纠缠熵和量子模拟.ppt

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Adiabatic quantum evolution for exact cover |0 |0 |0 |0 |1 |1 |1 |1 (|0+|1) (|0+|1) (|0+|1) …. (|0+|1) NP-complete NP problem as a non-local two-body hamiltonian! n=100 right solution found with MPS among 1030 states Non-critical spin chains S ~ ct Critical spin chains S ~ log2 n Spin chains in d-dimensions S ~ nd-1/d Fermionic systems? S ~ n log2 n NP-complete problems 3-SAT Exact Cover S ~ .1 n Shor Factorization S ~ r ~ n Physics vs. simulation Physics vs. simulation 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 2. Euclidean evolution Non-unitary evolution entails loss of norm are sums of commuting pieces Trotter expansion MPS Ex: iTEBD (infinite time-evolving block decimation) even odd A A A B B B A A A B B A ? B Translational invariance is momentarily broken MPS i) ii) iii) iv) MPS Schmidt decomposition produces orthonormal L,R states MPS Moreover, sequential Schmidt decompositions produce isometries = are isometries MPS Energy Read out Entropy for half chain MPS Heisenberg model ?=2 -S=.486 ?=4 -S=.764 ?=6 -S=.919 ?=8 -S=.994 ?=16 -.443094 S=1.26 Trotter 2 order, ?=.001 New ideas New ideas entropy energy Convergence MPS Local observables are much easier to get than global entanglement properties S M Perfect alignment MPS 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 A B Entropy is proportional to the boundary Contour A = L “Area law” Some violations of the area law have been identified PEPS As the contraction proceeds, the number of open indices grows as the area law PEPS 2D seemed out of reach t