Imaginary time evolution plays a crucial role in simulating thermal equilibrium on a computer, and the difficulty of the simulation largely depends on the entanglement growth under the imaginary time evolution. However, entanglement growth under the imaginary time evolution is not as well understood as its real-time counterpart for which the Lieb-Robinson bound is available. In fact, entanglement growth under imaginary time evolution can sometimes generate entanglement exponentially fast, and hence its classical simulation can be very difficult.
In this talk, we first make a connection between imaginary time evolution in gapless quantum many-body systems and wormholes in asymptotically AdS spacetime via the AdS/CFT correspondence. We will show that simple computations in the wormhole spacetime suggest that imaginary time evolution might be generically very slow.
Taking the observation from wormholes as a guiding hint, we present an analytic upper bound for the average entanglement Renyi entropy of imaginary time evolved product states. In 1D gapless systems, for example, the upper bound grows logarithmically, which is very slow. We also numerically testify the average entanglement entropy in lattice systems, and see it matches nicely with predictions from AdS/CFT.
As an application of our results, we show that the minimally entangled typical thermal state (METTS) algorithm is efficient in simulating thermal equilibrium at low temperatures in 1D gapless systems, and provides a significant speedup compared to other algorithms such as the purification method.
This talk is based on arXiv: 2309.02519 and ongoing works.
Hosts: M. Gonzalez, X. Yang, & J. Lap