Systematic improvement of neural network quantum states using Lanczos
Authors: Hongwei Chen
Douglas Gerard Hendry
Phillip E Weinberg
Adrian Feiguin
Venue: NeurIPS 2022 Accept (2022)

Abstract: The quantum many-body problem lies at the center of the most important open challenges in condensed matter, quantum chemistry, atomic, nuclear, and high-energy physics. While quantum Monte Carlo, when applicable, remains the most powerful numerical technique capable of treating dozens or hundreds of degrees of freedom with high accuracy, it is restricted to models that are not afflicted by the infamous sign problem. A powerful alternative that has emerged in recent years is the use of neural networks as variational estimators for quantum states. In this work, we propose a symmetry-projected variational solution in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one to restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers a solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg $J_1-J_2$ model on the square lattice, a paradigmatic problem under debate in condensed matter physics, and achieve state-of-the-art accuracy in the representation of the ground state.

Systematic improvement of neural network quantum states using Lanczos
Hongwei 1.23 Douglas Hendry' Phillip Weinberg' Adrian E. Feiguin'
123.3
*Department of Physics, Northeastern University, Boston, USA
2 Stanford Institute for Materials and Energy Sciences, Stanford University, Stanford, USA 3 Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, USA {chen. hongu, hendry. d, p weinberg, a. f.feiguin) wwwggiinnnooorrsttteeeeooeeeeeee edu
Abstract
The quantum many-body problem lies at the center of the most important open challenges in condensed matter, quantum chemistry, atomic, nuclear, and high- energy physics. While quantum Monte Carlo, when applicable, remains the most powerful numerical technique capable treating dozens or hundreds of degrees freedom with high accuracy, it is restricted to models that not afflicted by the infamous sign problem. powerful alternative that has emerged in recent years is the use of neural networks variational estimators for quantum states. In this work, we propose symmetry-projecteee variational solution in the form of linear combinations of simple restricted Boltzmann machines. This construction allows one to explore states outside of the original variational manifold and increase the representation power with moderate computational effort. Besides allowing one restore spatial symmetries, an expansion in terms of Krylov states using a Lanczos recursion offers solution that can further improve the quantum state accuracy. We illustrate these ideas with an application to the Heisenberg J/1 - J2 model on the square lattice, paradigmatic problem under debate in condensed matter physics, and achieve state-of- of-tte--rtttttt accuracy in the representation of the ground state.
Introduction
Understanding correlated quantum systems requires dealing with large configuration space: datasets comprised of all possible electronic configurations 5 and cannot be stored in the memory of the largest supercomputer. Hence, the quantum many-body problem can be interpreted as an "extreme data science" problem [13] from an information processing perspective. In quantum
function, each electronic or spin configuration has an associated complex amplitude (²()) determined by solving for the cigenvectors of the Hamiltonian operator. In particular, if one is interested in the zero temperature properties of the system, the solution given by the eigenvector with the smallest eigenvalue. Finding the exact solution of N quantum bit system with interactions requires solving for the eigenvectors of a N x 2 matrix. Alternatively, one can formulate the calculation as an optimization problem in which an "energy functional" E(bb) has minimized with respect all the 2 complex amplitudes.
Since the number of configurations d grows exponentially with the number of degrees of freedom (electrons, spins), this problem quickly becomes intractable. A solution consists of "compressing" the wave function by proposing suitable guess for the amplitudes based some variational parameters = (a1, Q22 ... am)) Typically, a functional form (6) f(ā, a') based on some physical intuition is utilized represent the amplitude of given configuration/state σ. The optimal parameters O; are determined by solving the system of equations V. E = O. The objective of this solution is to achieve the lowest possible energy with number of parameters << d.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Some relatively simple wave functions have enjoyed various degrees of success in the past, such as those of the Jastrow type where the amplitudes can be written as pair products f(ō.a) Hi U (aszoraz)) However, in recent years we have witnessed impressive developments based on the use of neural network (NN) wave functions as variational estimators [4], which have jump-started a new vibrant field of research dubbed "quantum machine learning". Notice that the optimization of the wave function parameters now translates into the "training" of the NN by minimizing the energy function that becomes cost function (we describe the training process below). The power of NN wave functions lies in the complex non-linear structure that provides them with remarkable expressivity to represent arbitrary complex many-body states by, at the same time, being completely agnostic to the physics.
Since restricted Boltzmann machines(RBM) were originally used as a variational ansatz for finding the ground state of the quantum many-body systems [4], there has been growing effort to investigate other forms of neural networks, including convolutional neural networks(CNN)19, 23], recurrent neural (owvors(RRNN[IUI network quantum (NNO))(NNQS) become the most appealing numerical alternative to treat quantum many body systems since they can be systematically improved by adding new layers or hidden variables, for instance. In addition to the ground state search, the application of NNQS ranges from classical simulation of quantum circuits[1, 5, 32), calculation of spectral function[17, 18), thermodynamies simulation[16) 16, 31], and quantum tomography(44).
graph networks(22], transformers(25] to mention a few. Thus, neural
Contributions In this work, we show how one can use a mathematically simple structure, a restricted Boltzmann machine (RBM), and yet obtain values of the ground state energy that beat all previous estimates by a range of numerical methods, including using convolutional neural networks. As we describe below, instead of increasing the number of layers or hidden variables, the solution lies on considering linear combinations of RBMs. The new wave function allows one to explore a much larger space of solutions. In particular, one can use this construction to restore spatial symmetries [40, 9, 28, 29]. In addition, we propose implementing a projection method based on Lanezos recursion using a "Krylov basis" of RBMs obtained by sequentially applying powers of the Hamiltonian operator.
