Numerical Linear and Multilinear Algebra in Quantum Tensor Networks

Transcription

1 Numerical Linear and Multilinear Algebra in Quantum Tensor Networks Konrad Waldherr October 20, 2013 Joint work with Thomas Huckle QCCC 2013, Prien, October 20,

2 Outline Numerical (Multi-) Linear Algebraic Approaches to......quantum Control...Quantum Tensor Networks i 1,k 1,k 2 i 2,k 2,k 3 i 3,k 3,k 4... i 1,m 1,m 2 i 2,m 2,m 3 i 3,m 3,m 4 QCCC 2013, Prien, October 20,

3 NMR Spectroscopy as Optimal Control Problem Goal: find optimal pulse amplitudes u(t) to steer nuclear spin states x(t) over the time interval [0, T ] Optimality can be described by a scalar function f: f(x, u) = f T (x(t )) }{{} quality Application: Synthesization of a unitary target operator U G : f(u(t ), u) = U(T ) U G 2 F The dynamics are governed by the equation of motion: [ ] M ẋ(t) = i H 0 + u m (t)h m x(t) m=1 } {{ } =H(t) QCCC 2013, Prien, October 20,

4 Gradient Ascent Pulse Engineering Algorithm: 1: Set initial control amplitudes u {0} m (t k ) for all times t k and U 0 2: for l = 0, 1, 2,... until convergence a: Calculate exponentials U k = e ih(t k) b: Compute the forward propagation U k:0 = U k U k 1 U 1 U 0 c: Compute the backward propagation for all k = 1,..., K for all k = 1,..., K W K+1:k+1 = U K+1 U K U k+1 U k for all k = K,..., 1 d: Calculate the gradient )] f u m(t k ) [trace (W = Re HK+1:k+1 ( ih m)u k:0 e: Update the controls u m (t k ) u {l+1} m (t k ) = u {l} m (t k ) + ɛ f u m(t k ) QCCC 2013, Prien, October 20,

5 GRAPE: Computational Challenges 1. Computation of the matrix exponentials U k = e i[h0+ m um(t k)h m] Challenges: Exponential of matrices are required in explicit form Operand matrices have certain properties (e.g., sparsity structure) QCCC 2013, Prien, October 20,

6 Matrix Exponentials NMR setting: Problem can be reduced to two problems of half size: cn 3 reduces to 2c ( ) n 3 2 = c n 3 4 = saves factor 4 QCCC 2013, Prien, October 20,

7 Matrix Exponentials Old version: Eigendecomposition method New version: Chebyshev method H = V ΛV H = ) e i th = V e i tλ V H = V diag (e i tλ1,..., e i tλn V H. Requires eigendecomposition of a typically sparse matrix. QCCC 2013, Prien, October 20,

8 Matrix Exponentials Old version: Eigendecomposition method New version: Chebyshev method ( ) m e i th H = a 0 I + 2 a k C k, a k = i k J k ( t H ). H k=1 Chebyshev recurrence translates into for k = m downto 0 do a k = i k J k ( t H ) D k = a k I + 2 H HD k+1 D k+2 e i th = D 0 D 2 Computation relies solely on products of the form sparse dense. = No expensive operations such as inversion/eigendecomposition required QCCC 2013, Prien, October 20,

9 GRAPE: Computational Challenges 2. Computation of the intermediate matrix products Compute the forward propagation U k:0 = U k U k 1 U 1 U 0 for all k = 1,..., K Compute the backward propagation W K+1:k+1 = U K+1 U K U k+1 U k for all k = K,..., 1 Instances of the matrix-multiplication prefix problem: ) k = 1 : K : (X 1 X k 1 X k Challenges: Parallelization Large number of matrices to be multiplied (e.g., K = 1024) Matrices comparably small (e.g., ) QCCC 2013, Prien, October 20,

10 Computation of the Matrix Multiplications Old version: coarse-grain parallel prefix tree New version: fine-grain 3D parallelization: p 0 p 1 p 2 p 3 X 1 X 8 X 2 X 7 X 3 X 6 X 4 X 5 X 1 X 7:8 X 1:2 X 7 X 3 X 5:6 X 3:4 X 5 X 1 X 5:8 X 1:2 X 5:7 X 1:3 X 5:6 X 1:4 X 5 X 1 X 1:8 X 1:2 X 1:7 X 1:3 X 1:6 X 1:4 X 1:5 = individual matrix multiplications are computed sequentially QCCC 2013, Prien, October 20,

