SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof. Thm B.2of Appendix B.4 shows that both TN and ARNN are special cases as ANTN and Thm B.3of Appendix B.4 shows that ANTN can be written as either a TN or an ARNN with an exponential (in system size) number of parameters. Thus, ANTN generalizes the expressivity over both TN and ARNN.
5.3 Symmetry Results of ANTN
Symmetry plays an important role in quantum many-body physics and quantum chemistry. Many of the symmetries can be enforced in ARNN via the two classes of the symmetries—mask symmetry and function symmetry.
Definition 5.1 (Mask Symmetry). A conditional wavefunction  $\psi(x_i | x_{<i})$  has a mask symmetry if  $\psi(x_i | x_{<i}) = 0$  for some  $x_i$  given  $x_{<i}$ .
Definition 5.2 (Function Symmetry). A conditional wavefunction tensor  $\psi(x_i | x_{<i})$  has a function symmetry over a function  $F$  if  $\psi(x_i | x_{<i}) = \psi(F(x_i; x_{<i}) | F(x_{<i}))$  where  $F(x_{\leq i}) := \{F(x_1), F(x_2; x_1), \ldots, F(x_i; x_{<i})\}$ .
Here, we list several symmetries that ANTN inherits from ARNN and show the proofs in Appendix B.4.
Theorem 5.4. Autoregressive Neural TensorNet inherits mask symmetry and function symmetry from autoregressive neural networks.
Corollary 5.4.1 (Global U(1) Symmetry). Autoregressive Neural TensorNet can realize global  $U(1)$  symmetry, which conserves particle number.
Corollary 5.4.2 ( $\mathbb{Z}_2$  Spin Flip Symmetry). Autoregressive Neural TensorNet can realize  $\mathbb{Z}_2$  spin flip symmetry such that the wavefunction is invariant under conjugation of the input.
Figure 2: Fidelity  $\uparrow$  on quantum state learning with 16 qubits for TN (MPS), ARNN (transformer) and ANTN (elementwise construction with transformer+MPS). (a) Learning random Bell states. (b) Learning real-valued depth-4 random circuit with and without sign rule. The error bar denotes the standard deviation (not the standard error of the mean) of the fidelities over the random states sampled from the corresponding distribution. The mean and standard deviation are calculated from 10 random states. The numbers inside the parentheses denote the bond dimension.
Corollary 5.4.3 (Discrete Abelian and Non-Abelian Symmetries). Autoregressive Neural TensorNet can realize discrete Abelian and Non-Abelian symmetries.
6.1 Quantum State Learning
As stated previously, ANTN generalizes both ARNN and TN to take advantage of both the expressivity and the inductive bias. Here, we test the ANTN on learning physically important quantum states to demonstrate this ability. In this task, we use the transformer neural network for ARNN and ANTN due to the 1D structure of the system.
Experiments on expressivity. We first test the expressivity of the ANTN by learning a class of well-known high-entangled states—random permuted Bell states. A 2-qubit Bell state is defined as  $(|00\rangle + |11\rangle) / \sqrt{2}$ , which has 1 bit of entanglement. For a system size of  $n$ , we randomly pair qubits in the first half of the system with the qubits in the second half of the system to be Bell states. This process creates quantum states with  $n / 2$  bit of entanglement between the first and second half of the system. It can be shown that a bond dimension of  $2^{n / 2}$  is required for MPS to fully represent such a system. In Fig. 2 (a), we plot the quantum fidelity of learning 16-qubit random Bell states. As the figure shows, MPS cannot represent the states without reaching the required bond dimension of  $256 = 2^{8}$ . The ARNN and ANTN, on the other hand, can represent such states without limitations from the bond dimension.
Experiments on inductive bias. We then test the physics inductive bias of the ANTN. One of the physics inductive biases of MPS is that it can represent wavefunctions with local or quasi-local sign structures that can be hard for neural networks to learn. These wavefunctions can have a fluctuating sign depending on the configuration  $x$ . Here, we use (real-valued) shallow random circuits to mimic the short-range interaction and generate states with sign structures. We test the algorithms on these states both with and without the "sign rule", which means that we explicitly provide the sign structure to the algorithm. As shown in Fig. 2 (b), the fidelity of MPS only depends weakly on the sign rule, whereas the fidelity of ARNN can change drastically. Our ANTN inherits the property of MPS and is thus not affected by the sign structure. Furthermore, being more expressive, ANTN with a bond dimension of 4 already performs better than MPS with a bond dimension of 16.
