Tensor networks and emergent geometry Sivaramakrishnan Swaminathan Theory Group Department of Physics, The University of Texas at Austin 11 August 2017 Research presentation @ Vicarious AI 

Based on B. Czech, P. H. Nguyen and S. Swaminathan A defect in holographic interpretations of tensor networks JHEP 1703, 090 (2017) doi:10.1007/JHEP03(2017)090 [arXiv:1612.05698 [hep-th]]. 

Quantum states and tensor networks 

Primer on tensors 

Quantum states ∼ probability distributions States are vectors in a Hilbert space, with ψ|ψ = 1 Alternately, density matrices ρ ≡ |ψ ψ| with Tr ρ = 1 Compute expectation values of operators O ≡ Tr [ρO] = ψ|O|ψ Entanglement ∼ Mutual Information Bell’s inequality 

Many-body states are high-dimensional Joint distribution on N random variables =⇒ dim. ∼ exp (N) Would be nice to handle infinite systems This is why QM is hard, even though it’s linear 

Exploit physical principles for efficiency? Goal: Model typical states eg: Lowest energy state; use “power method” Typical states form a tiny fraction of Hilbert space Locality =⇒ “area scaling” of entanglement Additional symmetries (translation, scale invariance) 

Tensor networks Approximate joint distributions by a variational ansatz which allows efficient representation and computation Number of variational parameters scale favorably 

Multiscale Entanglement Renormalization Ansatz 

MERA: Constraints = u† u i j k l i j k l = w† w i j i j 

MERA: Efficient computations Causal structure of influence simplifies computations 

Computing the variational parameters Iteratively improve tensors layer-by-layer, by Ascending the Hamiltonian (forward-propagation) Descending the state (back-propagation) Alternating minimization to optimize tensors Note: Recent developments demonstrate faster learning techniques 

Minimal Updates Proposal (MUP) Modeling scale invariant systems with a local defect Originally motivated by computational convenience We provide a principled justification (Boundary OPE) 

Searching for principles MERA works fabulously well in practice! Renormalization group flow Connections to quantum gravity Link between MERA and deep learning 

Quantum gravity 

A hard problem Holy grail of fundamental physics for the past half-century If we naively combine gravity and quantum mechanics “Infinities” from marginalizing over infinitely many DOFs (dependence on prior; loss of predictivity) Physicists care about answers being finite and unique, so they may be compared with experiment. 

Holographic quantum gravity Quantum Mechanics on "boundary" = Quantum Gravity in "bulk" (justifications from string theory) Figure from: https://commons.wikimedia.org/wiki/File:AdS3_(new).png 

MERA ??? ←→ Holography MERA models entanglement structure in quantum states Holographic spacetime maps entanglement structure Bulk geodesic length ∼ = boundary entanglement (Ryu-Takayanagi formula) Emergent direction encodes scale-dependence of entanglement (Renormalization group flow) 

Searching for a more direct relationship Lots of discussion over the last several years. . . I’ll summarize recent understanding, without detailing justifications MERA discretizes the integral transform of bulk geometry 

Simplest example: hyperbolic space H2 Full conformal symmetry Easy to verify correspondence with MERA (I’m happy to sketch the calculation if desired) 

Our generalization: defect geometries Reduced symmetry: more nuanced duality; harder computations Proposed a novel generalization of the MUP: Rayed MERA 

Summary Brief overview of tensor networks MERA networks Holographic quantum gravity Relation between tensor networks and holographic geometry 

Thank you! 

Other publications P. Agrawal, C. Kilic, S. Swaminathan and C. Trendafilova Secretly Asymmetric Dark Matter Phys. Rev. D 95, no. 1, 015031 (2017) doi:10.1103/PhysRevD.95.015031 [arXiv:1608.04745 [hep-ph]]. C. Kilic and S. Swaminathan Can A Pseudo-Nambu-Goldstone Higgs Lead To Symmetry Non-Restoration? JHEP 1601, 002 (2016) doi:10.1007/JHEP01(2016)002 [arXiv:1508.05121 [hep-ph]]