11 Computation of the Matrix Multiplications Old version: coarse-grain parallel prefix tree New version: fine-grain 3D parallelization: = compute loop over k sequentially and = parallelize individual matrix multiplications Total amount of work corresponds to a 3D cube. Each subblock corresponds to a block operation C i,l := C i,l + A i,j B j,l Find an appropriate domain decomposition. 3D parallelization features lowest communication overhead l j j Ci,l Ai,j i i l Bj,l j QCCC 2013, Prien, October 20,

12 Numerical Results (10 spins) computation time [s] gradient backward propagation forward propagation exponential 0 old new old new old new old new K = 128 K = 256 K = 512 K = 1024 Figure : Runtime of one GRAPE iteration on the HLRB2 using K 2 processors. QCCC 2013, Prien, October 20,

13 Numerical Results (11 spins) computation time [s] gradient backward propagation forward propagation exponential 0 old new old new old new old new K = 128 K = 256 K = 512 K = 1024 Figure : Runtime of one GRAPE iteration on the HLRB2 using K 2 processors. QCCC 2013, Prien, October 20,

14 Ground State of Quantum Systems Ground state: Compute lowest eigenvalue of the Hamiltonian H = M k=1 α k Q (k) 1 Q (k) N C2N 2 N Challenges: the vector space V = C 2N grows exponentially in N for large N (e.g., N = 100) traditional matrix methods fail Solution: data-sparse representation format that allows to overcome the curse of dimensionality, to describe the right corner of the Hilbert space, for efficient computations and algorithms. QCCC 2013, Prien, October 20,

15 Ground State of Quantum Systems Ground state: Compute lowest eigenvalue of the Hamiltonian H = M k=1 α k Q (k) 1 Q (k) N C2N 2 N Challenges: the vector space V = C 2N grows exponentially in N for large N (e.g., N = 100) traditional matrix methods fail Solution: data-sparse representation format that allows to overcome the curse of dimensionality, to describe the right corner of the Hilbert space, for efficient computations and algorithms. Two approaches: subspace method in the MPS format, symmetry-adapted variational ground-state search. QCCC 2013, Prien, October 20,

16 Matrix Product States In physical simulations, Matrix Product States (MPS) are in use: x = ( ) A (i1) 1 A (i2) 2 A (in ) N e i1i 2...i N. i 1,...,i N tr with D j D j+1 matrices A (ij) j. MPS representation requires 2ND 2 instead of 2 N entries. Minimization in terms of MPS vmps (ALS): j DMRG (MALS): j j + 1 QCCC 2013, Prien, October 20,

17 i 1 i 2 i 3 i N Tucker: α 1 α 2 α 3 α N α 1, α 2, α 3,..., α N i 1 i 2 i 3 i N CP: QCCC 2013, Prien, October 20, α

18 i 1 i 2 i 3 i N Tucker: α 1 α 2 α 3 α N α 1, α 2, α 3,..., α N i 1 i 2 i 3 i N TT/MPS: α 2 α 3 α 4 α N i 1 i 2 i 3 i N CP: QCCC 2013, Prien, October 20, α

19 The Lanczos Algorithm Algorithm: Lanczos iteration for Hermitian matrix H Choose initial guess r 0 0 Define β 0 = r 0 and q 0 = 0 for j = 1, 2,..., m do q j = r j 1 /β j 1 u j = Hq j α j = q H j u j r j = u j α j q j β j 1 q j 1 β j = r j T m = tridiag(β, α, β) (λ, y) = eig(t m ) v = Qy = (q 1,..., q m )y QCCC 2013, Prien, October 20,

20 The Lanczos Algorithm Algorithm: Lanczos iteration for Hermitian matrix H Choose initial guess r 0 0 Define β 0 = r 0 and q 0 = 0 for j = 1, 2,..., m do q j = r j 1 /β j 1 u j = Hq j α j = q H j u j r j = u j α j q j β j 1 q j 1 β j = r j T m = tridiag(β, α, β) (λ, y) = eig(t m ) v = Qy = (q 1,..., q m )y Computational tasks: QCCC 2013, Prien, October 20,