6.2 Variational Monte Carlo
We further test our algorithm on finding the ground state of the challenging 2D  $J_{1} - J_{2}$  Heisenberg model with open boundary condition. The model has a rich phase diagram with at least three different phases across different  $J_{2} / J_{1}$  values (Capriotti et al., 2004) (Lante & Parola, 2006). In addition, the
Energy per site ↓8 × 8
Algorithms	J2=0.2	J2=0.5	J2=0.8
RBM (NS)	-1.9609(16)	-1.5128(20)	-1.7508(19)
RBM (S)	-2.1944(17)	-1.8625(20)	-0.7267(29)
PixelCNN (NS)	-2.21218(16)	-1.77058(29)	-1.93825(16)
PixelCNN (S)	-2.23171(9)	-1.88902(19)	-1.83672(26)
Elementwise (8 NS)	-2.23690(4)	-1.93018(8)	-2.00036(16)
Elementwise (8 S)	-2.23688(4)	-1.93102(7)	-2.00262(11)
Blockwise (70 NS)	-2.23484(7)	-1.93000(7)	-1.99148(13)
Blockwise (70 S)	-2.23517(6)	-1.92880(9)	-1.99246(13)
Figure 3: Energy per site difference  $\downarrow$  between elementwise ANTN with bond dimension 8 and MPS with bond dimension 1024 optimized using DMRG algorithm for various  $J_{2}$  and different system sizes from  $4\times 4$  to  $12\times 12$ .
Table 1: Energy per site  $\downarrow$  for  $8 \times  8$  system with various algorithms where elementwise and blockwise are two constructions of ANTN (with PixelCNN + MPS). The bond dimensions for ANTN are labeled inside the parentheses. For each algorithm, we test it both with the sign rule (S) and without the sign rule (NS) The best energy is highlighted in boldface and the second in italic.
Energy per site ↓10 × 10
Algorithms	J2= 0.2	J2= 0.3	J2= 0.4	J2= 0.5	J2= 0.6	J2= 0.7	J2= 0.8
MPS (8)	-1.997537	-1.893753	-1.797675	-1.734326	-1.716253	-1.768225	-1.869871
MPS (70)	-2.191048	-2.069029	-1.956480	-1.866159	-1.816249	-1.854296	-2.007845
MPS (1024)	-2.255633	-2.138591	-2.031681	-1.938770	-1.865561	-1.894371	-2.062730
PixelCNN	-2.22462(24)	-2.12873(14)	-2.02053(14)	-1.74098(29)	-1.71885(27)	-1.81800(13)	-1.98331(17)
Elementwise (8)	-2.26034(6)	-2.14450(4)	-2.03727(7)	-1.94001(6)	-1.85684(10)	-1.88643(7)	-2.05707(43)
Blockwise (70)	-2.25755(8)	-2.14152(8)	-2.03319(10)	-1.93842(42)	-1.85270(12)	-1.87853(13)	-2.05088(14)
Energy per site ↓12 × 12
Algorithms	J2= 0.2	J2= 0.3	J2= 0.4	J2= 0.5	J2= 0.6	J2= 0.7	J2= 0.8
MPS (8)	-1.998207	-1.887531	-1.800784	-1.735906	-1.720619	-1.788652	-1.893916
MPS (70)	-2.185071	-2.059443	-1.944832	-1.851954	-1.812450	-1.853650	-2.030131
MPS (1024)	-2.264084	-2.141043	-2.027736	-1.931243	-1.858846	-1.913483	-2.093013
PixelCNN	-2.24311(102)	-2.12616(23)	-2.01768(21)	-1.74282(30)	-1.72637(16)	-1.85239(13)	-2.03226(59)
Elementwise (8)	-2.27446(27)	-2.15537(6)	-2.04437(7)	-1.94686(6)	-1.85443(15)	-1.91391(10)	-2.09457(10)
Blockwise (70)	-2.26152(50)	-2.15395(7)	-2.04225(8)	-1.94298(43)	-1.85176(15)	-1.90571(12)	-2.09113(43)
Table 2: Energy per site  $\downarrow$  for  ${10} \times  {10}$  and  ${12} \times  {12}$  system with various algorithms where elementwise and blockwise are two constructions of ANTN (with PixelCNN + MPS). The bond dimensions for MPS and ANTN are labeled in parentheses. The MPS is optimized with DMRG algorithm. The best energy is highlighted in boldface and the second best value is highlighted in italic.
complicated structure of its ground state makes it a robust model on which to test state-of-the-art methods. Here, we use the PixelCNN for ARNN and ANTN due to the 2D geometry of the system.
