-0.006966(2) projected	5	not yet (HEG3D model)
bcc cell, N = 27, rs = 30, polarized	-0.022541(5)	FermiNet	10.1103/PhysRevLett.130.036401	-0.0225620(2) projected	5	not yet (HEG3D model)
bcc cell, N = 27, rs = 5, polarized	-0.057691(4)	FermiNet	10.1103/PhysRevLett.130.036401	-0.057702(1) projected	5	not yet (HEG3D model)
bcc cell, N = 27, rs = 50, polarized	-0.014478(4)	FermiNet	10.1103/PhysRevLett.130.036401	-0.0144918(1) projected	5	not yet (HEG3D model)
bcc cell, N = 27, rs = 70, polarized	-0.010693(3)	FermiNet	10.1103/PhysRevLett.130.036401	-0.0107118(1) projected	5	not yet (HEG3D model)
bcc cell, N = 27, rs = 90, polarized	-0.008496(1)	FermiNet	10.1103/PhysRevLett.130.036401	-0.0085137(2) projected	11	not yet (HEG3D model)
bcc cell, N = 64, rs = 100, polarized	-0.0076803(2)	Slater-Jastrow-backflow	10.1103/PhysRevB.105.245135	-0.00769619(9) projected	8	not yet (HEG3D model)
fcc cell, N = 54, rs = 2, unpolarized	no variational energy yet; the first verified state sets it			0.00426(2) projected	1	not yet (HEG3D model)
fcc cell, N = 102, rs = 2, unpolarized	no variational energy yet; the first verified state sets it			0.00081(2) projected	1	not yet (HEG3D model)
fcc cell, N = 178, rs = 2, unpolarized	no variational energy yet; the first verified state sets it			0.00892(2) projected	1	not yet (HEG3D model)
fcc cell, N = 226, rs = 2, unpolarized	no variational energy yet; the first verified state sets it			0.00202(2) projected	1	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 10, unpolarized	-0.0537396(1)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 100, unpolarized	-0.00769287(1)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 30, unpolarized	-0.02272554(2)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 5, unpolarized	-0.0756267(6)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 50, unpolarized	-0.01449795(2)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 80, unpolarized	-0.00945620(2)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 86.6, unpolarized	-0.00879026(2)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 86.7, unpolarized	-0.00878092(1)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 86.8, unpolarized	-0.00877159(1)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 86.9, unpolarized	-0.00876228(1)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 87, unpolarized	-0.008753017(8)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
rect-3x3x2 cell, N = 36, rs = 87.1, unpolarized	-0.008743822(8)	MP-NQS	arXiv:2604.15888		4	not yet (HEG3D model)
sc cell, N = 7, rs = 1, polarized	1.120358(3)	WAP-net	10.1103/PhysRevB.107.235139		4	not yet (HEG3D model)
sc cell, N = 7, rs = 10, polarized	-0.0531436(2)	WAP-net	10.1103/PhysRevB.107.235139		4	not yet (HEG3D model)
sc cell, N = 7, rs = 100, polarized	-0.00774654(1)	WAP-net	10.1103/PhysRevB.107.235139		4	not yet (HEG3D model)
sc cell, N = 7, rs = 2, polarized	0.110850(1)	WAP-net	10.1103/PhysRevB.107.235139		4	not yet (HEG3D model)
sc cell, N = 7, rs = 20, polarized	-0.0322689(1)	WAP-net	10.1103/PhysRevB.107.235139		4	not yet (HEG3D model)
sc cell, N = 7, rs = 5, polarized	-0.0664225(5)	WAP-net	10.1103/PhysRevB.107.235139		4	not yet (HEG3D model)
sc cell, N = 7, rs = 50, polarized	-0.01466089(1)	WAP-net	10.1103/PhysRevB.107.235139		4	not yet (HEG3D model)
sc cell, N = 14, rs = 0.1, unpolarized	no variational energy yet; the first verified state sets it			106.89994(1) extrapolated	3	not yet (HEG3D model)
sc cell, N = 14, rs = 0.2, unpolarized	no variational energy yet; the first verified state sets it			25.4050004 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 0.3, unpolarized	no variational energy yet; the first verified state sets it			10.6954209 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 0.4, unpolarized	no variational energy yet; the first verified state sets it			5.6765448 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 0.5, unpolarized	3.41247(3)	FCIQMC	10.1063/1.4720076	3.41226(2) extrapolated	39	not yet (HEG3D model)
