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1、Entanglement entropyandthe simulation of quantum systemsOpen discussion with pde2007Jos Ignacio LatorreUniversitat de BarcelonaBenasque, September 2007PhysicsTheory 1Theory 2Exact solutionApproximated methodsSimulationClassical SimulationQuantum SimulationClassical Theory Classical simulation Quantu
2、m simulationQuantum Mechanics Classical simulation Quantum simulationClassical simulation of Quantum Mechanics is related to our ability to supportlarge entanglementClassical simulation may be enough to handle e.g. ground states: MPS, PEPS, MERAQuantum simulation needed for time evolution of quantum
3、 systemsand for non-local Hamiltonians Classical computerQuantum computer?IntroductionIntroductionIs it possible to classically simulate faithfully a quantum system?Quantum Ising modelrepresentevolvereadIntroductionIntroductionIntroductionThe lowest eigenvalue state carries a large superposition of
4、product statesEx. n=3Nave answer: NO Exponential growth of Hilbert spaceClassical representation requires dn complex coefficientsn A random state carries maximum entropyIntroductionIntroductioncomputational basisIs it possible to classically simulate faithfully a quantum system?Refutation Realistic
5、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 IntroductionIntroductionPlanMeasures of entanglementEfficient description of slight entanglementEntro
6、py: physics vs. simulationNew ideas: MPS, PEPS, MERAMeasures of entanglementOne qubitQuantum superposition Two qubitsQuantum superposition + several parties = entanglement Measures of entanglementMeasures of entanglement Separable statese.g. Entangled statese.g.Measures of entanglementLocal realism
7、is droppedQuantum non-local correlationsMeasures of entanglementPure states: Schmidt decomposition = Singular Value DecompositionA B =min(dim HA, dim HB) is the Schmidt numberMeasures of entanglementEntangled stateDiagonalise AMeasures of entanglementSeparable stateVon Neumann entropy of the reduced
8、 density matrixMeasures of entanglementMeasures of entanglementProduct statelargelargeVery entangled statee-bitMaximum Entropy for n-qubits Strong subadditivity theorem implies entropy concavity on a chain of spinsSLSL-MSL+MSmax=nMeasures of entanglementMeasures of entanglementEfficient description
9、for slightly entangled states A BSchmidt decompositionEfficient descriptionRetain eigenvalues and changes of basisEfficient descriptionSlight entanglement iff poly(n) 2.413-SAT with m clauses: easy-hard-easy around m=4.2nAdiabatic quantum evolution (Farhi,Goldstone,Gutmann)H(s(t) = (1-s(t) H0 + s(t)
10、 HpInicial hamiltonianProblem hamiltonians(0)=0s(T)=1tAdiabatic theorem:ifE1E0EtgminNP-completeAdiabatic quantum evolution for exact cover|0|0|0|0|1|1|1|1(|0+|1) (|0+|1)(|0+|1).(|0+|1)NP-completeNP problem as a non-local two-body hamiltonian!n=100 right solution found with MPS among 1030 statesNon-c
11、ritical spin chainsS ctCritical spin chainsS log2 nSpin chains in d-dimensionsS nd-1/dFermionic systems?S n log2 nNP-complete problems3-SAT Exact CoverS .1 nShor FactorizationS r nPhysics vs. simulationPhysics vs. simulationNew ideasMPS using Schmidt decompositions (iTEBD) Arbitrary manipulations of
12、 1D systems PEPS2D, 3D systemsMERAScale invariant 1D, 2D, 3D systemsNew ideasRecent progress on the simulation side2. Euclidean evolutionNon-unitary evolution entails loss of normare sums of commuting piecesTrotter expansionMPSEx: iTEBD (infinite time-evolving block decimation)evenoddAAABBBAAABBA B
13、Translational invariance is momentarily brokenMPSi)ii)iii)iv)MPSSchmidt decomposition produces orthonormal L,R states MPSMoreover, sequential Schmidt decompositions produce isometries=are isometriesMPSEnergyRead outEntropy for half chainMPSHeisenberg model=2-.42790793S=.486=4-.44105813S=.764=6-.4424
14、9501S=.919=8-.44276223S=.994=16-.443094S=1.26Trotter 2 order, =.001New ideasNew ideasentropyenergyConvergenceMPSLocal observables are much easier to get than global entanglement propertiesSMPerfect alignmentMPSNew ideasPEPS: Projected Entangled Pairsphysical indexancillaeGood: PEPS support an area l
15、aw!Bad: Contraction of PEPS is #PNew results beat Monte Carlo simulationsNew ideasABEntropy is proportional to the boundaryContour A = L“Area law”Some violations of the area law have been identifiedPEPSAs the contraction proceeds, the number of open indices grows as the area lawPEPS2D seemed out of
16、reach to any efficient representationContraction of PEPS is #PBuilding physical PEPS would solve NP-complete problemsYet, for translational invariant systems, it comes down to iTEBD !EEComparable to quantum Monte Carlo?EPEPSPEPSE becomes a non-unitary gatePEPSPEPSResults for 2D Quantum Ising model (
17、JOVVC07)MCPEPSMERAMERA: Multiscale Entanglement Renormalization AnsatzIntrinsic support for scale invariance!MERAMERAAll entanglemnent on one lineAll entanglemnent distributed on scalesMERAContraction = IdentityMERAUUpdateIf MPS, PEPS, MERA are a good representation of QM Approach hard problems Prec
18、isionCan 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 amountof entanglement Worst case (max entropy) remains at phase transition pointsPh
19、ysics vs. simulationQuantum Complexity ClassesQMAL is in QMA if there exists a fixed and a polynomial time verifier (V) such thatWhat is the QMA-complete problem?Feynman idea (shaped by Kitaev)QMAQMAQMA-complete problem Log-local hamiltonian 5-body 3-body 2-body (non-local interactions) 2-body (near
20、est neighbor 12 levels interaction)!Given H on n-party decide ifOpen 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 .