The 2D  $J_{1} - J_{2}$  Hamiltonian is given by
$$
\hat {\mathcal {H}} = J _ {1} \sum_ {\langle i, j \rangle} \vec {\sigma} _ {i} \cdot \vec {\sigma} _ {j} + J _ {2} \sum_ {\langle \langle i, j \rangle \rangle} \vec {\sigma} _ {i} \cdot \vec {\sigma} _ {j}, \tag {8}
$$
where subscript refers to the site of the qubits,  $\langle \cdot ,\cdot \rangle$  is the nearest neighbour and  $\langle \langle \cdot ,\cdot \rangle \rangle$  is the next nearest neighbour.  $\vec{\sigma}_i\cdot \vec{\sigma}_j = X_i\otimes X_j + Y_i\otimes Y_j + Z_i\otimes Z_j$  with  $X,Y$  and  $Z$  the Pauli matrices
$$
X = \left[ \begin{array}{l l} 0 & 1 \\ 1 & 0 \end{array} \right], \quad Y = \left[ \begin{array}{c c} 0 & - i \\ i & 0 \end{array} \right], \quad Z = \left[ \begin{array}{c c} 1 & 0 \\ 0 & - 1 \end{array} \right]. \tag {9}
$$
We will fix  $J_{1} = 1$  and vary  $J_{2}$  in our studies.
Experiments on inductive bias of  $J_{1} - J_{2}$  Heisenberg Model.
As shown previously, it can be challenging for neural networks to learn the sign structures. The ground state of the  $J_{1} - J_{2}$  model also has a sign structure, which can be partially captured by the Marshall sign rule (Marshall, 1955). The sign rule is exact at  $J_{2} = 0$  and becomes worse for large  $J_{2}$ . This could introduce bias to the neural network wavefunction if directly applied. We compare the restricted Boltzmann machine (RBM) (from NetKet (Vicentini et al., 2022)), our implementation of gated PixelCNN, and the two different constructions of ANTN with and without the sign rule.
The result is shown in Table 1. Since our approach is based on the variational principle discussed in Sec. 3, it provides an upper bound on the exact ground state energy; therefore, a lower energy implies a better state. As shown in the results, both RBM and PixelCNN improve significantly with the application of the sign rule at  $J_{2} = 0.2$  and  $J_{2} = 0.5$ , but the results deteriorate at  $J_{2} = 0.8$ . This is expected because the sign rule becomes increasingly inaccurate as  $J_{2}$  increases, especially past  $J_{2} = 0.5$ . Our ANTN, on the other hand, does not require the sign rule in both constructions. As an additional comparison, we note that both of our ANTN constructions achieved better results compared to the recently developed matrix product backflow state (Lami et al., 2022), which uses the sign rule and has a per site energy of  $-1.9267$  at  $J_{2} = 0.5$ .
Ablation study on  $J_{1} - J_{2}$  Heisenberg Model. We scan across many  $J_{2}$  for the model without using the sign rule. In this experiment, we compare the performance of six models to compute the ground state energy of the  $J_{1} - J_{2}$  Hamiltonian system with  $J_{1} = 1$  fixed and  $J_{2} = 0.2 - 0.8$  with 0.1 increments, covering three different phases of the model. The first three models are TN models, using the MPS with bond dimensions of 8, 70, and 1024. For up to  $4 \times 4$  system, the MPS results with bond dimension 1024 are exact and can be regarded as a benchmark. The fourth model is a PixelCNN pure neural network; the fifth and sixth models are elementwise and blockwise ANTN models. Thus the experiment compares the ANTN against the two previous state-of-the-art techniques.
Table 2 summarizes the per site ground state energy computed by different models at different  $J_{2}$ 's in three distinct phases. Our results provide strong evidence that ANTN integrating TN and ARNN outperforms each base model, achieving state-of-the-art results.
