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Introduction to Tensor Network States

Introduction to Tensor Network States. Sukhwinder Singh Macquarie University (Sydney). Contents. The quantum many body problem. Diagrammatic Notation What is a tensor network? Example 1 : MPS Example 2 : MERA. Quantum many body system in 1-D.

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Introduction to Tensor Network States

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  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

  4. How many qubits can we represent with 1 GB of memory?Here, D = 2.To add one more qubitdouble the memory.

  5. 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,

  6. Ground states of local Hamiltonians are special

  7. Properties of ground states in 1-D • Gapped Hamiltonian  • Critical Hamiltonian 

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

  9. Ground states of local Hamiltonians

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

  11. Tensors Multidimensional array of complex numbers

  12. Contraction = a a b c d

  13. Contraction = a a b c d

  14. Contraction a a = b c d a c

  15. Trace = a =

  16. Tensor product

  17. Decomposition = a a = b c d =

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

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

  20. Many-body state as a tensor

  21. Expectation values

  22. Correlators

  23. Reduced density operators

  24. Tensor network decomposition of a state

  25. Essential features of a tensor network • Can efficiently storethe 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))

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

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

  28. Matrix Product States

  29. Recall!

  30. Expectation values

  31. Expectation values

  32. Expectation values

  33. Expectation values

  34. Expectation values

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

  36. But is the MPS good for representing ground states? Claim: Yes! Naturally suited for gapped systems.

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

  38. In any MPS Correlations decay exponentiallyEntropy saturates to a constant

  39. Recall!

  40. Correlations in a MPS

  41. Correlations in a MPS

  42. Correlations in a MPS

  43. Correlations in a MPS

  44. Correlations in a MPS

  45. Correlations in a MPS

  46. Entanglement entropy in a MPS

  47. Entanglement entropy in a MPS

  48. Entanglement entropy in a MPS

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