Entanglement and computational complexity for 1D quantum many-body systems

Applied Math And Analysis Seminar

Thomas Barthel (Duke University, Physics Department)

Wednesday, March 29, 2017 -
12:00pm to 1:00pm
119 Physics

The Hilbert space dimension of quantum-many body systems grows exponentially with the system size. Fortunately, nature does usually not explore this monstrous number of degrees of freedom and we have a chance to describe quantum systems with much smaller sets of effective degrees of freedom. A very precise description for systems with one spatial dimension is based on so-called matrix product states (MPS). With such a reduced parametrization, the computation cost, needed to achieve a certain accuracy, is determined by entanglement properties (quantum non-locality) in the system.
I will give a short introduction to the notion of entanglement entropies and their scaling behavior in typical many-body systems. I will then employ entanglement entropies to bound the required computation costs in MPS simulations. This will lead us to the amazing conclusion that 1D quantum many-body systems can usually be simulated efficiently on classical computers, both for zero and finite temperatures, and for both gapless and critical systems.
In these considerations, we will encounter a number of mathematical concepts such as the theorem of typical sequences (central limit theorem), concentration of measure (Levy's lemma), singular value decomposition, path integrals, and conformal invariance.