21 The Lanczos Algorithm Algorithm: Lanczos iteration for Hermitian matrix H Choose initial guess r 0 0 Define β 0 = r 0 and q 0 = 0 for j = 1, 2,..., m do q j = r j 1 /β j 1 u j = Hq j α j = q H j u j r j = u j α j q j β j 1 q j 1 β j = r j T m = tridiag(β, α, β) (λ, y) = eig(t m ) v = Qy = (q 1,..., q m )y Computational tasks: 1 Application of the matrix to a vector QCCC 2013, Prien, October 20,

22 The Lanczos Algorithm Algorithm: Lanczos iteration for Hermitian matrix H Choose initial guess r 0 0 Define β 0 = r 0 and q 0 = 0 for j = 1, 2,..., m do q j = r j 1 /β j 1 u j = Hq j α j = q H j u j r j = u j α j q j β j 1 q j 1 β j = r j T m = tridiag(β, α, β) (λ, y) = eig(t m ) v = Qy = (q 1,..., q m )y Computational tasks: 1 Application of the matrix to a vector 2 Computation of inner products QCCC 2013, Prien, October 20,

23 The Lanczos Algorithm Algorithm: Lanczos iteration for Hermitian matrix H Choose initial guess r 0 0 Define β 0 = r 0 and q 0 = 0 for j = 1, 2,..., m do q j = r j 1 /β j 1 u j = Hq j α j = q H j u j r j = u j α j q j β j 1 q j 1 β j = r j T m = tridiag(β, α, β) (λ min, y) = eig(t m ) v = Qy = (q 1,..., q m )y Computational tasks: 1 Application of the matrix to a vector 2 Computation of inner products 3 Sums and linear combinations of vectors QCCC 2013, Prien, October 20,

24 Computation of Inner Products x H y = i 1,...,i N = i j k j ( (Ā(i )) ( ( )) tr 1) 1 Ā (in ) N tr B (i1) 1 B (in ) N a (i1) 1;k 1,k 2 a (i N ) N;k N,k 1 m j b (i1) 1;m 1,m 2 b (i N ) N;m N,m 1 = Find an efficient ordering of the summations m j, k j, i j. Consider the tensor network i 1,k 1,k 2 i 2,k 2,k 3 i 3,k 3,k 4... i 1,m 1,m 2 i 2,m 2,m 3 i 3,m 3,m 4 QCCC 2013, Prien, October 20,

25 i1,k1,k2 i2,k2,k3 i3,k3,k4 i1,k1,k2 i2,k2,k3 i3,k3,k4... i1... i1,m1,m2 i2,m2,m3 i3,m3,m4 i1,m1,m2 i2,m2,m3 i3,m3,m4 k2 k1,m1,k2,m2 i2,k2,k3 i3,k3,k4 k1,m1,k2,m2 i2,k2,k3 i3,k3,k i2,m2,m3 i3,m3,m4 i2,m2,m3 i3,m3,m4 k1,m1,i2,m2,k3 i3,k3,k4 k1,m1,k3,m3 i3,k3,k4 i2 m i2,m2,m3 i3,m3,m4 i3,m3,m4 QCCC 2013, Prien, October 20,

26 Computation of the Sum of Two MPS Sum of two MPS vectors x = ( ) A (i1) 1 A (i2) 2... A (in ) N y = i 1,...,i N tr i 1,...,i N tr e i1...i N ( ) B (i1) 1 B (i2) 2... B (in ) N e i1...i N (x + y) i1,...,i N = [( A (i1) 1 0 tr 0 B (i1) 1 ) ( A (i2) B (i2) 2 )... ( )] A (in ) N 0 0 B (in ) N Adding MPS leads to an increase of the bond dimensions: x MPSD + y MPSD = z MPS2D QCCC 2013, Prien, October 20,

