Insights into Energy Spectra of Matrix Product States and Their Implications for Quantum Simulations

In recent research, authors J. Maxwell Silvester, Giuseppe Carleo, and Steven R. White have explored the energy spectra of matrix product states, particularly in the context of strongly correlated quantum systems. Their paper, titled "Unusual energy spectra of matrix product states," was submitted on August 24, 2024, and is available on arXiv (arXiv:2408.13616).

The study focuses on how the decomposition of approximate solutions into exact eigenstates of the Hamiltonian affects the performance of simulations in quantum mechanics. The authors found that, contrary to expectations, the energy spectra of approximate matrix product state ground states exhibit a roughly constant contribution to the spectra at surprisingly high energies. This behavior persists even as the bond dimension increases, which reduces the amplitude but not the extent of these high-energy tails.

This finding has significant implications for sampling-based methods in quantum simulations. The authors noted that estimating the energy variance through sampling yields larger fluctuations than anticipated. They propose that bounding the most extreme samples can reduce noise in variance estimates, although this introduces a bias. Nevertheless, they discovered that this biased variance estimator serves as an effective surrogate for the variance when extrapolating the ground-state energy, outperforming other extrapolation methods in both accuracy and computational efficiency.

The results of this research could enhance the understanding of matrix product states and improve the efficiency of simulations in quantum physics, which is crucial for advancing quantum computing technologies. The full paper can be accessed at arXiv:2408.13616.