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 simulations with tensor network state related methods, while the average sign measures the difficulty in many-body simulations 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 temperatures. The results show that they are indeed correlated. The entanglement entropy (average sign) shows a peak (dip) around $20\%$ doping, indicating that it is a difficult region for both methods. The vicinity of $20\%$ doping is also the most intriguing region in both the Hubbard model and cuprate high-$T_c$ superconductors where competing states with close energy intertwine with each other. Recognizing the correlation between entanglement and average sign provides insights into our understanding of the difficulty in the simulation of strongly correlated quantum many-body systems.

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• Entanglement entropy
• Density matrix renormalization group
• Quantum Monte Carlo