Entanglement entropy and the simulation of quantum systems Open discussion with pde2007

1. Entanglement entropyandthe simulation of quantum systemsOpen 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. Introduction Introduction • Classical Theory • Classical simulation • Quantum simulation • Quantum Mechanics • Classical simulation • Quantum simulation Classical computer ? Quantum computer 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

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

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

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

7. Introduction Introduction • 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

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

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

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

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

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

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

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

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

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

17. Efficient description Efficient computation of a local action U

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

19. Efficient description Intelligent way to represent, manipulate, read-out entanglement Adaptive representation for correlations among parties 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

20. i2=1 i2=2 i2=3 i2=4 Efficient description Spin-off: Image compression | i2 i1 | i1 105| 2,1  i1=1 i1=2 i1=3 i1=4 RG addressing level of grey pixel address

21. Efficient description • 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 Max  = 81  = 1 PSNR=17  = 4 PSNR=25  = 8 PSNR=31

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

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

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

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

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

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

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

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

30. NP-complete Entanglement for NP-complete problems Exact Cover A clause is accepted if 001 or 010 or 100 Exact Cover is NP-complete 0 1 1 0 0 1 1 0 instance For every clause, one out of eight options is rejected • 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. NP-complete t s(T)=1 H(s(t)) = (1-s(t)) H0 + s(t) Hp s(0)=0 Adiabatic quantum evolution (Farhi,Goldstone,Gutmann) Inicial hamiltonian Problem hamiltonian Adiabatic theorem: if E E1 gmin E0 t

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

33. n=100 right solution found with MPS among 1030 states

34. Physics vs. simulation Physics vs. simulation

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

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

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

38. MPS i) ii) iii) iv)

39. MPS Schmidt decomposition produces orthonormal L,R states

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

41. MPS Read out Energy Entropy for half chain

42. New ideas New ideas Heisenberg model Trotter 2 order, =.001

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

44. MPS S Perfect alignment M

45. New ideas 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

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

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

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

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

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