T ensor N et works. I ztok Pizorn Frank Verstraete. University of Vienna M ichigan Quantum Summer School
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1 T ensor N et works I ztok Pizorn Frank Verstraete University of Vienna 2010 M ichigan Quantum Summer School
2 Matrix product states (MPS) Introduction to matrix product states Ground states of finite systems Ground states of infinite systems Real-time evolution Projected dynamics
3 I ntroduction to M PS S Östlund & S Rommer PRL (95); M Fannes, B Nachtergaele, RF Werner (92) in total: 2n matrices
4 M PS bond dimension D~ exp(n) Hilbert Space D=8 D=3 D=2 Exact description of an arbitrary quantum state Matrix product states only describe a certain subset requires exponentially large matrices A of the full Hilbert space But what kind of states are we really interested in?
5 Physical background F Verstraete, D Porras & JI Cirac, PRL (04) Bond dimension D puts a bound to entanglement: entanglement entropy at most log2(d) Area law: typically A l l gapped 1d syst ems at zer o t emper at ur e can be w el l described by matrix product states! Fi nd mat r i ces A such t hat t he t ot al ener gy i s mi ni mal!
6 W hy t ensor net works Density matrix renormalization group (DMRG) successful simulation of strongly correlated 1d systems 1D: matrix product states ~ DMRG 2D: PEPS & friends early stage but promising quantum monte carlo & sign problem Obsolete when we get a quantum computer (Chris Monroe s talk)
7 On notation or or...
![such mat r i ces A[j] t](/docs-images/72/66366945/images/8-11.jpg "hat t he ener gy i s")
8 Observables and M PS One-site operators Simulate ground states Two-site operators Fi nd such mat r i ces A[j] t hat t he ener gy i s minimal Expectation value of any (local) operator can be calculated efficiently
9 V ariat ional approach F Verstraete, D Porras & JI Cirac, PRL (04) All tensor elements of contained in
10 Variational optimization F Verstraete, D Porras & JI Cirac, PRL (04) 1. Choose a fixed site j 2. Find tensor that minimizes the energy 3. Move to the next site T echni cal det ai l s: Re-gauge the M PS such that Neff = I and solve an eigenvalue problem H x = E x 4. Repeat 2-3 until convergence reached Quadratic form in t ensor el ement s!
11 Variational optimization 1. Choose a fixed site j 2. Find tensor that minimizes the energy 3. Move to the next site Quadratic form in t ensor el ement s! n= Repeat 2-3 until convergence reached
12 Variational optimization 1. Choose a fixed site j 2. Find tensor that minimizes the energy 3. Move to the next site Quadratic form in t ensor el ement s! n= Repeat 2-3 until convergence reached
13 Variational optimization 1. Choose a fixed site j 2. Find tensor that minimizes the energy Non-cri t i cal model s (gapped): Finite bond dimension for any n Quadratic form in t ensor el ement s! n= Cri Move ti cal models to the (non-gapped): next site Bond dimension scales polynomially with n 4. Repeat 2-3 until convergence reached
14 Gauge t ransformat ions =
15 Imaginary time evolution G Vidal, PRL (03) Evolve in small time steps how to decompose into local time steps? Suzuki-Trotter decomposition All we need is a smart decomposition of H
16 Imaginary time evolution G Vidal, PRL (03) Suzuki-Trotter decomposition
17 M PS t ime evolut ion truncate SVD contract Sufficiently long time yields the ground state...
18 Real t ime evolut ion A state gets entangled and bond dimension explodes! P Calabrese & J Cardy, JSM (05) Efficient in very limited cases...
19 Infinite chains Assumption: invariance under shifts by (1,2,...) sites G Vidal, PRL (07) Much fewer parameters!
20 imps simulation Basi c el ement s Only local tensor updates required!