The paper organized as follows: In Sec.2.1 we describe the quantum many-body problem in the context of the Heisenberg model; in Sec.2.2 we summarize prior attempts to study this problem using NNQS; in Sec.3 we review the basic formalism, including the structure of neural network wave functions, how to incorporate the symmetries of the problem into the quantum many-body state, and the numerical training procedure to optimize it. In Sec.4 we present results of -fate-of-the-ar calculations for the J1 - J/ Heisenberg model on the square lattice and compare to other numerical techniques. We finally close with a summary and conclusions.
2
The quantum many-body problem
2.1 Model
In the following, we will focus quantum spin problems where the degrees of freedom d, can assume two possible values 1/2 (or "up" and "down"). Similarly, one can think of them as generic two-level systems or "quubit".. In particular, we will benchmark our methods in the context of the spin antiferromagnetic Heisenberg model with nearest and next nearest neighbor interactions, the so-called J1 J2 model defined by the Hamiltonian:
A = J, > 5,
S,
+ J2
Σ S, (01)
(1)
(50)
where 5= (Š5 $8,, Š²) are spin operators, the first term runs over nearest neighboring sites (iz) on square lattice and the second term runs over next nearest pairs ({i))) along the diagonals of the plaquettes. For convenience, in the following, we set Jj = as the unit of energy. In this problem, the number of possible configurations grows as d = 2 However, the ground state wave function
lies on the sector with the same number of up and down spins, constraining our search to a smaller subset of states, albeit still exponentially large.
Without the J2 term, the problem can be numerically solved for hundreds of spins using quantum Monte Carlo (QMC) [38]. However, the method cannot be applied to problems with frustration since

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it is noticeably affected by the infamous sign problem[24]. In our case, this is due to the presence of the J2 term that makes some transition probabilities ill-defined (negative). The ground states of this model are well established in two extreme cases: at small J1 the system antiferromagnetically orders with wave vector q= (I,, :); at large J1 spins prefer columnar order q = (x. 0), (0, x), in which they aligned antiparallel in one direction, but ferromagnetically in the other. However, in the maximally frustrated regime J1 ~ 0.5J2. the system does not display any apparent order and the nature of this spin liquid state remains controversial despite significant research efforts over the past three decades[3, 6, 11, 10, 35, 39, 41, 34, 37, 26, 20, 15, 21, 45).
Therefore, we choose this Hamiltonian for two reasons: (i) it realizes quantum spin liquid in parameter regime near J/2 ~ 0.5.J and (ii) conventional Monte Carlo methods fail, making the model an ideal testing ground to benchmark new techniques. Variational Monte Carlo(VMMC) provides suitable alternative that can be scaled up to large two-dimensional systems without being affected by the sign problem. The quest for relatively simple yet powerful variational states has focused on neural network states, which have shown a great deal of promise. The complexity of the problem lies in the fact that many states with similar energy have very different physical properties. Therefore, an accurate representation of the ground state becomes the key to studying the nature of the quantum phase.
2.2 Related work
Before the concept of NNQS became popular new alternative for simulating many-bode systems, the most successful numerical techniques to treat the 2D J1 - 12 model have been the density matrix renormalization group (DMRG)155,, VMC based on projected fermionic ansatz[20), and tensor product states[45]. Recently, some research has focused on improving the accuracy of NNQS by using deep neural networks such CNN[9] and group-CNN[36] The idea of applying quantum number projection to recover the symmetries of the wave function[40, 46) has proven to be effective in improving the performance of NNQS[9, 28, 29, 36). In addition, other alternatives that enhance the quality of the approximations consist of combining NNQS with Gutzwiller- purojeeler-projeceed fermionic wave functions[ 22, or pair-product wave funetioos{30)]
3 Method
3.1 Neural Network Wave Function with symmetry
An RBM wave function takes a spin configuration a sequence of values #1/2 and returns a complex coefficient corresponding to the wave function amplitude. In other words, it is a function // (11/2, +1/23 + C. This function is highly non-linear and is parametrized by biases ã. b and weights W as:
$5 a, b) = E EN a.oo II cosh
Wyoz dz)-
In this expression, the number of "hidden variables" M is tunable parameter. While RBMs have remained a simple example of a basic neural network for many decades, it was only recently that their potential as variational wave functions was appreciated [4]. In this case, unlike conventional machine learning applications, the biases and weights are complex valued.
It is possible to account for certain symmetries [27] of the problem directly within the internal mathematical structure of the RBM. In particular:
Spin flip symmetry: the z-component of the total magnetization is zero of = 0), the global spin flip operation of - - preserves this property. Notice that since cosh(x) is an even function, we can easily restore the global flip symmetry in RBM wave function by removing the "magnetic field" terms associated to biases ā, b in E2.( Thus, the RBM wave function coefficients become:
(d" W)
2 cosh Wyjoz).
