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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 Román Orús, Artur García-Sáez, Frank Verstraete, Miguel Aguado, Ignacio Cirac

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

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

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

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

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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

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

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

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

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Graphic representation of a MPS Efficient computation of scalar products operations

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Local action on MPS U

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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

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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 Crazy ideas: Image compression pixel addresslevel of grey RG addressing

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= 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

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

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Success of MPS will depend on how much entanglement is present in the physical state PhysicsSimulation If MPS is in very bad shape Back to the central idea: entanglement support

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Exact entropy for a reduced block in spin chains At Quantum Phase TransitionAway from Quantum Phase Transition

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Maximum entropy support for MPS Maximum supported entanglement

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

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Physics VLRK02-03 For 3-SAT OL04 Simulation LLRV04 Exact RG on statesVCLRW05 Lipkin model OLRV05 100-qubit Ex-cover instance BOLP05 OLEC06 Image compression L05 Area law RL06 Laughlin ILO06 Continuous variables ILO06

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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 problemsS ~.1 n Shor FactorizationS ~ r ~ n Local (12 levels), nearest neighbor H is QMA-complete!!AGK07

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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? 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

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

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

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m=2

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

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

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= are isometries Energy

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Heisenberg model.1-.42474325.01-.425759.001-.427876149.0001-.427904824.00001-.427907692 Trotter 2 nd order =2 -.42790793S=.486 =4 -.44105813S=.764 =6 -.44249501S=.919 =8 -.44276223S=.994 =16 -.443094S=1.26 Trotter 2 order, =.001 Exponential distribution λ Poorness of DMRG

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

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To compute block entropies, use exact coarse graining of MPS 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 Local basis

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Exact solution for =2 = min S=.485704202

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

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S M Perfect alignement

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

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So far Physics Simulation technique representation evolution observables EntanglementEntanglement support Exploit MPS, PEPS, MERA NEXT

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

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

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