21 Project ed ent angled pair st at es (PEPS) Generalization of matrix product states to two spatial dimensions Entangled pairs between neighboring sites Success not entirely guaranteed by the area law More costly than MPS
22 PEPS Ansatz F Verstraete, JI Cirac (unpublished) Projector Entangled pair
23 PEPS: T ensor N et work 5-leg tensors parameters/site 4-leg tensors Tr = Tr
24 Observables and PEPS Expectation value of a local operator O Exact contraction straight forward but costly!
25 T wo layers t o a single double layer Bond dimension D 2
26 Efficient contraction of PEPS F Verstraete & JI Cirac (unpublished) 1. Merge two rows together 2. Truncate 3. Replace the two rows
27 Variational minimization H ow t o mi ni mi ze ener gy? Solve a generalized eigenvalue problem and mov e t o t he next si t e...
28 (Imaginary) time evolution Finite-site PEPS: Time evolution very costly i-peps: Imaginary time evolution the way to go!
29 (Imaginary) time evolution V Murg, F Verstraete & JI Cirac, PRB (09) Finite-site PEPS: Time evolution very costly i-peps: Imaginary time evolution the way to go!
30 T here s more... infinite PEPS (ipeps) J Jordan, R Orus, G Vidal, F Verstraete, & JI Cirac, PRL (08) multiscale entanglement renormalization (MERA) G Vidal, PRL (07) quantum monte carlo + tensor networks N Schuch, MM Wolf, F Verstraete & JI Cirac, PRL (08) A Sandvik & G Vidal, PRL (07) tensor renormalization HH Zhao et al, PRB (10)
31 Fermionic syst ems 1-dimensional fermionic systems Jordan-Wigner transformation 2-dimensional fermionic systems fermionic PEPS
32 Fermionic 1-D syst ems Jordan-W i gner T ransformati on Locality of interactions is preserved!
33 PEPS and fermions 1. Fermionic (contraction) order is important 2. Fock space 3. Jordan-Wigner transformation destroys locality 4. Parity preservation fpeps = PEPS + parity + contraction order
34 fpeps CV Kraus, N Schuch, F Verstraete & JI Cirac PRA (10) Q s (anti-)commute H s and V s commute Entangled pair How to get a tensor network from this mess?
35 f-contraction order Choose a contraction order IP & F Verstraete PRB (10) Further changes to the contraction order produce signs
36 f-observables
37 Fermionic swap rule P Corboz et al (09) T Barthel, C Pineda & J Eisert (10) IP & F Verstraete (10)... Example but
38 f-observables IP & F Verstraete PRB (10) Physical sites in <bra and ket> can now be brought together!
39 f-observables: double-layer IP & F Verstraete PRB (10)
40 Conversion to sign-free T N IP & F Verstraete PRB (10) site&product operator dependent sign factors (globally determined)
41 Si gn-free contraction of fpeps IP & F Verstraete PRB (10) All fermionic signs are accounted for locally! Technical details It s not very pretty - but it s local!
42 fpeps algorithm 1. Choose a site (i,j) and calculate effective operators 2. Find A[i,j] minimizing the total energy 3. Move to the next site and repeat Once w e v e obt ai ned a si gn-f r ee t ensor net w or k, al l i nt er medi at e st eps ar e w el l know n.
43 fpeps algorithm IP & F Verstraete PRB (10) 1. Choose a site (i,j) and calculate effective operators 2. Find A[i,j] minimizing the total energy 3. Move to the next site and repeat Once w e v e obt ai ned a si gn-f r ee t ensor net w or k, al l i nt er medi at e st eps ar e w el l know n.
44 fpeps algorithm IP & F Verstraete PRB (10) 1. Choose a site (i,j) and calculate effective operators 2. Find A[i,j] minimizing the total energy 3. Move to the next site and repeat Once w e v e obt ai ned a si gn-f r ee t ensor net w or k, al l i nt er medi at e st eps ar e w el l know n.