(3)
3

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Table 1: Character table of the Cav point group for square lattice.
E
2C4
C2
20T,
2
A
I
[
A2
1
1
-1
B1
-1
1
-1
B2
1
-1
1
-1
E
2
-2
0
Translational symmetry: translationally invariant systems, the ground state of the Hamil- tonian is expected to preserve the translational symmetry of the lattice. By applying the momentum projection, one can construct a variational wave function that preserved the symmetry with well-defined momentum K:
UK
Σ
-R-R e. FR),,,)
(4)
where the translation operator TR shift all particles by distance R. For the 2D square lattice,
R mx + ny.
(5)
The operator TR will perform a translation by m steps in the & direction, and steps in the y direction. The coefficient t'K now satisfies the translational symmetry at the cost of requiring computation time N time larger.
Lattice point symmetry: As our target model is on the 2D square lattice, the point group symmetries, which consist of rotation and reflection operations, may also be included:
K. (TRR)) W),
(6)
R
where & is the symmetry operation in the C42 point group, and x(C) is the character of the irreducible representation for the symmetry operation L. Since there are operations in the CAn point group, as shown in table 1, thC is 8 times more expensive to calculate compared to 1/K-
Notice that even though the computational cost of optimizing and evaluating observables with the symmetrized wave function has increased, the resulting state has much larger expressivity than the original one, translating into remarkable accuracy as shall demonstrate. We should highlight here that the new states, by being linear combinations RBMs, are no longer RBMs, and therefore allow one to explore much larger space outside the original manifold defined by v.. (2).
3.2
Wave Function Optimization
The goal of the calculation is minimize the cost function defined by the expectation value of the energy:
Woe (ukk)
Evar
(wre (//x)
Σ Ps Fioc(())
(8)
where the probability distribution is determined by the normalized wave function coefficients
|5 w
(9)
Σg (6") wKk))
and the local energy is given by
(0 H (Kke)
Eioe
(10)
(ō v/kc)
By formulating the problem in probabilistic terms, one can resort to Metropolis-Hastingg Markoy
Chain Monte Carlo evaluate the averages. The sampling over the spin configurations 5 is carried

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out by randomly flipping pairs of anti-aligned spins. and using von Neumann rejection according to transition probability W = KKnnwll / (
The wave function optimization can be implemented by a variety of methods. Since the energy landscape is extremely complex, simple gradient descent tends get trapped into metastable solutions. More sophisticated strategies are usually employed, such as natural gradient descent or "stochastic reconfiguration" [[2]]. In contrast to the "standard" natural gradient descent method, the Fubini-study metric[33], which is the complex-valued form of Fisher information, is used to measure the "distance" between wave functions (15) and (Ø):
( (+0 (olo)
( .)- = arccos
(11)
The procedure to update variational parameters using natural gradient descent is well described in literature[4, 8, 30), and we hereby summarize it. The optimization is done by minimizing the Fubini-study metric between le -- (0)) and v0 + 50)) where dr is a small step in imaginary time and can be viewed as learning rate in the training of neural network. The optimal choice for 50 is given by the solution system of equations:
Σ
[kttow> (Ot) 80w = -d - (oi)
-
(12)
a010) where the log derivative O₂- víoj 0e
and (...)) means average over samples. We update the parameters by 0g = 0% + 50, and repeat until convergence is reached.
3.3 Lanczos recursion
Using the symmetrized RBM wave function combined with the stochastic reconfiguration method, a good approximation of the ground state can be achieved after hundreds or thousands of iterations. However, due to the limited representation power of neural network wave functions, and the errors stemming from the Monte Carlo sampling and the optimization method, the true ground state of the Hamiltonian H can still differ significantly from the variational one. One possible way to increase the expressivity of the wave function is to introduce additional hidden variables or layers. However, an alternative to systematically improve the neural network wave function consists of applying modified Lanezos recursion [14, 2, 20j. The procedure begins with a (normalized) trial wave function voo which in our case is an initial guess for the ground state, 3/o d'c Then, a new state 1/1 is constructed by applying the Hamiltonian on v/o and subtracting the projection over vo in order to preserve orthogonality:
H (H)
1
(13)
(1H3) (H)
where (H") = (wo (")))))))))))))))))))))))))) Notice that Ui is orthogonal to th and also normalized. In the usual Lanczos method, this recursion can be continued such that new complete orthogonal basis can be constructed. In this representation, the Hamiltonian will have a tri-diagonal form. However, we only use t/u and 1h as our basis, and thus the Hamiltonian will be 2 x 2 matrix.
The eigenvector vo that corresponds to the lowest eigenvalue Eo of this matrix will be a better approximation of the true ground state of Hamiltonian compared vh. The lowest eigenvalue and corresponding eigenvector are
Eg = (H) va,
(14)
50u (1 + 02)112 Vu + a
(1 + 22 ui.
(15)
where
a = ((112) - (2)
(16)
r= (H²) - -((H2) + 2
(17)
2(H2) 22)
a= (r2 + 1/2
(18)
5

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Table Ground state energy per site E/N and the spin structure factor S(q) obtained by our RBM wave function, CNN[9], RBM+PP[30J, and exact diagonalization for the J1 - Jz model on 6 x 6 square lattice. p represents the number of Lanczos steps applied.
