Brute-force positivization of $J_1-J_2$ model ground states

TL;DR

This work examines how to positivize the ground states of the one-dimensional $J_1-J_2$ Heisenberg chain in the strong-frustration regime by brute-forcing single-qubit $R^z( heta)$ rotations and, when beneficial, adding two-qubit CZ gates. It confirms the Marshall-Peierls rule as exact for $J_2/J_1\in[0,0.5]$ and identifies parity- and boundary-condition–dependent positivization protocols for $0.5< J_2/J_1\le 2$, including odd/even $N/2$ schemes and an enhanced 18-spin method. The CZ-augmented approach (MPR+CZ) significantly improves positivity at the cost of introducing multi-spin interactions in the transformed Hamiltonian, while also increasing entanglement. These findings have practical implications for sign-structure–aware variational methods and neural quantum states, potentially enhancing ground-state estimates and optimization efficiency.

Abstract

Exploring sign structures of quantum wave functions attracts considerable attention due to the potential for advances in modeling complex phases of matter. This stimulates developing different optimization procedures for imitating and manipulating sign structures of quantum states. In this work, utilizing a brute force approach based on a set of single-qubit transformations we evaluate protocols enabling positivization of the one-dimensional $J_1 -J_2$ model ground states in the regime of strong frustration. Based on the obtained positivization results, we show the difference between the cases of periodic and open boundary conditions, and also establish the dependence of the sign structure on parity of the simulated spin chains.

Figures (11)

• Figure 1: Schematic of the one-dimensional $J_1-J_2$ model with open (a) and periodic (b) boundary conditions. Two variants of subsystem partition for the limiting cases of $J_2=0$ (c) and $J_1=0$ (d). The black and green lines denote the magnetic interactions between nearest ($J_1$) and next-nearest ($J_2$) neighbours, respectively.
• Figure 2: Schematic of one-qubit gates protocols used in this work for positivization of the ground states of the one-dimensional $J_1-J_2$ model. ${\rm R}^{z} (\theta)$ denotes the rotation operator about the z axis by $\theta$. (a) Marshall-Peierls scheme derived in Ref.Marshall for the case of $J_{2} = 0$. (b) Odd and even protocols obtained in this work with brute-force positivization procedure for quantum systems characterized by different parity of $N_{\frac{1}{2}}$. (c) The protocol introduced in Ref.Torlai_positivization by using an optimization procedure in the case of the one-dimensional $J_1 - J_2$ model with periodic boundary conditions at $J_2 / J_1 = 2$ for even $N_{\frac{1}{2}}$. The blue line marks the spins belonging to the sublattice A, the red line --- B.
• Figure 3: Values of sign function, Eq.\ref{['metric']} calculated for the pozitivized ground states of the $J_1-J_2$ model with odd (a) and even (b) $N_{\frac{1}{2}}$. The red line indicates the shift of the minimum value of the sign function with the increase of the system size.
• Figure 4: Overlap between the ground states of the $J_1 -J_2$ model calculated at different $J_2$ and three distinct solutions of the same model: (a) the eigenstate obtained at $J_1 = 1$ and $J_2 = 0$, (b) the eigenstate obtained at $J_1 = 1$ and $J_2 = 0.5$ and (c) the eigenstate obtained at $J_1 = 0$ and $J_2 = 1$. These calculations were done for the system with open boundary conditions.
• Figure 5: Dependence of the fraction of the basis states with negative amplitudes $N_n/N_a$ on $J_{2}$. (a) and (b) correspond to the cases of odd and even $N_{\frac{1}{2}}$, respectively.