![Find A[i,j] minimizing the total energy 3.](/docs-images/72/66366945/images/45-8.jpg "Move to the next site and repeat Once w e v e obt ai ned a si gn-f r ee")
45 fpeps algorithm IP & F Verstraete PRB (10) 1. Choose a site (i,j) and calculate effective operators 2. Find A[i,j] minimizing the total energy 3. Move to the next site and repeat Once w e v e obt ai ned a si gn-f r ee t ensor net w or k, al l i nt er medi at e st eps ar e w el l know n.
46 PEPS & fpeps fpeps basically same complexity as PEPS no sign problem (quantum monte carlo feature) with fermionic systems frustrated spin systems relatively small bond dimensions accessible presently (D~8)
47 Ot her t ensor net works Tree tensor networks Quantum chemistry Momentum space MERA String states Continuous matrix product states
48 I I I : T ime evolut ion & M ixed st at es Time evolution of matrix product states: a projected evolution approach Mixed states and time evolution Systems in a thermal equilibrium Systems far from the equilibrium Time evolution in Heisenberg picture
49 Project ed dynamics Time evolution But translation invariant MPS breaks the T-invariance!
50 Project ed dynamics Let s choose quite often gradient projection Requirement: must stay in the same T- class
51 Project ed dynamics projection gradient IP, TJ Osborne, K Temme & F Verstraete (in prep)
52 I nfinite translation-invariant chains IP, TJ Osborne, K Temme & F Verstraete (in prep)
53 I nfinite translation-invariant chains IP, TJ Osborne, K Temme & F Verstraete (in prep)
54 Essent ials of im PS Leading eigenvectors only! Scale A such that
55 Essent ials of im PS
56 I nfinite translation-invariant chains IP, TJ Osborne, K Temme & F Verstraete (in prep)
57 I nfinite translation-invariant chains IP, TJ Osborne, K Temme & F Verstraete (in prep)
58 Project ed dynamics Preserves the translation invariance properties Infinite & finite systems Imaginary time evolution (fast & easy) Real time evolution (ODE solvers) Slower than the usual MPS time evolution Continuous Matrix Product States (cmps): the way to go!
59 Matrix product operators F Verstraete, JJ Garcia-Ripoll & JI Cirac, PRL (04) T Prosen & M Znidaric PRE (07) Any operator can be written in this form (if D is sufficiently large)
60 W hy operators? A system can be in a mixed state Thermal equilibrium Far from the equilibrium Can tensor networks be used for these systems?
61 Systems in a thermal equilibrium Zero temperature Infinite temperature An operator is just a state in the operator space
62 On operator space T Prosen & IP, PRA (07) How can we get Simpl e t i me ev ol ut i on!
63 Example 1. start with a product state 1> 2. evolve MPS in time
64 Is it efficient? M Znidaric, T Prosen & IP, PRA (08) Quantum mutual information critical non-critical (1/6) comes from the central charge in CFT Same behavior as for the ground states!
65 Dynamics of mixed states Schrödinger equation Liouville equation Isolated systems Liouvillian operator is a linear operator Again: time evolution (in operator space)
66 Open syst ems Lindblad master equation Non-equi l i br i um st eady st at e NESS shouldn t be too correlated! T Prosen & IP, PRL (08) But it is sometimes.
67 Open syst ems Convergence, spin current, XXZ T Prosen & M Znidaric, JSM (09) Lindblad master equation Non-equi l i br i um st eady st at e NESS shouldn t be too correlated! T Prosen & IP, PRL (08) But it is sometimes.
68 H eisenberg pict ure
69 H eisenberg pict ure
70 H eisenberg pict ure Growth of effective MPO-bond di mensi on T Prosen & M Znidaric, PRE (07) non-integrable integrable (infinite index) integrable (finite index)
71 H eisenberg pict ure T Prosen & IP, PRA (07) time evolution in operator space
72 W hy? Local operators are very simple! T Prosen & IP, PRA (07) IP & T Prosen, PRB (09) time Still, it only works in integrable models!
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