6 x 6
J2 0.5
J2 0.55
J2 0.6
Energy(Fxact)
-0.5038T0
-0.495178
-0,493239
Energy(CINN)
-0.50185(1)
-0.49067(2)
-0.49023(1))
EBMPP)
0.503365(1)
-0.495075(1)
Energy(RBM)
-0.0036((2)
-0.49501(1)
-0.49298(5)
Energy(RBM)
-0.5076(3)
-0,49512(4)
-0.49313(5)
Energy(RBM) p
-0.00378(4)
0049514(()
-0.49318(5)
S(T, x (EEact)
1.16989
0.89452
0.5545
S(m, m (RRBM
1.177(8)
0,902(6)
0.555(4)
S(m, O)(Exact)
0.201907
0.2489
0,48412
S(T, ()RBM)
0.200(2)
0.246(2)
0,486(6)
The eigenvector th. being linear combination of vo and 1/i, is the improved neural network wave function, and Eg is the new improved variational energy. By considering 4h the new trial wave function replacing Vo. this method can be repeated to further improve the wave function. The neural network wave function obtained during the Lanczos recursion can be generalized as
M = (1 Σ B,H*)
(19)
where p is the maximum number of Lanezos steps, and B is the wave function coefficient corre- sponding to H (Vo) In this form, one can easily identify the wave function as an expansion on Krylov basis.
In practice, taking into account the fact that the computational complexity increases dramatically with increasing p. only few steps be calculated for large quantum many body- system. In this study, and for illustration purposes, we shall consider only the p = 1 or = cases.
3.4 Implementation details
In this work, we focus on the 2D J1 J2 Heisenberg model on L x L square lattices where L is an simulations and consider three different
even number. For the neural network, we use t'к values for the number of hidden variables M consisting of 2, 2.5, and 3 times of the number of spins == the system. The parameters W in the RBM are initialized be randomly chosen random numbers with a uniform distribution between |-0011, 0.01] for both real and imaginary parts. The ground state can belong to the A₁ or B1 irreducible representations of the C.e point group, depending the transition between the spin liquid phase and the columnar phase with K = (TT, 0), e. for J2/ J1 2 0.5.
on the value of J2/ J1. In caleulations consider both cases
Due to large number of parameters and the numerical noise in sampling, we implement the conjugate gradient method solve the system of equations, (12). To stabilize the method, &i For each training step, we collect 10000 samples to evaluate averages as mentioned Sec. 3.2 including the variational energy and log derivatives. Since the adjacent states in the Markov chain are highly correlated, the number of the skipped states between samples Nskip chosen according to this relation Nskip 5 x 1.0/r, where the acceptance rate in the previous training step. The typical value for Nakip is from 30 to 100. As for evaluation, we collect 2 x 10 samples to calculate the average and statistical error. The learning rate used in the training ranges from 10-4 x 1 Once we observe that variational energy is not decreasing, a smaller learning rate(half of the previous one) is used instead. For large L, to save training time, we initialize parameters W in thç using parameters trained by means of the cheaper wave function vh. All simulations are performed using Eigen and Intel MKL. on Intel 25-268004 and AMD Rome 7702 CPU nodes. Source code will be available at: https:// /gi ///ggttuu thub. com httm///chen2211/1aaaa
we introduce ridge parameter À 10 10-6
Neural Network, "huantun_Statt State.

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Table 3: Comparison of the ground state energy per site E/Nand the spin structure factor S(q)
with QMC[38] results for the 2D Heisenberg model(J2 0) on 6 x 6, 8 x 8 and x 10 square lattice.
System Size
S(n, ((MC))QMM))
Energy(RBM)
S( ()BBM
6 6
-0667887(()
2.51799(6)
-0.678868(2)
2.51(2)
8 8
-0.673487(4)
3.7939(2)
-0.673482(3)
3.79(4)
10 x 10 0.671549(4)
5.3124(3)
-0.671519(4)
5.38(6)
Table 4: Ground state energy per site E/Nand the spin structure factor S(q) obtained by our RBM wave function, CNN[9], RBM+PP[30), VMC[20], and DMRG[15] for the J1 J2 model on 10 10 square lattice. p represents the number of Lanczos steps applied.
10 x 10
== 0.45
J2 J2= 0.5
J2 = 0.55
Energy (VC))
-0.50811()
-0.49521(1)
-0.48335(1)
Energy(DMRG)
-0.507976
-0.495530
-0.485434
Anergy(RBBM+P)
-0449769(11
Energy(CNN)
(0.50905(1)
-0.49516(1
-0.48277(1)
Energy(RBM)
-0.50916(2)
-0.49580(2)
-0.48410(3)
Energy(RBM) =
-0.5099(5)
-0.4968(4)
-0.4859(5)
S(T, (RBM)
2.06(3)
1.56(2)
1.18(2)
S(T, ()RBM)
0.186(1)
0.191(2)
0.200(2)
4 Results
4.1 Comparison with Exact Diagonalization
We benchmark the accuracy of the neural network wave functions for the ground state mainly on the 6 x and x 10 square lattices with periodic boundary conditions. For the 6 lattice, the JI - J2 is numerically soluble by enumerating the possible spin configurations, constructing the Hamiltonian matrix, and explicitly solving the eigenvalue problem [39]. Once the ground state (or its variational approximation) is obtained, the wave function can be used to calculate other physical quantities besides the energy. Here, for illustration, we compute the spin structure factor, that defines the sublattice magnetization squared for finite system
5( No
Σ (ooof)e 1.
