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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

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  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]].

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  3. Quantum states and tensor networks

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  4. Primer on tensors

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  5. 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

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  6. 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

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  7. 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)

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

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  9. Multiscale Entanglement Renormalization Ansatz

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

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  11. MERA: Efficient computations
    Causal structure of influence simplifies computations

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  12. 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

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

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

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  15. Quantum gravity

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  16. 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.

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  17. 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

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  18. 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)

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  19. 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

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

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

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

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  23. 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]]

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