1. Introduction to Tensor Network States Sukhwinder Singh Macquarie University (Sydney)

2. Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

3. Quantum many body system in 1-D

5. H ow many qubits can we represent with 1 GB of memory? Here, D = 2. To add one more qubit double the memory.

6. But usually, we are not interested in arbitrary states in the Hilbert space. Typical problem : To find the ground state of a local Hamiltonian H,

7. Ground states of local Hamiltonians are special

8. 1)Gapped Hamiltonian  2)Critical Hamiltonian  Properties of ground states in 1-D

9. We can exploit these properties to represent ground states more efficiently using tensor networks.

10. Ground states of local Hamiltonians

11. Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

12. Multidimensional array of complex numbers Tensors

13. Contraction a bc a d =

14. Contraction a bc a d =

15. Contraction a bc a d = a c

16. Trace = = a

17. Tensor product

18. Decomposition a bc a d = = =

19. Decomposing tensors can be useful = Number of components in M = Number of components in P and Q = Rank(M) =

20. Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

21. Many-body state as a tensor

22. Expectation values

23. Correlators

24. Reduced density operators

25. Tensor network decomposition of a state

26. Essential features of a tensor network 1)Can efficiently store the TN in memory 2) Can efficiently extract expectation values of local observables from TN Total number of components = O(poly(N)) Computational cost = O(poly(N))

27. Number of tensors in TN = O(poly(N)) is independent of N

28. Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

29. Matrix Product States

31. Recall!

32. Expectation values

37. But is the MPS good for representing ground states?

38. Claim: Yes! Naturally suited for gapped systems.

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

40. In any MPS Correlations decay exponentially Entropy saturates to a constant

41. Recall!

42. Correlations in a MPS

48. Entanglement entropy in a MPS

54. 1.Variational optimization by minimizing energy 2. Imaginary time evolution MPS as an ansatz for ground states

55. Contents The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

57. Summary The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA

58. Thanks !