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arXiv:2611.00603

Implications of the PXP ground-state sign structure for neural-network variational Monte Carlo

Jun Sur Richard Park, Kwangmin Yu

quant-phSubmitted Oct 31, 2026• Announced Nov 3, 2026
arXivHTMLPDF

Abstract

Positive-definite neural-network quantum states cannot directly represent the ground state of the PXP Hamiltonian, since the ground-state wavefunction carries a nontrivial relative-sign structure. Building on the chiral symmetry of the PXP Hamiltonian, we show that this sign structure is fixed exactly by the parity of the total up-spin number: a diagonal unitary transformation that assigns a sign to each spin configuration according to this parity maps the PXP Hamiltonian to its negative, rendering the transformed Hamiltonian stoquastic on the Rydberg-blockaded Hilbert space. Since the configuration graph of allowed configurations is connected, the Perron-Frobenius theorem guarantees a unique, strictly positive ground state of the transformed Hamiltonian, yielding an exact sign rule for the ground state of the original Hamiltonian in terms of the parity of the up-spin number. Variational Monte Carlo performed directly on the transformed Hamiltonian with a positive-definite ansatz therefore requires no architecture-specific modification to represent the true ground state. We evaluate the performance of this formulation using five representative neural-network quantum state architectures, SineKAN, reflection-symmetrized SineKAN, RBM, MLP, and LSTM, for different chain lengths.

