Benchmarking Computational Methods for Hydrodynamics of Noisy Quantum Chains

In ergodic quantum spin chains, locally conserved quantities such as energy or particle number evolve according to hydrodynamic equations as they relax to equilibrium. Qualitatively the hydrodynamics is typically diffusive; however, quantitatively predicting the diffusion constant is generally challenging. We investigate the complexity and accuracy of computational techniques to compute diffusion constants and the approach to hydrodynamics using tensor network time-evolution of operators [1]. Density matrix truncations [2] greatly improve the convergence of such simulations, resulting in more precise estimates of the diffusion constant than previously achieved. The predictions and computational cost of the simulations are compared with computations using Krylov methods and the universal operator growth hypothesis [3]. We also use these methods to simulate hydrodynamics in spin chains with depolarizing noise, which is unavoidably present in experimentally accessible quantum systems. The noisy dynamics is well-described by a diffusion equation with an added slow decay of the conserved quantities and a noise-modified diffusion constant, and can be simulated with reduced computational complexity due to the destruction of long-range entanglement.

[1] Vidal, Phys. Rev. Lett. 91, 147902 (2003)

[2] White et al., Phys. Rev. B 97, 035127 (2018)

[3] Parker et al., Phys. Rev. X 9, 041017 (2019)