2. Introduction to MERA Sukhwinder Singh Macquarie University

3. Multidimensional array of complex numbers Tensors

4. Cost of Contraction a bc a d =

6. Made of layers

7. Disentanglers & Isometries

8. Different ways of looking at the MERA 1.Coarse-graining transformation. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.A specific realization of the AdS/CFT correspondence.

9. Coarse-graining transformation Length Scale

10. Coarse-graining transformation

11. Layer is a coarse-graining transformation

12. Coarse graining of operators

19. Scaling Superoperator

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

22. Types of MERA

23. Binary MERATernary MERA

24. Different ways of looking at the MERA 1.Coarse-graining transformation. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.A specific realization of the AdS/CFT correspondence.

25. Expectation values from the MERA

26. “Causal Cone” of the MERA

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

28. Recall! 1)Gapped Hamiltonian  2)Critical Hamiltonian 

29. In any MERA Correlations decay polynomially Entropy grows logarithmically

30. Correlations in the MERA

32. Entanglement entropy in the MERA

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

40. Different ways of looking at the MERA 1.Coarse-graining transformation. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.A specific realization of the AdS/CFT correspondence.

43. Time Space

44. Different ways of looking at the MERA 1.Coarse-graining transformation. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.A specific realization of the AdS/CFT correspondence.

45. Figure Source: Evenbly, Vidal 2011

47. MERA and spin networks

49. (Wigner-Eckart Theorem)

50. MERA and spin networks

53. Summary – MERA can be seen as.. 1.As defining a RG flow. 2.Efficient description of ground states on a classical computer. 3.Quantum circuit to prepare ground states on a quantum computer. 4.Specific realization of the AdS/CFT correspondence.