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Positive-definite neural-network quantum states cannot directly represent the ground state of the PXP Hamiltonian, since the ground-state wavefunction carries a nontrivial relative-sign structure. Building on the chiral symmetry of the PXP Hamiltonian, we show that this sign structure is fixed exactly by the parity of the total up-spin number: a diagonal unitary transformation that assigns a sign to each spin configuration according to this parity maps the PXP Hamiltonian to its negative, rendering the transformed Hamiltonian stoquastic on the Rydberg-blockaded Hilbert space. Since the configuration graph of allowed configurations is connected, the Perron-Frobenius theorem guarantees a unique, strictly positive ground state of the transformed Hamiltonian, yielding an exact sign rule for the ground state of the original Hamiltonian in terms of the parity of the up-spin number. Variational Monte Carlo performed directly on the transformed Hamiltonian with a positive-definite ansatz therefore requires no architecture-specific modification to represent the true ground state. We evaluate the performance of this formulation using five representative neural-network quantum state architectures, SineKAN, reflection-symmetrized SineKAN, RBM, MLP, and LSTM, for different chain lengths.
Variational Monte Carlo with neural network quantum states has proven to be a promising avenue for evaluating the ground state energy of spin Hamiltonians. Based on anecdotal evidence, it has been claimed repeatedly in the literature that neural network quantum state simulations are insensitive to the sign problem. We present a detailed and systematic study of restricted Boltzmann machine (RBM) based variational Monte Carlo for quantum spin chains, resolving exactly how relevant stoquasticity is in this setting. We show that in most cases, when the Hamiltonian is phase connected with a stoquastic point, the complex RBM state can faithfully represent the ground state, and local quantities can be evaluated efficiently by sampling. On the other hand, we identify a number of new phases that are challenging for the RBM Ansatz, including non-topological robust non-stoquastic phases as well as stoquastic phases where sampling is nevertheless inefficient. Our results suggest that great care needs to be taken with neural network quantum state based variational Monte Carlo when the system under study is highly frustrated.
Strongly interacting quantum systems described by non-stoquastic Hamiltonians exhibit rich low-temperature physics. Yet, their study poses a formidable challenge, even for state-of-the-art numerical techniques. Here, we investigate systematically the performance of a class of universal variational wave-functions based on artificial neural networks, by considering the frustrated spin- 1/2 1 / 2 J_1-J_2 J 1 − J 2 Heisenberg model on the square lattice. Focusing on neural network architectures without physics-informed input, we argue in favor of using an ansatz consisting of two decoupled real-valued networks, one for the amplitude and the other for the phase of the variational wavefunction. By introducing concrete mitigation strategies against inherent numerical instabilities in the stochastic reconfiguration algorithm we obtain a variational energy comparable to that reported recently with neural networks that incorporate knowledge about the physical system. Through a detailed analysis of the individual components of the algorithm, we conclude that the rugged nature of the energy landscape constitutes the major obstacle in finding a satisfactory approximation to the ground state wavefunction, and prevents learning the correct sign structure. In particular, we show that in the present setup the neural network expressivity and Monte Carlo sampling are not primary limiting factors.
We study Heisenberg antiferromagnets through a graph-theoretic representation of their Hamiltonians. In the computational basis of a fixed-magnetization sector, the Heisenberg Hamiltonian equals a constant minus the weighted Laplacian of a graph whose vertices are spin configurations and whose edges connect configurations related by exchanging two antiparallel spins on a lattice bond. We call this graph the Hilbert graph (HG) which is isomorphic to the token graph of the lattice over which the Hamiltonian is defined. The exact ground-state energy is then determined by the largest eigenvalue of the HG Laplacian, which is the spectral relaxation of weighted Max-Cut on the HG. With the amplitudes fixed, the phase-dependent part of the variational energy defines an antiferromagnetic classical XY model on the HG, with couplings set by the amplitudes. Restricting the variational state to be real, i.e., restricting its phases to {0,π}\{0,π\}{0,π}, reduces the fixed-amplitude energy minimization to solving an antiferromagnetic Ising model and hence a weighted Max-Cut on the HG. When the weights are generated from non-degenerate ground-state amplitudes, the optimal phases give the ground-state sign structure which can be recovered from the weights in time polynomial in the size of the HG. On the Hamiltonian side, a diagonal sign transformation removes all positive off-diagonal matrix elements if and only if the HG is bipartite, which reproduces the Marshall sign rule on bipartite lattices through the graph two-coloring problem; otherwise, the minimal remaining positive weight equals the sum of HG edge weights left uncut by the Max-Cut. This graph formulation establishes a common framework for understanding wave function sign structure, Hamiltonian stoquasticity, non-convexity of phase optimization in variational Monte Carlo, and the average sign in quantum Monte Carlo.
Strongly interacting quantum systems described by non-stoquastic Hamiltonians exhibit rich low-temperature physics. Yet, their study poses a formidable challenge, even for state-of-the-art numerical techniques. Here, we investigate systematically the performance of a class of universal variational wave-functions based on artificial neural networks, by considering the frustrated spin-1/2 J1 − J2 Heisenberg model on the square lattice. Focusing on neural network architectures without physics-informed input, we argue in favor of using an ansatz consisting of two decoupled real-valued networks, one for the amplitude and the other for the phase of the variational wavefunction. By introducing concrete mitigation strategies against inherent numerical instabilities in the stochastic reconfiguration algorithm we obtain a variational energy comparable to that reported recently with neural networks that incorporate knowledge about the physical system. Through a detailed analysis of the individual components of the algorithm, we conclude that the rugged nature of the energy landscape constitutes the major obstacle in finding a satisfactory approximation to the ground state wavefunction, and prevents learning the correct sign structure. In particular, we show that in the present setup the neural network expressivity and Monte Carlo sampling are not primary limiting factors.
Deep neural networks have been very successful as highly accurate wave function Ansätze for variational Monte Carlo calculations of molecular ground states. We present an extension of one such Ansatz, FermiNet, to calculations of the ground states of periodic Hamiltonians, and study the homogeneous electron gas. FermiNet calculations of the ground-state energies of small electron gas systems are in excellent agreement with previous initiator full configuration interaction quantum Monte Carlo and diffusion Monte Carlo calculations. We investigate the spin-polarized homogeneous electron gas and demonstrate that the same neural network architecture is capable of accurately representing both the delocalized Fermi liquid state and the localized Wigner crystal state. The network converges on the translationally invariant ground state at high density and spontaneously breaks the symmetry to produce the crystalline ground state at low density, despite being given no a priori knowledge that a phase transition exists.
We investigate the Shastry-Sutherland model (SSM), i.e., spin-1/2 quantum Heisen- berg model on a Shastry-Sutherland lattice, using a newly emerging approach exploiting well-developed machine learning techniques. We utilize neural networks as variational quantum states in quantum Monte Carlo investigations of the ground state properties. We first focus on SSM without an external magnetic field. For small lattices accessible via exact diagonalization, we compare the precision of various architectures based on re- stricted Boltzmann machines (RBM) or group-convolutional neural networks. The most versatile and precise architecture, namely complex-valued RBM, is then applied for larger lattices. Here we investigate the frustrated regime. We show that the RBM is able to represent all three of the major and fundamentally different phases of SSM. Finally, we apply the complex-valued RBM for SSM in a finite external magnetic field. We find that it cannot capture the intriguing steps in magnetization typical for SSM correctly due to its tendency to prefer more ordered states with higher magnetization. 1