sc cell, N = 14, rs = 0.6, unpolarized	no variational energy yet; the first verified state sets it			2.2154921 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 0.7, unpolarized	no variational energy yet; the first verified state sets it			1.5130365 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 0.8, unpolarized	no variational energy yet; the first verified state sets it			1.0698103 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 0.9, unpolarized	no variational energy yet; the first verified state sets it			0.7746685 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 1, unpolarized	0.56864(4)	FCIQMC	10.1063/1.4720076	0.56850(3) extrapolated	39	not yet (HEG3D model)
sc cell, N = 14, rs = 10, unpolarized	-0.0552126(6)	MP-NQS	10.1103/PhysRevB.110.035108	-0.0552180(5) projected	20	not yet (HEG3D model)
sc cell, N = 14, rs = 100, unpolarized	-0.00773024(1)	WAP-net	10.1103/PhysRevB.107.235139	-0.00773430(2) projected	6	not yet (HEG3D model)
sc cell, N = 14, rs = 15, unpolarized	-0.0408619(1)	FermiNet	arXiv:2604.15888		6	not yet (HEG3D model)
sc cell, N = 14, rs = 2, unpolarized	-0.0084275(7)	FermiNet	10.1103/PhysRevLett.130.036401	-0.00873(3) extrapolated	39	not yet (HEG3D model)
sc cell, N = 14, rs = 20, unpolarized	-0.0324553(2)	MP-NQS	10.1103/PhysRevB.110.035108	-0.0324578(2) projected	17	not yet (HEG3D model)
sc cell, N = 14, rs = 3, unpolarized	no variational energy yet; the first verified state sets it			-0.07432(2) extrapolated	8	not yet (HEG3D model)
sc cell, N = 14, rs = 30, unpolarized	no variational energy yet; the first verified state sets it			-0.0203291 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 4, unpolarized	no variational energy yet; the first verified state sets it			-0.0792519 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 40, unpolarized	no variational energy yet; the first verified state sets it			-0.0155875 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 5, unpolarized	-0.0798630(2)	FermiNet	arXiv:2604.15888	-0.08002(2) projected	44	not yet (HEG3D model)
sc cell, N = 14, rs = 50, unpolarized	-0.01462631(6)	MP-NQS	10.1103/PhysRevB.110.035108	-0.0146287(1) projected	10	not yet (HEG3D model)
sc cell, N = 14, rs = 6, unpolarized	no variational energy yet; the first verified state sets it			-0.0706647 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 60, unpolarized	no variational energy yet; the first verified state sets it			-0.0106188 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 7, unpolarized	no variational energy yet; the first verified state sets it			-0.0651069 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 70, unpolarized	no variational energy yet; the first verified state sets it			-0.0091195 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 8, unpolarized	no variational energy yet; the first verified state sets it			-0.0599259 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 80, unpolarized	no variational energy yet; the first verified state sets it			-0.0080403 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 9, unpolarized	no variational energy yet; the first verified state sets it			-0.0553862 extrapolated	1	not yet (HEG3D model)
sc cell, N = 14, rs = 90, unpolarized	no variational energy yet; the first verified state sets it			-0.0071484 extrapolated	1	not yet (HEG3D model)
sc cell, N = 19, rs = 1, polarized	1.046242(3)	WAP-net	10.1103/PhysRevB.107.235139	1.0462250(5) projected	5	not yet (HEG3D model)
sc cell, N = 19, rs = 10, polarized	-0.0528624(2)	MP-NQS	10.1103/PhysRevB.110.035108	-0.0528652(3) projected	5	not yet (HEG3D model)
sc cell, N = 19, rs = 100, polarized	-0.00771899(1)	Iterative backflow	10.1103/PhysRevB.107.235139	-0.00772142(1) projected	5	not yet (HEG3D model)
sc cell, N = 19, rs = 2, polarized	0.096303(1)	WAP-net	10.1103/PhysRevB.107.235139	0.096302(2) projected	5	not yet (HEG3D model)
sc cell, N = 19, rs = 20, polarized	-0.03201095(5)	WAP-net	10.1103/PhysRevB.107.235139	-0.0320167(1) projected	5	not yet (HEG3D model)
sc cell, N = 19, rs = 5, polarized	-0.0672510(2)	WAP-net	10.1103/PhysRevB.107.235139	-0.0672540(5) projected	5	not yet (HEG3D model)
sc cell, N = 19, rs = 50, polarized	-0.01456685(3)	MP-NQS	10.1103/PhysRevB.110.035108	-0.01456920(4) projected	5	not yet (HEG3D model)
