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Tensor networks and emergent geometry

Tensor networks and emergent geometry

Siva Swaminathan

August 11, 2017

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  1. 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
  2. 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]].
  3. 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
  4. 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
  5. 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)
  6. Tensor networks Approximate joint distributions by a variational ansatz which

    allows efficient representation and computation Number of variational parameters scale favorably
  7. MERA: Constraints = u† u i j k l i

    j k l = w† w i j i j
  8. 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
  9. Minimal Updates Proposal (MUP) Modeling scale invariant systems with a

    local defect Originally motivated by computational convenience We provide a principled justification (Boundary OPE)
  10. Searching for principles MERA works fabulously well in practice! Renormalization

    group flow Connections to quantum gravity Link between MERA and deep learning
  11. 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.
  12. 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
  13. 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)
  14. 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
  15. Simplest example: hyperbolic space H2 Full conformal symmetry Easy to

    verify correspondence with MERA (I’m happy to sketch the calculation if desired)
  16. Our generalization: defect geometries Reduced symmetry: more nuanced duality; harder

    computations Proposed a novel generalization of the MUP: Rayed MERA
  17. Summary Brief overview of tensor networks MERA networks Holographic quantum

    gravity Relation between tensor networks and holographic geometry
  18. 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]]