DARTMOUTH COLLEGE PHYSICS AND ASTRONOMY

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Presentation transcript:

DARTMOUTH COLLEGE PHYSICS AND ASTRONOMY WHITFIELD GROUP DARTMOUTH COLLEGE  PHYSICS AND ASTRONOMY Ψ

Approximating Matrix Product States with Machine Learning Sam Greydanus, James Whitfield 4/25/2019

Objective Obtain a compressed representation of a system of entangled particles 4/25/2019 Document reference

What is entanglement? Simplest case, 2 sites Cannot describe one without the other Bell state Simplest case, 2 sites 4/25/2019 opentextbc.ca/chemistry/ wikimedia commons

opentextbc.ca/chemistry/ What is entanglement? 4/25/2019 opentextbc.ca/chemistry/ Wikimedia commons

What is entanglement? More generally… n-site cases How to partition? s1, s2, …, sN How to partition? N=8 4/25/2019 wikimedia commons phys.org/news/

Why is entanglement useful? Find lowest energy state Makes computation hard… 1. dwavesys.com/blog 4/25/2019

Curse of dimensionality Hamiltonian N entangled sites -> 2N dimensional matrix Simulation is hard Memory, computation 4/25/2019

A better idea: Matrix Product States Most entanglements are local 1D vs 2D and up How to find compressed representation? 4/25/2019 phys.org/news/

Computing MPS Density Matrix Renormalization Group (DMRG) Machine learning? 4/25/2019

1. Density Matrix Renormalization Groups 1D algorithm Finite, infinite 4/25/2019

1. Density Matrix Renormalization Groups Enlarge system 4/25/2019

1. Density Matrix Renormalization Groups Compute Hblock This is the DMRG step! 4/25/2019

1. Density Matrix Renormalization Groups min 4/25/2019

1. Density Matrix Renormalization Groups Rotate and truncate any operator! 4/25/2019

1. Density Matrix Renormalization Groups 4/25/2019 1. cond-mat/0603842v2

2. Deep Learning “…multilayer feedforward networks…are capable of approximating any Borel measurable function from one finite dimensional space to another to any desired degree of accuracy” (Stinchcombe and White 1989) 4/25/2019 doi: 10.1016/0893-6080(89)90020-8

2. Deep Learning Neural networks 4/25/2019 1. greydanus.github.io

2. Deep Learning Neural nets as function approximators Is small Arbitrary function of any dimension Is small 4/25/2019

2. Deep Learning Our training objective We know how to solve this! 4/25/2019

Progress Finished To do Working finite DMRG Working infinite DMRG Get MPS from DMRG Get MPS for AKLT state Approximate MPS with NN Evaluate model 2D case? Profit  4/25/2019

DARTMOUTH COLLEGE PHYSICS AND ASTRONOMY WHITFIELD GROUP DARTMOUTH COLLEGE  PHYSICS AND ASTRONOMY Ψ

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