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Introduction to MERA

Introduction to MERA. Sukhwinder Singh Macquarie University. Tensor s. M ultidimensional array of complex numbers. Cost of Contraction. =. a. a. b. c. d. Made of layers. Disentanglers & Isometries. Different ways of looking at the MERA. Coarse-graining transformation.

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Introduction to MERA

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  1. Introduction to MERA Sukhwinder Singh Macquarie University

  2. Tensors Multidimensional array of complex numbers

  3. Cost of Contraction = a a b c d

  4. Made of layers

  5. Disentanglers & Isometries

  6. Different ways of looking at the MERA • Coarse-graining transformation. • Efficient description of ground states on a classical computer. • Quantum circuit to prepare ground states on a quantum computer. • A specific realization of the AdS/CFT correspondence.

  7. Coarse-graining transformation Length Scale

  8. Coarse-graining transformation

  9. Layer is a coarse-graining transformation

  10. Coarse graining of operators

  11. Coarse graining of operators

  12. Coarse graining of operators

  13. Coarse graining of operators

  14. Coarse graining of operators

  15. Coarse graining of operators

  16. Coarse graining of operators

  17. Scaling Superoperator

  18. Scaling Superoperator

  19. MERA defines an RG flow Wavefunction on coarse-grained lattice with two sites

  20. Types of MERA

  21. Types of MERA Ternary MERA Binary MERA

  22. Different ways of looking at the MERA • Coarse-graining transformation. • Efficient description of ground states on a classical computer. • Quantum circuit to prepare ground states on a quantum computer. • A specific realization of the AdS/CFT correspondence.

  23. Expectation values from the MERA

  24. “Causal Cone” of the MERA

  25. But is the MERA good for representing ground states? Claim: Yes! Naturally suited for critical systems.

  26. Recall! • Gapped Hamiltonian  • Critical Hamiltonian 

  27. In any MERA Correlations decay polynomiallyEntropy grows logarithmically

  28. Correlations in the MERA

  29. Correlations in the MERA

  30. Entanglement entropy in the MERA

  31. Entanglement entropy in the MERA

  32. Entanglement entropy in the MERA

  33. Entanglement entropy in the MERA

  34. Entanglement entropy in the MERA

  35. Entanglement entropy in the MERA

  36. Entanglement entropy in the MERA

  37. Therefore MERA can be used a variational ansatz for ground states of critical Hamiltonians

  38. Different ways of looking at the MERA • Coarse-graining transformation. • Efficient description of ground states on a classical computer. • Quantum circuit to prepare ground states on a quantum computer. • A specific realization of the AdS/CFT correspondence.

  39. Time Space

  40. Different ways of looking at the MERA • Coarse-graining transformation. • Efficient description of ground states on a classical computer. • Quantum circuit to prepare ground states on a quantum computer. • A specific realization of the AdS/CFT correspondence.

  41. Figure Source: Evenbly, Vidal 2011

  42. MERA and spin networks

  43. MERA and spin networks

  44. MERA and spin networks (Wigner-Eckart Theorem)

  45. MERA and spin networks

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