sc cell, N = 27, rs = 1, polarized	1.051771(4)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 27, rs = 10, polarized	-0.0526015(2)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 27, rs = 2, polarized	0.098460(2)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 27, rs = 30, polarized	-0.0228184(1)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 27, rs = 5, polarized	-0.0665523(6)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 27, rs = 50, polarized	-0.01457976(5)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 27, rs = 70, polarized	-0.01074986(3)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 27, rs = 90, polarized	-0.00853156(2)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 38, rs = 0.5, unpolarized	no variational energy yet; the first verified state sets it			3.20691(3) extrapolated	1	not yet (HEG3D model)
sc cell, N = 38, rs = 1, unpolarized	no variational energy yet; the first verified state sets it			0.52478(3) extrapolated	1	not yet (HEG3D model)
sc cell, N = 38, rs = 2, unpolarized	no variational energy yet; the first verified state sets it			-0.01422(5) extrapolated	1	not yet (HEG3D model)
sc cell, N = 54, rs = 0.5, unpolarized	3.221226(2)	DeepSolid	10.1038/s41467-022-35627-1	3.2202(2) extrapolated	17	not yet (HEG3D model)
sc cell, N = 54, rs = 1, unpolarized	0.52973(1)	MP-NQS	10.1103/PhysRevB.110.035108	0.52967(2) projected	25	not yet (HEG3D model)
sc cell, N = 54, rs = 10, unpolarized	-0.054448(1)	MP-NQS	10.1103/PhysRevB.110.035108		21	not yet (HEG3D model)
sc cell, N = 54, rs = 100, unpolarized	-0.00769203(5)	MP-NQS	10.1103/PhysRevB.110.035108		3	not yet (HEG3D model)
sc cell, N = 54, rs = 15, unpolarized	no variational energy yet; the first verified state sets it			-0.0363669 extrapolated	1	not yet (HEG3D model)
sc cell, N = 54, rs = 2, unpolarized	-0.014046(8)	MP-NQS	10.1103/PhysRevB.110.035108	-0.01429(3) projected	16	not yet (HEG3D model)
sc cell, N = 54, rs = 20, unpolarized	-0.0320524(5)	MP-NQS	10.1103/PhysRevB.110.035108		20	not yet (HEG3D model)
sc cell, N = 54, rs = 5, unpolarized	-0.079090(2)	MP-NQS	10.1103/PhysRevB.110.035108		22	not yet (HEG3D model)
sc cell, N = 54, rs = 50, unpolarized	-0.0145015(1)	MP-NQS	10.1103/PhysRevB.110.035108		7	not yet (HEG3D model)
sc cell, N = 66, rs = 1, unpolarized	0.5748(3)	Slater-Jastrow	10.1103/PhysRevB.58.6800	0.57290(3) projected	3	not yet (HEG3D model)
sc cell, N = 66, rs = 10, unpolarized	-0.05318(1)	Slater-Jastrow	10.1103/PhysRevB.58.6800	-0.053728(3) projected	3	not yet (HEG3D model)
sc cell, N = 66, rs = 2, unpolarized	no variational energy yet; the first verified state sets it			-0.00208(4) projected	2	not yet (HEG3D model)
sc cell, N = 66, rs = 20, unpolarized	-0.031515(5)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 66, rs = 5, unpolarized	-0.07583(2)	Slater-Jastrow	10.1103/PhysRevB.58.6800	-0.076676(2) projected	3	not yet (HEG3D model)
sc cell, N = 114, rs = 0.5, unpolarized	no variational energy yet; the first verified state sets it			3.48453(8) projected	1	not yet (HEG3D model)
sc cell, N = 114, rs = 1, unpolarized	0.6039(3)	Slater-Jastrow	10.1103/PhysRevB.58.6800	0.59877(6) projected	2	not yet (HEG3D model)
sc cell, N = 114, rs = 10, unpolarized	-0.05276(1)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 114, rs = 2, unpolarized	no variational energy yet; the first verified state sets it			0.00487(6) projected	1	not yet (HEG3D model)
sc cell, N = 114, rs = 20, unpolarized	-0.031390(5)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 114, rs = 5, unpolarized	-0.07433(1)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 128, rs = 110, unpolarized	-0.0071121(2)	MP-NQS	10.1103/PhysRevB.110.035108		2	not yet (HEG3D model)
sc cell, N = 128, rs = 200, unpolarized	-0.0040615(1)	MP-NQS	10.1103/PhysRevB.110.035108		3	not yet (HEG3D model)
sc cell, N = 162, rs = 1, unpolarized	0.5581(2)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 162, rs = 10, unpolarized	-0.053210(5)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 162, rs = 20, unpolarized	-0.031350(5)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 162, rs = 5, unpolarized	-0.07619(1)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 246, rs = 1, unpolarized	0.5969(2)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 246, rs = 10, unpolarized	-0.052740(5)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 246, rs = 20, unpolarized	-0.031375(5)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)
sc cell, N = 246, rs = 5, unpolarized	-0.07443(2)	Slater-Jastrow	10.1103/PhysRevB.58.6800		1	not yet (HEG3D model)

