1. Entanglement entropy and the simulation of quantum systems Open discussion with pde2007 José Ignacio Latorre Universitat de Barcelona Benasque, September 2007

2. Physics Theory 1 Theory 2 Exact solution Approximated methods Simulation Classical Simulation Quantum Simulation

3. Classical Theory Classical simulation Quantum simulation Quantum Mechanics Classical simulation Quantum simulation Classical simulation of Quantum Mechanics is related to our ability to support large entanglement Classical simulation may be enough to handle e.g. ground states: MPS, PEPS, MERA Quantum simulation needed for time evolution of quantum systems and for non-local Hamiltonians Classical computer Quantum computer ? Introduction

4. Is it possible to classically simulate faithfully a quantum system? Quantum Ising model represent evolve read Introduction

5. The lowest eigenvalue state carries a large superposition of product states Ex. n=3

6. Naïve answer: NO Exponential growth of Hilbert space Classical representation requires d n complex coefficients n A random state carries maximum entropy Introduction computational basis Is it possible to classically simulate faithfully a quantum system?

7. Refutation Realistic quantum systems are not random symmetries (translational invariance, scale invariance) local interactions little entanglement We do not have to work on the computational basis use an entangled basis Introduction

8. Plan Measures of entanglement Efficient description of slight entanglement Entropy: physics vs. simulation New ideas: MPS, PEPS, MERA

9. Measures of entanglement One qubit Quantum superposition Two qubits Quantum superposition + several parties = entanglement Measures of entanglement

10. Separable states e.g. Entangled states e.g. Measures of entanglement Local realism is dropped Quantum non-local correlations

11. Measures of entanglement Pure states: Schmidt decomposition = Singular Value Decomposition A B  =min(dim H A, dim H B ) is the Schmidt number Measures of entanglement Entangled state Diagonalise A Measures of entanglement Separable state

12. Von Neumann entropy of the reduced density matrix Measures of entanglement Product state large Very entangled state e-bit

13. Maximum Entropy for n-qubits Strong subadditivity theorem implies entropy concavity on a chain of spins SLSL S L-M S L+M S max =n Measures of entanglement

14. Efficient description for slightly entangled states A B Schmidt decomposition Efficient description Retain eigenvalues and changes of basis Efficient description

15. Slight entanglement iff  poly(n)<< d n Representation is efficient Single qubit gates involve only local updating Two-qubit gates reduces to local updating Readout is efficient Vidal 03: Iterate this process efficient simulation Efficient description

16. Graphic representation of a MPS Efficient computation of scalar products operations Efficient description

17. Efficient computation of a local action U Efficient description

18. Matrix Product States i α Approximate physical states with a finite  MPS canonical form PVWC06 Efficient description

19. Intelligent way to represent, manipulate, read-out entanglement Classical simplified analogy: I want to send 16,24,36,40,54,60,81,90,100,135,150,225,250,375,625 Instruction: take all 4 products of 2,3,5 MPS= compression algorithm Efficient description Adaptive representation for correlations among parties

20. i 1 =1 i 1 =2 i 1 =3 i 1 =4 | i 1  i 2 =1 i 2 =2 i 2 =3 i 2 =4 | i 2 i 1  105| 2,1  Spin-off: Image compression pixel addresslevel of grey RG addressing Efficient description

21.  = 1 PSNR=17  = 4 PSNR=25  = 8 PSNR=31 Max  = 81 QPEG Read image by blocks Fourier transform RG address and fill Set compression level:  Find optimal gzip (lossless, entropic compression) (define discretize Γ’s to improve gzip) diagonal organize the frequencies and use 1d RG work with diferences to a prefixed table Efficient description

22. Note: classical problems with a direct product structure! Spin-off: Differential equations Efficient description

23. Matrix Product States for continuous variables Harmonic chains MPS handles entanglementProduct basis Truncate  tr d tr Efficient description

24. Nearest neighbour interaction Minimize by sweeps Choose Hermite polynomials for local basis optimize over a Efficient description

25. Results for n=100 harmonic coupled oscillators (lattice regularization of a quantum field theory) d tr =3  tr =3 d tr =4  tr =4 d tr =5  tr =5 d tr =6  tr =6 Newton-raphson on a Efficient description Error in Energy

26. Success of MPS will depend on how much entanglement is present in the physical state PhysicsSimulation IfMPS is in very bad shape Back to the central idea: entanglement support Physics vs. simulation

27. Exact entropy for a reduced block in spin chains At Quantum Phase TransitionAway from Quantum Phase Transition Physics vs. simulation