27 Matrix Product Operators (MPO) The MPS concept x i1,i 2,...,i N can be generalized to operators: O i 1,...,i N j 1,...,j N = tr ( ) = tr A (i1) 1 A (i2) 2 A (in ) N ( W (i1,j1) 1 W (i2,j2) 2 W (in,jn ) N Application of such an MPO to an MPS leads to [ ( ) ( ) ] tr W (i1,j1) 1 A (j1) 1 W (in,jn ) N A (jn ) N, j N j 1 i.e., an MPS of larger size D MPS D MPO. Usually exact representation with small ranks Not dependent on N ) QCCC 2013, Prien, October 20,

28 Augmentation of ranks Increase of the bond dimension during the Lanczos algorithm Lanczos algorithm Bond dimension for j = 1, 2,..., m q j = r j 1 /β j 1 D (j) u j = Hq j D O D (j) α j = qj Hu j r j = Hq j α j q j β j 1 q j 1 D O D (j) + D (j) + D (j 1) = β j = r j = D (j+1) v = Qy = (q 1,..., q m )y D (1) + D (2) + + D (m) QCCC 2013, Prien, October 20,

29 Augmentation of ranks Increase of the bond dimension during the Lanczos algorithm Lanczos algorithm Bond dimension for j = 1, 2,..., m q j = r j 1 /β j 1 D (j) u j = Hq j D O D (j) α j = qj Hu j r j = Hq j α j q j β j 1 q j 1 D O D (j) + D (j) + D (j 1) = β j = r j = D (j+1) v = Qy = (q 1,..., q m )y D (1) + D (2) + + D (m) Solution: keep the ranks D (j) bounded = Projection required QCCC 2013, Prien, October 20,

30 Compression/Projection of MPS P D : y MP SD arg min A [ tr SVD-based truncation: ymp arg min x 2 SD MP S D = x MP SD 2 ) ( B (i1) 1 B (in ) N ( )] tr A (i1) 1 A (in ) N i 1,...,i N Suppose A 1,..., A r 1 are already constructed. B r is D D Define A (ir) r ( ) B (0) r B r (1) = ( ) U (0) r U r (1) Σ r V H trunc r = Ũ r (ir), a D D matrix proceed with the D D matrix Σ r Ṽr HB(i r+1) r+1 (Ũ ) (0) r Σ Ũ r (1) r Ṽ H r 2 2 QCCC 2013, Prien, October 20,

31 Restarted Lanczos Algorithm: Restarted Lanczos iteration Choose initial guess r 0 0 ; for iter = 1, 2,... do Define β 0 = r 0 and q 0 = 0 ; for j = 1, 2,..., m do q j = r j 1 /β j 1 ; u j = Hq j ; α j = q j H u j ; r j = u j α j q j β j 1 q j 1 ; β j = r j ; T m = tridiag(β, α, β) ; (λ min, y) = eig(t m ) ; v = Qy = (q 1,..., q m )y ; r 0 = P D (v) QCCC 2013, Prien, October 20,

32 Numerical Results (Ising Hamiltonian N = 25) restart parameter m = 3 λ1 λ1 / λ outer iteration D = 2: D = 3: D = 4: D = 5: D = 6: Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos (full vector) QCCC 2013, Prien, October 20,

33 Numerical Results (Ising Hamiltonian N = 25) restart parameter m = 4 λ1 λ1 / λ outer iteration D = 2: D = 3: D = 4: D = 5: D = 6: Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos (full vector) QCCC 2013, Prien, October 20,

34 Numerical Results (Ising Hamiltonian N = 25) restart parameter m = 5 λ1 λ1 / λ outer iteration D = 2: D = 3: D = 4: D = 5: D = 6: Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos DMRG Lanczos (full vector) QCCC 2013, Prien, October 20,

35 Matrix and Vector Symmetries A R n n is symmetric, if a i,j = a j,i A R n n is skew-symmetric, if a i,j = a j,i A R n n is persymmetric, if a i,j = a n+1 j,n+1 i A R n n is skew-persymmetric, if a i,j = a n+1 j,n+1 i A R n n is symmetric persymmetric, if a i,j = a j,i = a n+1 i,n+1 j A vector v R n is called symmetric, if v i = v n+1 i. A vector v R n is called skew-symmetric, if v i = v n+1 i. A = a b c d b e f g c f h i d g i j QCCC 2013, Prien, October 20,