(20)
where the wave vector q determines spatial structure of the magnetic order. Notice that in all the tables shown here, we display the results times a factor N for readability.
We first focus on the symmetrized RBM wave function without the Lanczos optimization, and we start by comparing the ground state energy for 6 x 6 lattice as function of J2/ //J1, shown in Fig. In this figure we calculate the relative error as Fun Frezaee |Feaact] using the exact ground state energy from Ref. [39]. We also include the relative error of the ground state energy obtained using a convolutional neural network wave function from Ref. While the relative error of the CNNs in order of 10-3 our RBM wave function achieves an accuracy of 10-4 in the frustrated regime. Even comparing other recent works using CNNs[43, 36, 7J, our RBM wave function still outperforms the CNN wave function. Besides the ground state energy, the spin structure factor computed from optimized wave functions agree very well with the exact solution as shown in Fig. 2, where the differences are smaller than the symbol size, and in data table
4.2 Comparison with state-of-the-ard quantum Monte Carlo
For larger lattices, the problem numerically intractable. However, as mentioned before, can be solved using QMC[38] for J2 0. Thus, for the case without frustration we can compare with QMC results for several different lattice sizes. From table we can see that even on the x lattice the energy difference is about 3 x 10-5 showing the extraordinary accuracy of our RBM wave function.
7

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6 X 6
RBM
10-2
CNN
RBM (p 23
16-8
AE/B
10-
10
0.0
0.2
014
0.6
0.8
1.0
1.2
1.4
Ja/ Jr
Figure 1: Relative error the ground state energy obtained with variational Monte Carlo using symmetrized RBM wave functions (this work) and convolutional neural network (CNN) wave functions, from Ref. [9].
For the frustrated case, J2 # 0, compare other methods, such those obtained with CNN wave functions as well as results using the density matrix renormalization groump(MMRG) method with (2) symmetry from Ref. [15] and VMC using Abrikosov-fermios mean field with Z2 gauge structure from Ref. [20]. From the data tables 4 and 5, we observe that our RBM wave function outperform the CNN wave function again in the entire range of J2/J1. In the frustrated regime, comparisons with VMC and DMRG using all the data available in literature demonstrate that the RBM wave functions still yield competitive ground state energies except at J2/ J1 0.55 where DMRG yields a lower value.
0.08
0.07
RBV 17,00
0.06
Disect (77.1)
ERE00 30 10
0.05
0.04
0.03
0.02
0.01
0.00
0.2
0.4
0.6
U.8
1.2
Jof J1
Static spin structure factor for ordering wave vectors q (I,, TT) and () obtained with VMC using symmetrized RBM wave functions, compared to numerically exact results on a 6 x lattice function of J2/J, J1. We also include VMC results for 10 10. Monte Carlo sampling emors are smaller than the symbol size.
Figure 2:
4.3 Lanczos optimization
Since the most interesting regime lies around the maximally frustrated point J2 ~ 0.5J,, we choose different values of J2/ J1 using x 6 and 10 x 10 lattices and perform few Lanczos steps to further

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Table Ground state energy per site E/N and the spin structure factor S(q) obtained by our RBM wave function, exact (iagonnalization39]] and CNN[9] for the J1 - J22 model on x 6 and 10 x 10 square lattice.
6x6 6
J2 0.0
J2 0.2
J2 = 0.4
J2 0.45
J2 = 0.7
Energy(Fxact)
-0.678872
-0.599046
-0.529745
-0.51565739
-0.530001
Energy(CNNN
-0.67882(1)
-0.59895(1)
-0.52936(1)
-0.51452(1)
Energy(RBM)
0.678868(2)
-0. (99904443)
0.529687(())
-0.51552(1)
-0.529921
, (Exact)
2.5180
2.22946
1.6604
0,15605
S(T, (RBM)
2.51(2)
2.25(2)
1.632)
1.412(8)
0.1567(7)
S(T, ()(Exctt)
0.167453
0.1691
0.17784
1.26743
S(T, ()RBM)
0.1678(7)
0.1681(7)
0.1774(8)
0.1845(8)
1.26(2)
6
J2 0.8
J2 1.0
J2 -== 1.2
J2 1.5
Energy(Exact)
0.586487
-0.714360
-0.848364
-1.05268
Energy(CNN)
0.58590(()
0..1335((I)
Energy(RBM)
0.586411(9)
-0.71429(I)
-0.848301)
-100224409)
S(n, (T) (Exact)
0.07726
0.02318
0.01115
0.00557
S(x, (RBM)
0.0779(4)
0.0233(2)
0.0110(1)
0.00556(5)
S(T, ()xecct
1.4402
1,54318
1.5729
1.59
S(T, ())BM)
1.43(2)
1.53()
1.57(2)
1.59(2)
10 x 10
J2 =2=00 0.0
J2 0.2
J2 0.4
J2 = 0.6
J2 = 0.7
Energy
0.67135(I)
0.59275(1)
0.5237I(1)
-0.47604(1)
(ReMg
-0.671519(4)
0.592847(9)
-0.52388(2)
-0.47662(3)
-0.51889(2)
S(x, (RBM)
5.38(6)
4.24(5)
2.59(4)
0.408(3)
0.1594(9)
S(r, (RBB)
0.1684(9)
0.1702(8)
0.180(2)
0.99(2)
2.33(7)
10 x 10
J2 1 0.8
J2 12-10 1.0
J2 = 1.2
J2 = 1.5
Energy(CNN)
0.57383(1)
0.69636(1)
Energy(RBMr
-0.57404()
-0,69670()
-0.22565(3)
-1.02371())
S(x, (RBM)
0.1004(5)
0.0480(2)
0.0212(2)
0.00760(6)
S(T, (RBM)
2.48(7)
2.59(5)
2.76(5)
3.07(5)
improve the ground state energy. From data table and 4, we see that the Lanczos steps are very effective regardless of the system size. Remarkably, by performing better energy at 11/ = 0.55 for the 10 x 10 lattice that improves significantly the best available data using state-of-the-arr DMRG, shown in data table 4. Besides, compared to the "RBM+PP" results[30], which is generally considered as the start-of-the-art NNQS method, we obtain slightly lower variational energy at J2 0.5, 0.55 on their variational energy is 8 x 10 4 lower than ours. Additionally, with the help of the Lanczos recursion, a better estimate of the energy can be obtained by carrying out variance extrapolation as illustrated in Ref. [2, 20J. We also try to improve the estimation of spin structure factor using Lanczos, but the Monte Carlo sampling error makes the improvement not obvious.