Electron gas, 2D#

6 instances, energies as E/N (Ha)

lattice size

instance	energy to beat	method	source	record (estimate)	results	verifier
square cell, N = 37, rs = 5, polarized	-0.14373(4)	Neural canonical transformation	10.21468/SciPostPhys.14.6.154		1	not yet (HEG2D model)
square cell, N = 90, rs = 30, unpolarized	-0.0317373(2)	Slater-Jastrow	10.1103/PhysRevB.110.245145	-0.031919(1) projected	7	not yet (HEG2D model)
triangular cell, N = 16, rs = 30, polarized	-0.031990687(3)	Fermi Sets	arXiv:2607.25872	-0.03199636(7) projected	6	not yet (HEG2D model)
triangular cell, N = 91, rs = 30, polarized	-0.03193763(3)	Self-attention NQS	arXiv:2607.08616		3	not yet (HEG2D model)
triangular cell, N = 91, rs = 40, polarized	-0.02443814(2)	Self-attention NQS	arXiv:2607.08616		3	not yet (HEG2D model)
triangular cell, N = 169, rs = 30, polarized	-0.03192077(3)	Self-attention NQS	arXiv:2607.08616		3	not yet (HEG2D model)

Citing QMBL: Open Problems

Cite QMBL: Open Problems by its concept DOI, which always resolves to the latest release:

T. Zerweck and J. Carrasquilla, QMBL: Open Problems, Zenodo (2026). 10.5281/zenodo.23439787

Individual energies should cite the primary paper named on the row, not this site.