The role of positivity in tensor network contraction

Description

It is well known that quantum systems with negative entries in their Hamiltonian are generally much harder to simulate than those with positive entries only. I will show that a very similar transition in hardness also shows up in tensor network simulations: Random tensor networks with negative entries are generally much harder to contract than those with positive entries only. While such a transition is expected from a sampling argument -- akin to the Monte Carlo negative sign problem -- I will show that the actual hardness transition occurs much earlier, namely already for a vanishingly small positive bias. I will discuss two independent and entirely different explanations which remarkably both yield the same transition point -- first, by relating the problem to a transition in the scaling of correlations (entanglement) in the boundary, and second, through a generalization of Barvinok's algorithm for approximating permanents.

Period	17 Oct 2024
Event title	Random Tensors 2024 Conference at IHP
Event type	Conference
Location	Paris, FranceShow on map
Degree of Recognition	International