Lanczos steps, obtain
6 lattice while for 10 10 lattice J/ 0.5.
Conclusion
Neural network wave functions hold a great deal of promise due to their ability to compress complex quantum many-body states within a relatively simple mathematical structure that, owing to its non- lincarity, can encode an exponentially large amount of information with polynomial resources. In particular, RBM wave functions, initially deemed too simple, can be used as building blocks for systematically improved wave functions. These improved states obey the internal symmetries of the model and the point group symmetries of the lattice. In addition, they may contain contributions from the state living in a "tangent space" to the original RBM manifold. These tangent vectors are spanned in terms of powers of the Hamiltonian and form Krylov basis.
We have demonstrated that we can achieve stateof--heeartt accuracy that improves previous results using convolutional neural networks with minimal amount of extra computational cost compared to simple RBMs. The combination problems previously beyond the reach of the most powerful numerical techniques and provide the
Lanezos and symmetrization offer an effective solution to

---

means to bypass the sign problem. These ideas can seamlessly translate to other areas of research ranging from materials science to quantum chemistry. Besides, our variational solution can be adopted to calculate the excitation spectrum of quantum many-body system| 17, 18], providing valuable information that can be directly compared experiments.
Limitations The computational cost of single training step scales as O X M where the number of hidden variables M is usually proportional to the system size N. Thus, the computation time of calculation may be bottleneck for its application on larger lattices. In particular, we find that even though the results for the energy are very accurate, correlation functions have relatively larger errors. This behavior might be improved by using variational forms with better representation power. Besides, the Lanczos step procedure is not size consistent, which means that the energy improvement with respect to the original wave function (ebo vanishes for fixed and N - oc. Also, the Lanczos correction will be smaller and smaller as p increases. Nevertheless, a sizable improvement obtained even for rather large clusters with 100 sites as shown in the data table 4.
Negative Societal Impact Our work presents the theoretical simulation of the quantum many-body problems without any foreseeable negative societal impacts.
Acknowledgments and Disclosure of Funding
AEF and HC acknowledge the National Science Foundation for support under grant No. DMR- 2120501. DH is partially supported by Northeastem Tier grant.
10

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes] See abstract and introduction.
(b) Did you describe the limitations of your work? [Yes] See conclusion.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] See
conclusion.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [N/A]
(b) Did you include complete proofs of all theoretical results? [N/A]
3. If you ran experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main ex- perimental results (either in the supplemental material or URL)? [Yes] See Sec. 3.4.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Sec. 3.4.
(c) Did you report error bars (e-g., with respect the random seed after running experi- ments multiple times)? [Yes] We report error bars in the figures and data tables. See Sec. 4.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] See Sec. 3.4.
4. If you using existing assets (e.g., code, data, models) or curating/releasing
assets...
