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Quantum Information and the simulation of quantum systems

Quantum Information and the simulation of quantum systems. José Ignacio Latorre Universitat de Barcelona Perugia, July 2007. In collaboration with: Sofyan Iblisdir, Luca Tagliacozzo Arnau Riera, Thiago Rodrigues de Oliveira, José María Escartín, Vicent Picó

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Quantum Information and the simulation of quantum systems

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  1. Quantum Informationandthe simulation of quantum systems José Ignacio Latorre Universitat de Barcelona Perugia, July 2007 In collaboration with: Sofyan Iblisdir, Luca Tagliacozzo Arnau Riera, Thiago Rodrigues de Oliveira, José María Escartín, Vicent Picó Román Orús, Artur García-Sáez, Frank Verstraete, Miguel Aguado, Ignacio Cirac

  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 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 Quantum simulation needed for typical evolution of Quantum systems (linear entropy growth to maximum)

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

  5. Misconception: NO Exponential growth of Hilbert space n Classical representation requires dn complex coefficients A random state carries maximum entropy

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

  7. e.g: efficient description for slightly entangled states Schmidt decomposition A B  = min(dim HA, dim HB) Schmidt number A product state will have

  8. Vidal 03: Iterate this process A product state iff • Slight entanglement iff poly(n)<<dn • Representation is efficient • Single qubit gates involve only local update • Two-qubit gates reduces to local updating efficient simulation

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

  10. Graphic representation of a MPS Efficient computation of scalar products operations

  11. Local action on MPS U

  12. Intelligent way to represent and manipulate entanglement Classical 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

  13. i2=1 i2=2 i2=3 i2=4 Crazy ideas: Image compression | i2 i1 | i1 105| 2,1  i1=1 i1=2 i1=3 i1=4 RG addressing level of grey pixel address

  14. 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

  15. Crazy ideas: Differential equations Crazy ideas: Differential equations Crazy ideas: Shor’s algorithm with MPS Crazy ideas: Shor’s algorithm with MPS Constructed: adder, multiplier, multiplier mod(N) Note: classical problems with a direct product structure!

  16. 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

  17. Exact entropy for a reduced block in spin chains At Quantum Phase Transition Away from Quantum Phase Transition

  18. Maximum entropy support for MPS Maximum supported entanglement

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

  20. Physics Simulation VLRK02-03 OL04 For 3-SAT LLRV04 Exact RG on states VCLRW05 OLRV05 Lipkin model 100-qubit Ex-cover instance BOLP05 Image compression L05 OLEC06 RL06 Area law ILO06 Laughlin ILO06 Continuous variables

  21. Local (12 levels), nearest neighbor H is QMA-complete!! AGK07

  22. Keep in mind: Area law << Volume law Translational symmetry and locality have reduced dramatically the amount of entanglement Worst case (max entropy) remains at phase transition points • MPS and PEPS are a good representation of QM • Approach new problems • Precision • Can we do any better than DMRG? • e.g.: Faithfull numbers for entropy? Exact solutions? Smaller errors? • Can we simulate better than lattice Monte Carlo? • Are MPS and PEPS the best simulation solution?

  23. Simulation of the Laughlin wave function Local basis: a=0,..,n-1 Dimension of the Hilbert space Analytic expression for the reduced entropy ILO06

  24. Exact MPS representation of Laughlin wave function Clifford algebra Optimal solution! (all matrices equal but the last!)

  25. m=2

  26. Example: Normalization of wave function for m=2 So far, we have not managed to exploit the product structure

  27. Translational invariant spin chains Vidal05: iTEBD translationaly invariant infinite system algorithm commute commute All even gates can be performe simultaneously All odd gates can be performe simultaneously Use Trotter to combine them

  28. are isometries = Energy

  29. Trotter 2nd order Heisenberg model Trotter 2 order, =.001 Exponential distribution λ Poorness of DMRG

  30. Advantage: clean results for infinite half chain entropy Problem: Poor convergence of entropy entropy energy Maximum half-chain entanglement for Heisenberg model Consistent with central charge c=1 Attention to spontaneous symmetry breaking

  31. To compute block entropies, use exact coarse graining of MPS Local basis Optimal choice! VCLRW  remains the same and locks the physical index! After L spins are sequentially blocked Entropy is bounded Exact description of non-critical systems

  32. Exact solution for =2 min = S= .485704202

  33. Numerics Precision for entropy requires some extra effort Trotter higher order Random seeds (avoiding hysteresis cycles associated to the minimization procedure) Boost

  34. S Perfect alignement M

  35. MPS support of entropy obeys scaling law!! S χ ??

  36. So far • Simulation technique • representation • evolution • observables Physics Entanglement Entanglement support Exploit MPS, PEPS, MERA NEXT

  37. Contraction of PEPS is #P Yet, for translational invariant systems, it comes to iTEBD JOVVC07 Beats quantum Montecarlo!!

  38. VIDAL Beyond MPS: Entanglement RG MERA Unitary networks Building the program: detailed check vs MPS

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