Simulating Quantum Systems through Matrix Product States. Laura Foini SISSA Journal Club

Transcription

1 Simulating Quantum Systems through Matrix Product States Laura Foini SISSA Journal Club

2 Motivations Theoretical interest in Matrix Product States Wide spectrum of their numerical applications Eventual relations with methods which work on the same geometries

3 Motivations

4 Motivations Quantum information perspective Quantum Many-Body simulations Equilibrium Out of Equilibrium - Quenches - Time-dependent Hamiltonian Simplified models (spin chains, Bose-Hubbard..) capture the relevant physics, but still they are not simple!

5 Why are quantum many-body systems so hard to simulate? Hilbert space grows exponentially with the system size Ex: Bose-Hubbard model Ĥ = J ˆb iˆb j + U ˆn i (ˆn i 1) 2 i,j i N particles, M sites # states: W (N, M) = (M + N 1)! (M 1)!N! N M W ~ ~10^8

6 SCHMiDT Decomposition ψ = Constant depending on the partition A χ A B ψ = λ α φ A α φ B α α=1 ˆρ A φ A α = λ α 2 φ A α ˆρ B φ B α = λ α 2 φ B α Schmidt number = max A A For a general state 1 χ 2 n/2 λ α 2 =1 α Measure of the entanglement

7 Schmidt decomposition from SVD ψ = ij c ij i A j B Singular Value Decomposition C = UDV C mxn matrix U,V unitary. D=d ii δ ij d kk u ik i A v kj j B k i j = k λ k φ A k φ B k

8 MPS ψ = Matrix Product state ψ = c i1..i n = i 1,..,i n c i1..i n i 1.. i n χ α 1..α n =1 Product state Site numbers State index Γ [2]i 2 α 1 α λ [2] Γ [3]i 3 2 α 2 α 2 α 3 1 Γ [1]i 1 α 1 λ [1] α 1 Γ [2]i 2 α 1 α 2 λ [2] α 2 Γ [3]i 3 α 2 α 3..Γ [n]i n α n 1 0 ψ = α λ α φ A={1,2} α φ B={3,..,n} α

9 Γ [1]i 1 α 1 Γ [2]i 2 α 1 α 2 Γ [n]i n α n 1 c i1,..,i n i 1 λ [1] α 1 λ [n 1] α n 1 i 1 i n i n We have expressed 2 n coefficients with (2χ 2 + χ)n We want to retaine a fixed χ number of states. In order to have a small error he choice of χ depends on the entanglement entropy S L = α λ α 2 log λ α 2 χ 2 S L if S L O(n δ ) nothing gained if S L const we can exploit small entanglement for general systems in D>1 true for gapped systems in D=1

10 Single site operations only the corresponding Γ tensor has to be updated (χ 2 operations) c i1..i n = α 1..α n 1 Γ 1,i1 α λ 1 1 α Γ 2,i 2 1 α 1,α λ 2 2 α Γ 3,i 3 2 α 2,α 3..Γ n,i n α n 1 Two-site operations only the corresponding Γ tensors and λ vector has to be updated (χ 3 operations)

11 Time Evolution Ĥ = k Ĥ 2k,2k+1 + k Ĥ 2k+1,2k+2 Suzuki-Trotter decomposition e iδtĥ e iδtĥodd e iδtĥeven + O(δt 2 ) δt e i 2 Ĥeven e iδtĥodd δt e i 2 Ĥeven + O(δt 3 ) Imaginary Time Evolution Ground State Properties ψ τ = e τĥ ψ 0 e τĥ ψ 0

12 Algorithm 1. Apply e -iδt H 2k,2k+1 on all even bonds 2. Carry out a Schmidt decomposition and retain a fixed number of states 3. Repeat point (1) and (2) for the odd bounds 4. This completes one Trotter time step which has to be iterated Translationally Invariant Infinite Chain α λ B λ B λa λ Γ A B λa Γ A λ A Γ B U i j β i ΓB j λ B Two kinds of tensors are enough to represent the infinite chain Computational cost reduction of a factor n due to parallelized, local updates

13 Matrix Product States on trees Bethe lattice MPS: write ψ> in terms of Γ α1α2α3 for each site and λ αk for each bound

14 Matrix Product States on trees Bethe lattice MPS: write ψ> in terms of Γ α1α2α3 for each site and λ αk for each bound

15 Matrix Product States on trees Bethe lattice A ψ = α λ α φ A α φ B α B MPS: write ψ> in terms of Γ α1α2α3 for each site and λ αk for each bound

16 The infinite Bethe lattice Consider a translationally invariant system Define two kind of tensor on the bipartite graph, Γ A Γ B and an incoming direction Update in two steps (Analogous to the even-odd update of the chain) Γ A Γ B

17 Correlation Functions In general Correlations over MPS are computed as contractions of tensors one can look at the second eigenvalues of a given matrix B function of the tensors Γ ÔiÔj Ôi Ôj µ i j 2 ξ = 1/ log µ 2 μ 2 second eigenvalue of B ξ correlation length μ 2 1. As far as μ 2 <1 correlations falls exponentially

18 Some results for the quantum transverse field s 2 Ising model (1 ˆσ z i ˆσ z)+ j ij c(1 s) 2 (1 ˆσ x) i i Infinite Chain

19 Some results for the quantum transverse field s 2 Ising model (1 ˆσ z i ˆσ z)+ j ij c(1 s) 2 (1 ˆσ x) i i

20 Some results for the quantum transverse field s 2 Ising model (1 ˆσ z i ˆσ z)+ j ij c(1 s) 2 (1 ˆσ x) i i Bethe Lattice

21 Some results for time evolving quantities Ĥ = r The infinite chain. At t=0 h z =10 h z =3 ˆσ r xˆσ r+1 x + h z ˆσ r z magnetization m z (t) χ=40 χ=20 χ= χ = 60 χ=100 χ=150 χ= exact 10 errors ε 10 4 χ= εχ=20 ε χ=40 χ χ= time [1/h] ε χ=100 ε χ=150 ε χ= Main sources of errors Suzuki-Trotter decomposition Truncation error (χ) The error grows with time

22 Comments & Conclusions General requirement: the entanglement must not grow too fast with time evolution in order to have a reliable simulation The algorithm establishes that any quantum evolution involving restricted amount of entanglement can be performed with classical computers It can be integrated into DMRG codes Lots of applications Spin models D. Gobert, C. Kollath, U. Schollwöck, G. Schütz, Phys. Rev. E 71, (2005). Quenches dynamics in BH model, C. Kollath, A. Läuchli, and E. Altman, Phys. Rev. Lett. 98, (2007). Andreev-like Reflections A. J. Daley, B. Trauzettel, and P. Zoller, Phys. Rev. Lett. 100, (2008). Three-body loss A. J. Daley, J. M. Taylor, S. Diehl, M. Baranov, and P. Zoller, Phys. Rev. Lett. 102, (2009) (and many others...) Many generalizations (D>1, critical systems, finite temperature,...)

23 References 1)G. Vidal, Efficient Simulation of One-Dimensional Quantum Many-Body-Systems, PRL 93, (2004) 1) G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension, Phys. Rev. Lett. 98, (2007). [cond-mat/ v2] 2) D. Nagaj, E. Farhi, J. Goldstone, P. Shor, and I. Sylvester, The Quantum Transverse Field Ising Model on an Infinite Tree from Matrix Product States, Phys. Rev. B 77, (2008). [cond-mat/ v1