(a) If your work uses existing assets, did you cite the creators? [N/A]
(b) Did you mention the license of the assets? [N/A]
(c) Did you include any assets either in the supplemental material URL? [N/A]
(d) Did you discuss whether and how consent was obtained from people whose data you're 're using/curating? [N/A]
(c) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, applicable? [N/A]
(b) Did you describe any potential participant risks, with links Institutional Review Board (IRB) approvals, if applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
14


Reviews:
[{"review_id": "jgbrbPQ889k", "reviewer": "Reviewer_37nZ", "summary": "This paper proposes a symmetry-projected variational solution in the form of linear combinations of simple restricted Boltzmann machines, which allows one to explore states outside of the original variational manifold and then increase representation power. Also, an expansion in terms of Krylov states using a Lanczos recursion is used to further improve quantum state accuracy. Experiments is conducted on Heisenberg J1-J2 model on square lattice. ", "questions": "I was aware of another recent work on solving quantum many-body problem over various lattices [1]. Have authors tried the proposed methods on other types of lattices, such as triangular and honeycomb, etc? How is the performance of proposed neural network wave function on these lattices?\n\n[1] Kochkov, Dmitrii, et al. \"Learning ground states of quantum Hamiltonians with graph networks.\" arXiv preprint arXiv:2110.06390 (2021).\n", "limitations": "Limitation of scaling up to larger systems is well addressed. Could authors explain more on the generality of this method on different type of lattices or even random graphs?", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n- *clarity*: This paper is overall well written and easy to follow. The method and results are clearly presented.\n- *significance*: The proposed neural network quantum state can obey the internal symmetries of the quantum model and the point group symmetries of the lattice. This could make the neural network being more effective to represent quantum state by satisfying internal constraints of quantum many-body problem. Also, Heisenberg J1-J2 model has the infamous sign problem due to frustration that cannot be handled well by traditional numerical methods. So applying neural network to this problem is of great importance to quantum physics.\n\n**Weaknesses:**\n- *originality*: Based on my background and the paper itself I cannot judge if the method of this work is novel, but apparently this work misses some relevant works, such as the following:\n  - *Liang, Xiao, et al. \"Solving frustrated quantum many-particle models with convolutional neural networks.\" Physical Review B 98.10 (2018): 104426.*\n  - *A. Szabó and C. Castelnovo. Neural network wave functions and the sign problem. Physical Review Research, 2(3):033075, 2020.*\n  - *Kochkov, Dmitrii, et al. \"Learning ground states of quantum Hamiltonians with graph networks.\" arXiv preprint arXiv:2110.06390 (2021).*\n\n   And authors don’t explain the difference between the proposed method and those works (mentioned in related works) that also try to restore symmetry. And how is the performance comparison with these methods also considering symmetry, not just one CNN method.\n\n\n- *quality*: Authors primarily compare the results with a specific CNN method. Could authors explain why only compare with this specific deep learning method?. Also, experiments are only performed on square lattice. Since the compared CNN method has performed so good on square lattice, the proposed method can only show very small improvements. I think authors could show the advantage of the proposed method over other methods on some other difficult quantum systems, such as several different lattices where the ground state is harder to learn, etc.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "GyurOK0ZrjP", "reviewer": "Reviewer_ZGgm", "summary": "The paper demonstrates that by leveraging symmetrization of variational states and employing Lenczos recursion, neural network quantum states can surpass the current state-of-the-art accuracy for representing the ground state of the J1-J2 model on the square lattice. The main finding is that instead of using more intricate NN anzatz (e.g., ConvNets, Graph NN, or Transformers), the paper proposes that using the much simpler RBM anzatz coupled with symmetrization and Lanczos steps is sufficient to represent complex wave-functions. With that method they show that they can obtain much better ground state energy estimation across the entire range of the J2/J1 parameter and specifically near the point of highest frustation (~ 0.5).", "questions": "* On the topic of symmetries it would have been helpful for the authors to discuss other approaches to brute-force symmetrization, e.g., symmetric constructions of RBMs / ConvNets / Graph NN. It is especially interesting that applying brute-force symmetrization to RBMs worked better than the ConvNet used in [1] which follows translational symmetry by construction and uses the same brute-force symmetrization only for the C4 symmetries. It would be great if the authors could comment on why that might be the case given that both models follow the same symmetries and one (the ConvNet) is supposedly more expressive (prior to using Lanczos). Moreover, given that the set of methods advocated by the paper could be applied to any anzatz, have the authors considered using ConvNets or any other NN architecture to verify whether the improved results are due to the use of simpler RBMs, or comparing the use of brute-force symmetrization vs. inherently symmetric models (whether using symmetric RBMs, or symmetric ConvNets)?\n\n* There is a slight mismatch between the usual meaning of invariance to symmetries and how you define your symmetrized anzatz. Specifically, one usually refers to symmetric invariant function as one where f(x) = f(Tx) for all T in some set of symmetric operators, but in this case it appears that this requirement is soften to be equal in amplitude but it does allow for a shift in phase, and so it is more akin to equivariant property. Could the authors please elaborate on this point to clarify what is the property they wish to enforce and why?\n\n* \"point group symmetries\" might not be clear for all readers, and it would be great to be specific on what it means in this context (rotations + reflections).\n\n* While not mandatory, given that it is a method defined by prior works, it would be helpful to include a derivation of equations 14-18 in the appendix to this work.", "limitations": "The authors have properly addressed the limitations of their approach, and the difficulty of scaling it to larger lattices. My only comment would be that the limitations paragraph at the end should also include the cost of symmetrization in the training step (so it should be N^3 not N^2), and that they should also repeat the cost of the Lencoz step. I would also add that it might be challenging to scale this method to larger NN, which might be necessary to solve certain cases with high accuracy.", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well written and offers a simple introduction to the subject matter that is a bit outside of the common domain of ML. The main strength of the paper are its impressive results, improving on the previous best results by a large factor on a problem that is considered exceptionally difficult. Moreover, it is noteworthy that these improvements are obtained by using relatively simple RBMs, as opposed to more intricate architectures (e.g., ConvNets or Transformers).