28. Maximum entropy support for MPS Maximum supported entanglement Physics vs. simulation

29. Faithfullness = Entanglement support Spin chains MPS Spin networks Area law Computations of entropies are no longer academic exercises but limits on simulations PEPS Physics vs. simulation

30. Exact Cover A clause is accepted if 001 or 010 or 100 Exact Cover is NP-complete 0 1 1 0 For every clause, one out of eight options is rejected instance NP-complete Entanglement for NP-complete problems 3-SAT is NP-complete k-SAT is hard for k > 2.41 3-SAT with m clauses: easy-hard-easy around m=4.2n

31. Adiabatic quantum evolution (Farhi,Goldstone,Gutmann) H(s(t)) = (1-s(t)) H 0 + s(t) H p Inicial hamiltonianProblem hamiltonian s(0)=0 s(T)=1 t Adiabatic theorem: if E1E1 E0E0 E t g min NP-complete

32. Adiabatic quantum evolution for exact cover |0> |1> (|0>+|1>) …. (|0>+|1>) NP-complete NP problem as a non-local two-body hamiltonian!

33. n=100 right solution found with MPS among 10 30 states

34. Non-critical spin chainsS ~ ct Critical spin chainsS ~ log 2 n Spin chains in d-dimensionsS ~ n d-1/d Fermionic systems?S ~ n log 2 n NP-complete problems 3-SAT Exact Cover S ~.1 n Shor FactorizationS ~ r ~ n Physics vs. simulation

35. New ideas MPS using Schmidt decompositions (iTEBD) Arbitrary manipulations of 1D systems PEPS 2D, 3D systems MERA Scale invariant 1D, 2D, 3D systems New ideas Recent progress on the simulation side

36. 2. Euclidean evolution Non-unitary evolution entails loss of norm are sums of commuting pieces Trotter expansion MPS

37. Ex: iTEBD (infinite time-evolving block decimation) even odd AAABB BA A AB B A  B Translational invariance is momentarily broken MPS

38. i) ii) iii) iv) MPS

39. Schmidt decomposition produces orthonormal L,R states MPS

40. Moreover, sequential Schmidt decompositions produce isometries = are isometries MPS

41. Energy Read out Entropy for half chain MPS

42. Heisenberg model  =2 -.42790793S=.486  =4 -.44105813S=.764  =6 -.44249501S=.919  =8 -.44276223S=.994  =16 -.443094S=1.26 Trotter 2 order,  =.001 New ideas

43. entropy energy Convergence MPS Local observables are much easier to get than global entanglement properties

44. S M Perfect alignment MPS

45. New ideas PEPS: Projected Entangled Pairs physical index ancillae Good: PEPS support an area law!! Bad: Contraction of PEPS is #P New results beat Monte Carlo simulations New ideas

46. A B Entropy is proportional to the boundary Contour A = L “Area law” Some violations of the area law have been identified PEPS

47. As the contraction proceeds, the number of open indices grows as the area law PEPS 2D seemed out of reach to any efficient representation Contraction of PEPS is #P Building physical PEPS would solve NP-complete problems

48. Yet, for translational invariant systems, it comes down to iTEBD !! EE Comparable to quantum Monte Carlo? E PEPS E becomes a non-unitary gate PEPS

49. Results for 2D Quantum Ising model (JOVVC07) MC PEPS

50. MERA MERA: Multiscale Entanglement Renormalization Ansatz Intrinsic support for scale invariance!! MERA

51. All entanglemnent on one line All entanglemnent distributed on scales MERA

52. Contraction = Identity MERA U Update

53. If MPS, PEPS, MERA are a good representation of QM Approach hard problems Precision Can we simulate better than Monte Carlo? Are MPS, PEPS and MERA the best simulation solution? Spin-off? Physics: Scaling of entropy: Area law << Volume law Translational symmetry and locality reduce dramatically the amount of entanglement Worst case (max entropy) remains at phase transition points Physics vs. simulation

54. Quantum Complexity Classes QMA L is in QMA if there exists a fixed  and a polynomial time verifier (V) such that What is the QMA-complete problem? Feynman idea (shaped by Kitaev) QMA

55. QMA-complete problem Log-local hamiltonian 5-body 3-body 2-body (non-local interactions) 2-body (nearest neighbor 12 levels interaction)! Given H on n-party decide if

56. Open problems Separability problem (classification of completely positive maps) Classification of entanglement (canonical form of arbitrary tensors) Better descriptions of quantum many-body systems Spin-off of MPS?? Rigorous results for PEPS, MERA Need for theorems for gaps/correlation length/size of approximation Exact diagonalisation of dilute quantum gases (BEC) Classification of Quantum Computational Complexity classes ….