Quantum simulation with string-bond states: Joining PEPS and Monte Carlo

Transcription

1 Quantum simulation with string-bond states: Joining PEPS and Monte Carlo N. Schuch 1, A. Sfondrini 1,2, F. Mezzacapo 1, J. Cerrillo 1,3, M. Wolf 1,4, F. Verstraete 5, I. Cirac 1 1 Max-Planck-Institute for Quantum Optics, Garching, Germany 2 University of Padua, Italy 3 University of Barcelona, Spain 4 Niels-Bohr-Instititute, Copenhagen, Denmark 5 University of Vienna, Austria

2 The key to quantum simulations Aim: Simulation of quantum many-body systems Hilbert space dimension is huge: exponentially in the number of spins. But: The system is governed by a local Hamiltonian i.e. it can be characterized by a polynomial number of parameters. Requirements on variational ansatz: it should capture the states of interest it should allow for efficient computation of energies etc.

3 Matrix Product States (MPS): = tr [ A 1 n1 A n2 n 1,, n L Matrix Product States 2 L A nl ] n 1,, n L D D matrices Alternatively: = n 1,, n L c n1,, n L n 1,, n L, with c n1,, n L A 1 A 2 n 1 n 2 A 3 n 3 A L Tensor network states n N Notation A A B A A B MPS approximate well the ground states of gapped 1D Hamiltonians [Hastings, JSTAT '07]

4 Computing expectation values on MPS On MPS, expectation values can be computed efficiently Local observables O: O = A i A i O I O I D D 2 Computation time D 4 ( D 6 ) for OBC (PBC) [can be improved to D 3 ( D 5 ) ]. Energies H, corr. functions O i O j can be computed efficiently.

5 Variational method with MPS: DMRG is linear in any A i X : X = tr [ A n1 The energy in state X which minimizes X E X is 1 L X nk A n L E X = X H X X X ] n 1,, n L = X M X X N X can be found efficiently. Density Matrix Renormalization Group (DMRG) method: Sweep through the MPS and minimize tensors locally. [White, PRL '92] h i DMRG algorithm performs extremely well! Accuracy is tuned by increasing D (computation time D 3 ). Idea extends to periodic boundary conditions (scales as ). D 5

6 Extending DMRG to two dimensions The MPS ansatz naturally generalizes to two dimensions. Tensor Product States (TPS), Projected Entangled Pair States (PEPS) [Nishino 90ies] [Verstraete & Cirac, PRA '04] PEPS form a complete family. PEPS are well suited for ground and thermal states in higher dimensions. [Hastings, PRB '06]

7 Extending DMRG to two dimensions The MPS ansatz naturally generalizes to two dimensions. D 2 Tensor Product States (TPS), Projected Entangled Pair States (PEPS) [Nishino 90ies] [Verstraete & Cirac, PRA '04] PEPS form a complete family. Problem: Computing expectation values on 2D PEPS seems to be hard: The size of the tensor grows exponentially in the perimeter! PEPS are well suited for ground and thermal states in higher dimensions. [Hastings, PRB '06] [Schuch, Wolf, Verstraete & Cirac, PRL '07]

8 PEPS algorithm with truncation Solution: proceed column-wise and truncate the bond dimension Corresponds to a DMRG-like optimization D 2 D 4 D 2 Works well in practice (due to symmetries of the system?) Known error [Verstraete & Cirac, cond-mat/ ] [Murg, Verstraete & Cirac, PRA '07] [Murg, Verstraete & Cirac '09] Works very well & outperforms other methods e.g. for frustrated systems D 12 ( D 18 Computation time scales as for PBC) can be reduced to D 10 ( D 16 ) Up to D=5 (lattice up to ). Not practical for PBC or higher dim. lattices, or irregular geometries.

9 The quest for other methods DMRG works well in 1D. PEPS extend DMRG to 2D and give very good results. PEPS contain the relevant states (ground and thermal states) However: - scaling of PEPS in D is unfavorable, in particular for PBC - truncation relies on lattice structure not applicable to irregular systems Is it possible to find a smaller/different class of states - for which expectation values can be computed efficiently - which has a favorable scaling in the accuracy parameter - which does not rely on the geometry of the system - which well approximates the states of interest

10 H = n Variational Monte Carlo Given a state on L spins, how can we sample exp. values? n n H = n n 2 n H n p n n and n H need to be efficiently computable! n= n 1,, n L basis state in some local basis e.g. 0=, 1= H = h i local terms n h i = few n' n' n' n H = few more n ' n' n' n needs to be efficiently computable!... the same works for products of Paulis, e.g. z z

11 Classes for which this holds n needs to be efficiently computable! Examples of such states: - Matrix Product States: n =tr [ A n1 1 L A n L (note: computation of expectation values scales as D 2 [D 3 ] instead of D 3 [ D 5 ] could be used to speed up DMRG or PEPS methods; see also Sandvik & Vidal, PRL '07) ] n 1 n 2 n 3 n 4 n 5 - coherent version of classical thermal states: = e H n /2 n for classical H H n /2 n =e Can we find new classes of suitable states (by extending these classes)?

12 Generalization of classical states What is the PEPS structure of classical states? n i = n = n i n j n is product of 0-dimensional objects! It can be computed efficiently! exp[ h ij n i, n j /2] Generalize to 1D objects! n i = n product of 1D objects can still be efficiently computed! Generalization of classical states (or of Matrix Product States!) subclass of PEPS with eff. Monte Carlo sampling

13 String-bond states (SBS) String-bond states (SBS) n = f i n defined on string (subset) of spins n i1,, n ik efficiently computabe (e.g. matrix product trace) n is a product of efficiently computable functions defined on subsets of spins matrix product traces tree tensor networks on small subsets: - arbitrary state - mini-peps... and any combination thereof! lines (horiz./vert./diagonal/...) 2x2 loops plaquettes (blocks) around each site

14 An example: Lines n 11 n 12 n 13 n 11 n 12 n 13 H 11 H 12 H 13 V 11 V 21 n 11 n 21 n 21 n 22 n 23 n 31 n 32 n 33 V 31 n 31 n =tr [V 11 n11 V 21 n21 V n ] tr [ H 11 n11 H 12 n12 H 13 n13 ]

15 String-Bond States Examples String-bond states form a complete family as they encompass MPS Any (generalized, weighted) graph state is a SBS: - the cluster state nn - any stabilizer state f ni, n j = 1 i The toric code state and the quantum double models are SBS: { j 1 for even parity 0 for odd parity

16 String-bond states: Properties SBS form a hierarchy of states which can be enlarged by - increasing D for the matrix products - adding new strings SBS include various common classes of states Computational resources scale favorably: D 2 (OBC) or D 3 (PBC) [cf. PEPS: D 12 (2D OBC), D 18 (2D PBC) ] SBS can thus be used for PBC or 3D systems, and much higher D's can be used no underlying geometry/locality necessary: also suitable for simulation of molecules, systems with non-local interactions, etc. each SBS is a PEPS, but PEPS-D exponential in #strings/site

17 Variational algorithm Basically as DMRG: pick one string, one tensor X on the string, and minimize E X = X H X X X = X M X X N X Idea: Determine M and N by MC sampling: X n n H X E X = X n' n' X = p 0 n X M= p 0 n b n a n N = p 0 n b n b n X H 2 H 3 H 4 i 1 i 2 i 3 i 4 b n a n X p 0 n' X b n' b n' X i 1 i 2 i 3 i 4 i 5 i 9 i 6 i 7 i 10 i 11 i 8 i 12 b n X n X := n X 0 n H a X n X := n X 0 p 0 n := n X 0 2 Reweighting: one distribution samples all

18 However... M = p 0 n b n a n N = p 0 n b n b n find X which minimizes E X = X M X X N X However, there are some problems : M and N are very large: dd 2 2 sampling rel. slow M and N less accurate far from X 0 due to reweighting sampling error MC sample length small errors in kernel of N can have big effect method unstable!

19 Gradient flow Solution: Move along gradient (in small steps), from initial value X =X 0 : X E X X =X 0 =2 [ M E X 0 N ] X 0 E X = X M X X N X X 0 N X 0 =1 well-behaved in N and M Gradient can be sampled directly: X E X X =X 0 =2 p 0 n b n [E n E X 0 ] Gradients indep. to first order: All tensors can be updated simultaneously (also allows for more eff. sampling) E n = n H X 0 n X 0 b n X := n X n X 0

20 The full algorithm Fix a string pattern & bond dimensions D. Choose initial configuration for the strings (random, guess,...) Repeat: 1) Monte Carlo sample gradients (& Energy) 2) Update tensors according to gradient and stepwidth (to this end: normalize gradients) 3) Stop if energy has converged, otherwise go to 1) Increase precision: either - decrease stepwidth - increase D - add new strings and restart algorithm from obtained optimum, until energy does not improve any more

21 How can we benchmark SBS? Testing the algorithm Quantum Monte Carlo + PBC, 3D possible - no frustrated systems PEPS + can do frustr. systems - only 2D OBC Note: Aim simulating systems neither QMC nor PEPS can do!

22 Implemented patterns lines loops diagonals

23 Comparison to QMC: transverse Ising model 2D transverse Ising model (PBC): QMC vs. lines and lines+loops

24 Comparison to PEPS: frustrated systems first comparisons: frustrated XX model SBS can outperform PEPS (D = 12 vs. D = 4). simulation of 2D frustrated PBC XX model w. transv. field: magnetization 2D field extensive comparison (and extension to PBC): J 1 J 2 model J 1 J 2 H =J 1 i, j i j J 2 i, j i j

25 Projected String Bond States Can we incorporate symmetries of H in the ansatz? SU(2) invariant Hamiltonian: There exists ground state with M 0 0 =0 (where M = i i z ) For any candidate 0, 0 = 0 0 is at least as good! Implementation: (this is a SBS n can be computed efficiently) enforce by sampling only from the M =0 subspace. In practice: start from configuration with and swap a random pair of spins M =0

26 Molecules Simulation of molecules: N e (e.g. atomic) orbitals modes c is H= T ij c is c js 1 i j s 2 i j k l s s ' V ijkl c is c is ' c ks' c ls Problem: no natural ordering (geometry) of modes

27 Molecules Ansatz: put strings between all pairs of modes = f, n, n c L n L c 1 n 1 vac n 1,, n L And: sample only in subspace with N e fermions Advantages: - all modes treated on equal footing - ansatz invariant under relabeling of modes gradient flow optimization of the f possible to include canonical transformations c i = O ij c j in the gradient optimization extension: use results soon! f n, n, n, n on quadruples of modes

28 Speeding up DMRG/PEPS with Monte Carlo The computational cost is D 2 / D 3 times the number of strings (each string can be evaluated by sequence of matrix multiplications) This is better than the scaling of DMRG (=a special SBS) D 3 / D 5. Reason: We don't have to contract the sandwich O but only n and n O. D D D O vs. O n 1 n 2 n 3 n 4 n 5 Monte Carlo sampling can thus be used to speed up DMRG and, more importantly, the PEPS algorithm ( D 6 instead of D 12 ). [see also: Sandvik and Vidal, PRL '07]

29 Summary Monte Carlo sampling can help in tensor contraction String-bond states: n String pattern can reflect geometry/ent. structure of system Enlarge family by larger D or more strings Cost of contraction: D 3 #strings prod. of eff. computable local objects SBS allow for efficient simulation of frustrated systems also with 2D PBC and in 3D Monte Carlo can speed up any tensor network based method Current & future direction: - better understand & characterize class of SBS - simulate molecules - string patterns using plaquettes - (imag.) time evolution - smarter sampling [N.Schuch, M. Wolf, F. Verstraete, I. Cirac, PRL '08] [A. Sfondrini et al., in preparation]