Comparison between ANTN and MPS. In all cases the ANTN surpasses the MPS with the corresponding bond dimension on which they are based. For example, even in the  $10 \times 10$  system with  $J_{2} > 0.5$ , where MPS with a bond dimension of 1024 outperforms the ANTN, the elementwise ANTN still significantly improves on the base MPS with a bond dimension of 8 and the blockwise ANTN improves on the base MPS with a bond dimension of 70. It is consistent with Theorem 5.3 and Theorem 5.2 that ANTN is more expressive than TN.
In addition, the ANTN models scale well for larger systems. Figure 3 visualizes the scalability of ANTN compared to MPS. The figure plots the difference in per site energy between the elementwise ANTN with bond dimension 8 and MPS with bond dimension 1024 for  $J_{2} = 0.2$ , 0.5, 0.8 where lower energies signify better performance for the elementwise ANTN. Compared to MPS, the ANTN models compute better ground state energies as the system size grows larger. As the system size increases, the elementwise ANTN with bond dimension 8 starts to outperform MPS with bond dimension 1024. Even at  $J_{2} = 0.6$ , where the elementwise ANTN is slightly worse than MPS, the difference gets significantly smaller at  $12 \times 12$  compared to  $10 \times 10$  (Table 2). In fact, the elementwise ANTN achieves such a performance using only  $\sim 1\%$  the number of parameters of MPS. We note that for large  $J_{2}$ , the system goes into a stripped phase (Nomura & Imada, 2021), which can be less challenging for MPS to represent. Nevertheless, in almost all cases our ANTN still outperforms MPS on the  $12 \times 12$  system. MPS has a cost dominated by the bond dimension (quadratic for memory and cubic for computational), which limits its use in practice for large system sizes that require large bond dimensions. According to the complexity analysis in Sec. 5.1, ANTN has lower complexity than TN and thus scales better for larger systems.
Comparison between ANTN and ARNN. The elementwise and blockwise ANTN models also consistently outperform the pure neural network PixelCNN model. This agrees with Theorem 5.3 that both elementwise and blockwise ANTN have a generalized expressivity compared to ARNN. In Appendix C we further compare ANTN with ARNN and find that: (a) it is challenging to surpass ANTN by ARNN even with an increased number of parameters; (b) the improvement from ANTN mainly comes from the effective inductive bias of MPS structure instead of the DMRG initialization; (c) ANTN has a favorable scaling compared to ARNN and MPS for reaching accurate results.
The details of all experiment setups and hyperparameters can be found in Appendix D.
In this paper, we developed Autoregressive Neural TensorNet, bridging the two state-of-the-art methods in the field, tensor networks, and autoregressive neural networks. We proved that Autoregressive Neural TensorNet is self-normalized with exact sampling, naturally generalizes the expressivity
of both tensor networks and autoregressive neural networks, and inherits proper physics inductive bias (e.g. sign structures) from tensor networks and various symmetries from autoregressive neural networks. We demonstrated our approach on quantum state learning and the challenging 2D  $J_{1} - J_{2}$  Heisenberg model with different system sizes and couplings. Our approach achieves better performance than both the original tensor network and autoregressive neural network while surpassing tensor networks with large bond dimensions as the system size increases. In addition, our approach is robust, independent of sign rule. Besides scientific applications, since both tensor networks and autoregressive neural networks have been applied to machine learning tasks such as supervised learning and generative modeling, our novel approach holds promise for better performance in these domains due to its exact sampling, expressivity, and symmetries.
Our Autoregressive Neural TensorNet, blending tensor networks and autoregressive neural networks, could advance our grasp of quantum phenomena, potentially fueling scientific breakthroughs. It enhances quantum material design and quantum computing through improved simulations and control of quantum states. This technology may also inspire new machine learning models for handling high-dimensional data. However, possible ethical and societal impacts, such as the use for chemical weapon development, require careful scrutiny.
The authors acknowledge support from the National Science Foundation under Cooperative Agreement PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi.org/). This material is based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. This material is also in part based upon work supported by the Air Force Office of Scientific Research under the award number FA9550-21-1-0317. The authors acknowledge the MIT SuperCloud (Reuther et al., 2018) and Lincoln Laboratory Supercomputing Center for providing (HPC, database, consultation) resources that have contributed to the research results reported within this paper. Some computations in this paper were run on the FASRC cluster supported by the FAS Division of Science Research Computing Group at Harvard University. ZC acknowledges the DARPA 134371-5113608 award and NSF 10434.