\n\nHowever, the presented methods are not novel and are based on previous works. Specifically, it appears that the symmetrization method is identical to those used by prior papers including with NN-based approaches [1,2], and to the extent of my knowledge it is a well-known widely-used technique in the field. Similarly, Lanczos iteration for improving the accuracy of a given ground-state approximation was suggested by other works and was specifically used in the past to improve results on the J1-J2 model [3]. It is worth noting that to the best of my knowledge this is the first use of this method with NN quantum states. While the authors do properly cite prior works, it is not sufficiently emphasized that they merely implement these ideas for NN quantum states -- something that should have been more clearly articulated in the introduction and abstract.\n\nAs a final note, while the paper clearly shows the superiority of their proposed method on the given task, there is not a lot of effort spent on attributing these gains to the various choices. Some abalation study (even if done only on the 6x6 case) could go a long way to clarify the contribution of each element (e.g., employing Lanczos on a non-symmetric RBM and the ConvNet model from [1], or using RBMs with a symmetric matrix to account for translation symmetry vs. brute-force symmetrization).\n\n[1] - Choo et al., Two-dimensional frustrated J1−J2 model studied with neural network quantum states, PRB, 100:125124, Sep 2019.\n\n[2] - Sharir et al., Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems, PRL 124:020503, Jan 2020.\n\n[3] - Iqbal et al., Spin liquid nature in the Heisenberg J1-J2 triangular antiferromagnet, PRB 93:144411, Apr 2016.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "E1g6mE7PZUD", "reviewer": "Reviewer_Cg58", "summary": "In the present paper, the authors introduce a method to model the wave function of quantum systems, in particular spin systems in the $J_1$-$J_2$ model. They use a restricted Boltzmann machine (RBM) and form linear combinations of them in order to incorporate certain symmetries of the investigated system. Furthermore, the authors introduce a method to determine the ground state utilizing the Lanczos recursion to iteratively improve the estimate.\n\nThey compare their method on spin lattices of various sizes to baseline methods such as Quantum Monte Carlo and a CNN based method on the basis of comparing the ground state energies and the spin structure factor obtained through the different procedures. Their method is competitive with or outperforms the baselines.", "questions": "* How much memory and compute do other methods use?\n* How do they scale with the system size?", "limitations": "The authors state that the computational cost of their method scales quadratically with the system size, which limits their ability to tackle larger systems.", "rating": 5, "confidence": 2, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\nIn general, the paper is concise and well written, making it pleasant to read and understand. \n\nThe authors utilized the physical knowledge about the symmetries of the system and incorporate it into the model, which is a common theme in applying machine learning models to problems in the natural sciences.\n\nDifferent concepts from machine learning (RBMs), advanced statistics (Fubini-study metric), and numerics (Lanczos recursion) are combined to form a new algorithm capable to compute the ground state energy of a spin system, which is competitive or outperforms its baselines.\n\n**Weaknesses**\n\nI am not familiar with algorithms for estimating the ground state energy of quantum systems, so I cannot judge how big or small the improvement of this procedure over existing baselines is. However, the difference of the results is often very small, only manifesting in the fourth or fifth digit, so it is not clear to me whether this is actually significant.\n\nThe authors mention that the key for such algorithms is that they use a small amount of memory and compute. A naive implementation would use an exponential amount of memory, while this procedure has a controllable number of variables. However, they do not discuss in detail how much compute and memory competing procedures use. They only mention in the conclusion that their method is slightly more expensive than the CNN-based method while having a better accuracy, but it is unclear what this means. \n\n**Conclusion**\n\nGiven my criticism and my lack of knowledge in the field, I am unsure whether to accept or reject this article. For me, it is important to clarify the amount of memory and compute needed by different methods, putting their respective performances in perspective. I am willing to raise my score if this concern is addressed appropriately. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "6bMujXr9d_n", "reviewer": "Reviewer_PiLy", "summary": "The paper applies Lanczos step improvements over a shallow neural quantum state based on RBMs. \nSymmetries are also exploited successfully, thanks to projections in the relevant symmetry sectors. \nThe authors report results that are highly competitive with the state of the art on a challenge benchmark (J1-J2 model in 2d). ", "questions": "1. While the authors report on a comparison on the 6x6 model, it is crucial to understand what happens on the larger 10x10 model, if they want to claim new SOTA results on the J1-J2 model. The paper by Nomura and Imada in Table 2 reports the relevant energy to compare to. How does the Lanczos-step approach compare? \n\n2. The effect of symmetries seems absolutely crucial to improve the bare RBM results. What does it happen if one imposes translation symmetries in the weights instead of summing over the group explicitly? ", "limitations": "yes", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "To my knowledge, this is the first application of the Lanczos-step style improvements to neural quantum states. This idea has been applied in the past to other variational states, and carries a cost that is exponential with the number of Lanczos steps. \n\nThe technique reported here allows to improve significantly on previously reported \"pure\" neural quantum states results. \nThe main limitation of the approach is the known lack of \"size extensive\" scaling of the Lanczos iterations, which make them ineffective for larger systems approaching in the thermodynamic limit. However, there could be cases of finite clusters where the improvement offered is still important. \n\nMy main criticism is the lack of comparison with the current best-known result on the model (more in the questions), reported in Nomura and Imada, PHYSICAL REVIEW X 11, 031034 (2021).  \n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept