Density Matrix Renormalization Group Algorithm - Optimization in TT-format

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1 Density Matrix Renormalization Group Algorithm - Optimization in TT-format R Schneider (TU Berlin MPI Leipzig 2010

2 Density Matrix Renormalisation Group (DMRG DMRG algorithm is promininent algorithm to treat many body quantum systems ( S White (1991 spin chains S White & Jourdan (1999 quantum chemistry G Chan & Head-Gordon (2002 electronic structure computation O Legeza et al or M Reiher et al DMRG provides an (approximate FCI algorithm in polynomial complexity dynamical correlation (short range - ee-cusp Coupled Cluster (CCSD- statical or strong correlation (multi-configurational CASPT-MRSCF - DMRG

3 Setting - Tensors H d = H := d ν=1 V ν V ν := R n d-fold tensor product Hilbert space, H {(x 1,, x d U(x 1,, x d R : x i = 1,, n} The function U H will be called an order d-tensor Here x 1,, x d will be called variables or indices k u(k 1,, k d = (u k1,,k d = u, k i = 1, n Operators (matriziation A =: ( ak l k,l, A = ( A k 1,,k l x 1,,x d := ( ax1,,x d,k 1,,k l

4 TT - Tensors r := k<j u kν = ν u kν = ( u xν,k ν 1,k ν r 1 k 0 =1 r j 1, r := k>j r j+1 +1 r d k d If k 0 := 1, k d := 1 (single index, a tensor U in the TT-format is U = r 1 k 0 =1 r d k d d ν=1 ν u kν k ν 1 = r d k d ν=1 u kν k ν 1 T T r u kν = (u kν x ν,k ν 1 xν,k ν 1 1 ν < d, could be orthonormalized For operators A : H d H d, A : H H: a kν k ν 1 = ( (a yν x ν kν k ν 1 = (a y ν,k ν x ν,k ν 1 xν,y ν

5 for general optimization problems F : H d R Espig & Hackbsuch & Rohwedder Optimization problems 1 u 2 j d u u u k k k k k 1 2 j 1 j d 1 TT- tensor U = k d ν=1 ukν k ν 1 Tensor optimization Approximate W H: by a tensor of (canonial rank r Linear elliptic equations U = argmin {F (V = 1 2 AV, V B, V : V T T r } eigenvalue problems U = argmin {F(V = 1 2 AV, V V, V = 1 : V T T r }

6 Optimization by relaxation in TT format Relaxation (see eg Gauss-Seidel, ALS: For j = 1,, d: 1 fix all tensors ν u kν k ν 1, ν {1,, d}\{j}, except the index j 2 Optimize u := j u Repeat the relaxation procedure (in the opposite direction j u j+1 u Relaxation scheme u kν k ν 1 This is NOT exactly the DMRG algorithm!

7 Relaxation in TT format where U = l kj 1 u kj r kj,, l := r j 1 u kν k ν 1, r kj := k<j ν=1 r d k<j ν=j+1 u kν k ν 1 operators in TT format A = l j 1 a l j l j 1 A Rlj l j 1,l j A L A L l j 1 = r l<j j 1 ν=1 alν l ν 1, A Rlj = r l>j d ν=j+1 alν l ν 1

8 Stationary condition For simplicity of representation, let us assume in the sequel: Rank A = 1, ie A = d ν=1 a ν For fixed j, the stationary condition is, l, A L l (au r kj, A R r k j = b R n, = 1,, r j 1, = 1,, r j, R n = span{e i }, where e i, b kj 1 := B, l e i r kj, i = 1,, n * Rank A = 1

9 Stationary condition Defining the tensors G := ( (g k j 1 = l, A L l k j 1, H := ( (h k j = rkj, A R r k j, and let u := (u kj 1,x j,, b := (b kj 1,x j, For fixed j, the stationary condition is à u = (G a H u = b * Rank A = 1

10 Update j j + 1 j u j+1 u j j + 1: update j G j+1 G =: G, g k j = n r j 1 k u j x j, g a y j x j,y j =1, =0 x j u k j x j, shortly j+1 G := u T ( j G au, ( u x j, := ( u x j, For ν = 1,, d, ν H are computed once and stored * Rank A = 1

11 Eigenvalue problem where A = I M := ( P I Q P = ( (p = l, l, Q := ( (q = rkj, r k j If l kν and/or r kν are orthonormal, then P = I, Q = I (EVP: Generalized eigenvalue problem Ãu := ( G a H u = λ Mu = λ ( P I Q u * Rank A = 1

12 Stationary condition -general case l j 1,l j where, l l, A j 1 L l (a l j l j 1 u r kj, A Rlj r k j = b R n, = 1,, r j 1, = 1,, r j, R n = span{e i }, e i, b kj 1 := B, l e i r kj, i = 1,, n Defining the tensors G := ( (g lj 1 k j 1 = l, A L l j 1 l k j 1, H := ( (h lj k j = rkj, A Rlj r k j, the stationary condition is equivalent to the linear equation (g lj 1 k l 1 k l 1 (a l j l j 1,l j l j 1 u k j 1 (h lj k j = b,

13 A modified relaxation algorithm Disadvantage of simple relaxation: 1 u +1 not orthogonal, 2 r = (r 1, r d is fixed Idea: optimize (w xj,x j+1 +1 r j (u xj (v xj+1 +1 modified variant system 2 sites environment j u j+1 u repeat in opposite direction * Rank A = 1

14 A modified relaxation algorithm n small, eg n = 2, 4 etc (complexity O(n O(n 2 Modified - Relaxation: For j = 1,, d: 1 fix all tensors u kν k ν 1, ν {1,, d}\{j, j + 1}, except j, j Optimize w +1 := (w xj,x j Decimation by SVD: (w xj,x j+1 +1 r j (u xj kj (v xj update G = j+1 G, (since P = I j+1 G := u T ( j G au, ( u x j, := ( u x j, Repeat the relaxation procedure in the opposite direction Q = I * Rank A = 1

15 Conclusions 1 u k are ortho-normal P = I, Q = I, 2 the ranks could be adapted by the SVD 3 SVD provides indication about the errror Complexity: TT rank: R; = max{r ν } TT rank of A: R A, canonical rank of A: M A Storage : O(R 2 R A d + O(n 2 R 2, O(R 2 M A d + O(n 2 R 2 Computing time: O(n 2 R 3 R 2 A d+o(n2 R 3, O(n 2 R 3 M A d+o(n 2 R 3 d

16 Fock space Let Ψ µ := Ψ SL [ϕ k1,, ϕ kn ] = Ψ[k 1,, k N ] basis Slater det Labeling of indices µ I by an binary string of length d d eg: µ = (0, 0, 1, 1, 0, =: µ i 2 d i+1, µ i = 0, 1, µ i = 1 means ϕ i is (occupied in Ψ[ ] µ i = 0 means ϕ i is absend (not occupied in Ψ[ ] (discrete Fock space F is of dimf = 2 d, d F := c µ Ψ µ } i=1 N=0 V N FCI = {Ψ : Ψ = µ F {c : µ c(µ 1,, µ d = c µ, µ i = 0, 1, i = 1,, d} = configuration space N d : span{ϕ k } R 2 j=1 k=1 j=1 work in progress Holtz & S d i=1 R 2

17 Discrete annihilation and creation operators Let us observe A := B := A T A = ( ( , A T = ( , AA T = A 2 = 0, AA T + A T A = I, ( In order to obtain the correct phase factor, we define ( 1 0 S :=, 0 1 and the discrete annihilation operator a p A p := I A (p S S where A (p means that A appears on the p-th position in the product

18 Schrödinger operator The creation operator a p A T p := I A T (p S S We have the one-to-one correspondence a p A T p, a p A p Single particle operator, eg Fock operator F := r,p f q p a qa p p,q f p q A T p A p Schrödinger-Hamilton operator H = hpa q qa p + p,q H = d p,q=1 h q pa T p A q + p,q,r,s g p,q r,s a r a sa q a p d p,q,r,s=1 g p,q r,s A T r A T s A p A q

19 Particle number operator and Schrödinger eqn d d P := a pa p, P := A T p A p p=1 p=1 The space of N-particle states is given by d VFCI N := {Ψ F : PΨ = NΨ} VN := {c R 2 : Pc = Nc} Variational formulation of the Schrödinger equation c = (c(µ = argmin{ Hc, c : c, c = 1, Pc = Nc} i=1 Example Let R 2 = span{e 0, e 1 }, a single Slater determinant, eg HF determinant, is rank 1 in d i=1 R2, eg Ψ[1,, N] e 1 (0 e1 (N e0 (N+1 e0 (d VN

20 DMRG n small, eg n = 2, 4 etc (complexity O(n O(n 2 DMRG - Relaxation: For j = 1,, d: 1 fix all tensors u kν k ν 1, ν {1,, d}\{j, j + 1}, except j, j Optimize w +1 := (w xj,x j Decimation by SVD: (w xj,x j+1 +1 r j (u xj (v xj update G = j+1 G, (since P = I j+1 G := u T ( j G au, Repeat the relaxation procedure in opposite direction (Q = I * Rank A = 1

21 DMRG for the eigenvalue problem DMRG algorithm system 2 sites environment j u j+1 u Super system Ãw := ( G j a j+1 a H w = λiw Decimation (SVD update repeat in opposite direction * Rank A = 1

22 DMRG for the eigenvalue problem Warm up sweep: j P = I, j Q = I j and j H, 1 G are computed For j = 1,, d: 1 Optimize w +1 := (w xj,x j+1 +1, super-block Hamiltonian Ã: with particle number N and z-spin m z Ãw := ( G j a j+1 a H w = λiw 2 Decimation by SVD: (w xj,x j+1 +1 r j (u xj (v xj+1 +1, eg by diagonalizing density matrix Du := ( w T w u = µu, 3 update G = j+1 G, (since P = I G u T (G j au Repeat the relaxation procedure in the opposite direction j + 1 j if convergence λ i = E i, i = 0, * Rank A = 1

23 Quantum chemistry 1 For λ i = E i, the tensor u is only used for updating G (the full tensor has never been considered 2 u k are ortho-normal P = I, Q = I, 3 additional constraint conditions, like for fixed quantum numbers eg N could be applied 4 the ranks could be adapted by the SVD 5 SVD provides indication about the errror In quantum chemistry: n = 2 (n = 4, M A = O(d 2 (M A = O(d 3 R A = O(d???, For a fixed N electron system and increasing d storage: O(R 2 d 3, computing time: O(R 3 d 3 R O(d d 2??? ( measure for entangelement