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A Additional Background
A.1 Quantum State Learning
In this section, we discuss more details about quantum state learning. As mentioned in the main paper, the quantum fidelity
$$
F = \left| \langle \phi | \psi_ {\theta} \rangle \right| ^ {2} = \left| \sum_ {\boldsymbol {x}} \phi^ {*} (\boldsymbol {x}) \psi_ {\theta} (\boldsymbol {x}) \right| ^ {2} \tag {A.1}
$$
measures the closeness of the two (normalized) quantum states  $|\phi \rangle$  and  $|\psi_{\theta}\rangle$ , with * denoting complex conjugation. By minimizing  $-\log F$ , one can obtain the  $|\psi_{\theta}\rangle$  closest to the target state  $|\phi \rangle$  (see Appendix D for optimization details). For small system sizes ( $\lesssim 20$  qubits), it is possible to enumerate all the basis  $x$  and compute  $-\log F$  exactly as in the case of this work. As mentioned in the main paper, we learn the random Bell state and shallow random circuit to test the expressivity and physics inductive bias of the ansatz. More specifically, the states are generated as follows
Random Bell State. A two-qubit Bell state is defined to be the state  $(|00\rangle + |11\rangle) / \sqrt{2}$ . We use the following algorithm to generate a  $n$ -qubit random Bell state ( $n$  is a multiple of 2).
Algorithm 1 Random Bell State Generation
$\pmb{a}\gets$  shuffle  $([1,\dots \frac{n}{2} ])$ $\pmb {b}\gets$  shuffle  $([\frac{n}{2} +1,\dots n])$ $\Psi \leftarrow \mathbb{I}$ $i\gets 1$    
while  $i\leq n / 2$  do  $\Psi .$  append(bell_state  $(a_i,b_i))$ $i\gets i + 1$    
end while   
 $|\psi \rangle \leftarrow$  product_state  $(\Psi)$    
return  $|\psi \rangle$
where bell_state  $(\cdot, \cdot)$  creates a Bell state given the two qubits and product_state([]) creates the tensor product state for a list of individual states. Each two-qubit Bell state in the random Bell state forms between the left half and right half of the system, allowing a maximum entanglement across a cut in the middle of the system.
Shallow Random Circuit. The shallow random circuit generates random states using the following algorithm.
Algorithm 2 Shallow Random Circuit State Generation
$|\psi \rangle \leftarrow |0\rangle^{\otimes n}$ $l\gets 1$    
while  $l\leq n_l$  do  
 $i\gets 1$    
while  $i <   n$  do  
 $U\gets$  random_GATE  $(i,i + 1)$ $|\psi \rangle \leftarrow U|\psi \rangle$ $i\gets i + 2$    
end while  
 $i\gets 2$    
while  $i <   n$  do  
 $U\gets$  random_GATE  $(i,i + 1)$ $|\psi \rangle \leftarrow U|\psi \rangle$ $i\gets i + 2$    
end while  
end while  
return  $|\psi \rangle$
where  $n_l$  is the number of layers, and random_GATE( $\cdot$ ,  $\cdot$ ) generates a random unitary gate on the two qubits. The random_GATE( $\cdot$ ,  $\cdot$ ) function is realized by first generating a (real-valued) Gaussian random matrix of size  $4 \times 4$ , following a QR decomposition of the matrix to obtain the unitary part as the random unitary gate.
Each of the above algorithms defines a distribution of states, which we average over multiple realizations to compute the mean and standard deviation of the learned quantum fidelity.
However, for large system sizes, the Hilbert space is too large, so one has to evaluate the fidelity stochastically. This can be achieved by rewriting
$$
f := \sum_ {\boldsymbol {x}} \phi^ {*} (\boldsymbol {x}) \psi_ {\theta} (\boldsymbol {x}) = \sum_ {\boldsymbol {x}} | \phi (\boldsymbol {x}) | ^ {2} \frac {\psi_ {\theta} (\boldsymbol {x})}{\phi (\boldsymbol {x})} = \mathbb {E} _ {\boldsymbol {x} \sim | \phi | ^ {2}} \frac {\psi_ {\theta} (\boldsymbol {x})}{\phi (\boldsymbol {x})}. \tag {A.2}
$$
Which allows the fidelity to be evaluated by sampling from  $|\phi|^2$ . The gradient of  $-\log F$  can be written as