On the correlation between entanglement and the negative sign problem

In this work, we study the correlation between entanglement and the negative sign problem in quantum Monte Carlo for the simulation of low-dimensional strongly correlated quantum many body systems. Entanglement entropy characterizes the difficulty of many-body simulation with tensor network state related methods, while the average sign measures the difficulty in many-body simulation for a variety of quantum Monte Carlo methods. Although there exist cases where one type of method works better than the other, it is desirable to find the possible correlation between entanglement and average sign for general hard strongly correlated systems regarding computational complexity. We take the doped two-dimensional Hubbard model as an example and numerically calculate the doping evolution of both the entanglement in the ground state with Density Matrix Renormalization Group and the average sign in the Auxiliary Field Quantum Monte Carlo simulation at low temperature. The results show that they are indeed correlated. The entanglement entropy (average sign) shows a peak (dip) around 20% doping, indicating that it is the difficult region for both methods. The vicinity of 20% doping is also the most intriguing region in both the Hubbard model and cuprate high-Tc superconductors where competing states with close energy intertwine with each other. Recognizing the correlation between entanglement and average sign provides new insight into our understanding of the difficulty in the simulation of strongly correlated quantum many-body systems.

💡 Research Summary

In this paper the authors investigate the relationship between two seemingly unrelated measures of computational difficulty in strongly correlated quantum many‑body systems: the entanglement entropy that limits tensor‑network methods such as DMRG, and the average sign that quantifies the severity of the sign problem in quantum Monte Carlo (QMC) simulations. Using the doped two‑dimensional Hubbard model as a testbed, they perform systematic calculations of the ground‑state von Neumann entanglement entropy with the Density Matrix Renormalization Group (DMRG) and of the average sign in Auxiliary‑Field QMC (AFQMC) at low temperature. The model parameters are U/t = 8 on a 4 × 16 cylinder (periodic along the short direction, open along the long direction). Doping is varied from half‑filling (0 % holes) up to about 50 % holes, with particular focus on the region around 20 % doping.

The AFQMC simulations are carried out at three inverse temperatures β = 2.5, 3, 4. As expected, the average sign is unity at half‑filling (the model is sign‑problem free there) and decreases rapidly as holes are introduced. The minimum average sign occurs near 20 % doping; the dip becomes deeper at larger β, reflecting the exponential scaling ⟨sign⟩ ∼ exp(−c β N) typical of sign‑problematic systems. The DMRG calculations are performed with several bond dimensions (D = 7 000–26 000). The von Neumann entanglement entropy is extracted by extrapolating to zero truncation error; the truncation error itself serves as an independent indicator of simulation difficulty. The entropy is smallest at half‑filling, rises with doping, reaches a maximum around 20 % holes, and then declines for higher dopings. The truncation error mirrors this behavior inversely: it is minimal where the entropy is low and grows where the entropy peaks.

To further characterize the entanglement structure, the authors plot the entanglement spectrum for representative dopings (0, 0.0625, 0.1875, 0.375, 0.4375). The spectrum at 20 % doping (h ≈ 0.1875) decays most slowly, confirming that the state is the most highly entangled. By contrast, at half‑filling the spectrum falls off steeply, consistent with a low‑entropy state. Interestingly, at h ≈ 0.375 the entropy shows a modest bump, yet the truncation error remains small, indicating that DMRG can still handle this doping efficiently.

The authors interpret these parallel trends as evidence that high entanglement and a severe sign problem are both signatures of a complex low‑energy landscape where many competing phases are nearly degenerate. This is precisely the regime that is of greatest interest in cuprate high‑Tc superconductors, where around 20 % hole doping the system exhibits a “pseudogap” or “stripe” phenomenology with intertwined orders. The paper also discusses the theoretical background: the sign problem is known to be NP‑hard in general, and its severity depends on the choice of basis and Hubbard‑Stratonovich transformation. While certain transformations (e.g., non‑local KT transformations) can eliminate the sign problem for specific 1D models, such remedies are not expected to be universal. Likewise, tensor‑network simulability is linked to the notion of “non‑stabilizerness”: states that can be generated by Clifford circuits are classically easy despite potentially large entanglement, whereas states with high non‑stabilizerness are hard for both tensor‑network and QMC approaches.

In summary, the study provides concrete numerical evidence that the entanglement entropy (and associated truncation error) and the average sign are strongly correlated across doping in the 2D Hubbard model. The region around 20 % doping is identified as the most challenging for both DMRG and AFQMC, and simultaneously the most physically intriguing due to competing orders. The authors suggest that monitoring both quantities can serve as a practical diagnostic for locating difficult yet potentially rich regions in other strongly correlated models, and may guide the development of hybrid or new algorithms that mitigate both entanglement growth and sign cancellations.