36 Matrix and Vector Symmetries A R n n is symmetric, if a i,j = a j,i A R n n is skew-symmetric, if a i,j = a j,i A R n n is persymmetric, if a i,j = a n+1 j,n+1 i A R n n is skew-persymmetric, if a i,j = a n+1 j,n+1 i A R n n is symmetric persymmetric, if a i,j = a j,i = a n+1 i,n+1 j A vector v R n is called symmetric, if v i = v n+1 i. A vector v R n is called skew-symmetric, if v i = v n+1 i. A = 0 b c d -b 0 f g -c -f 0 i -d -g -i 0 QCCC 2013, Prien, October 20,

37 Matrix and Vector Symmetries A R n n is symmetric, if a i,j = a j,i A R n n is skew-symmetric, if a i,j = a j,i A R n n is persymmetric, if a i,j = a n+1 j,n+1 i A R n n is skew-persymmetric, if a i,j = a n+1 j,n+1 i A R n n is symmetric persymmetric, if a i,j = a j,i = a n+1 i,n+1 j A vector v R n is called symmetric, if v i = v n+1 i. A vector v R n is called skew-symmetric, if v i = v n+1 i. A = a b c d e f g c h i f b j h e a QCCC 2013, Prien, October 20,

38 Matrix and Vector Symmetries A R n n is symmetric, if a i,j = a j,i A R n n is skew-symmetric, if a i,j = a j,i A R n n is persymmetric, if a i,j = a n+1 j,n+1 i A R n n is skew-persymmetric, if a i,j = a n+1 j,n+1 i A R n n is symmetric persymmetric, if a i,j = a j,i = a n+1 i,n+1 j A vector v R n is called symmetric, if v i = v n+1 i. A vector v R n is called skew-symmetric, if v i = v n+1 i. A = a b c 0 e f 0 -c h 0 -f -b 0 -h -e -a QCCC 2013, Prien, October 20,

39 Matrix and Vector Symmetries A R n n is symmetric, if a i,j = a j,i A R n n is skew-symmetric, if a i,j = a j,i A R n n is persymmetric, if a i,j = a n+1 j,n+1 i A R n n is skew-persymmetric, if a i,j = a n+1 j,n+1 i A R n n is symmetric persymmetric, if a i,j = a j,i = a n+1 i,n+1 j A vector v R n is called symmetric, if v i = v n+1 i. A vector v R n is called skew-symmetric, if v i = v n+1 i. A = a b c d b e f c c f e b d c b a QCCC 2013, Prien, October 20,

40 Matrix and Vector Symmetries A R n n is symmetric, if a i,j = a j,i A R n n is skew-symmetric, if a i,j = a j,i A R n n is persymmetric, if a i,j = a n+1 j,n+1 i A R n n is skew-persymmetric, if a i,j = a n+1 j,n+1 i A R n n is symmetric persymmetric, if a i,j = a j,i = a n+1 i,n+1 j A vector v R n is called symmetric, if v i = v n+1 i. A vector v R n is called skew-symmetric, if v i = v n+1 i. v 1 v 2.. v n 1 v n = v n v n 1.. v 2 v 1 = } {{ } =:J v 1 v 2.. v n 1 v n QCCC 2013, Prien, October 20,

41 Matrix and Vector Symmetries A R n n is symmetric, if a i,j = a j,i A R n n is skew-symmetric, if a i,j = a j,i A R n n is persymmetric, if a i,j = a n+1 j,n+1 i A R n n is skew-persymmetric, if a i,j = a n+1 j,n+1 i A R n n is symmetric persymmetric, if a i,j = a j,i = a n+1 i,n+1 j A vector v R n is called symmetric, if v i = v n+1 i. A vector v R n is called skew-symmetric, if v i = v n+1 i. v 1 v 2.. v n 1 v n v n =.. v 2 v 1 v n 1 = } {{ } =:J v 1 v 2.. v n 1 v n QCCC 2013, Prien, October 20,

42 Symmetries of Spin- 1 2 Hamiltonians H = σ x = M k=1 α k Q (k) 1 Q (k) N, Q(k) i {I, σ x, σ y, σ z } C 2 2 ( ) 0 1, σ 1 0 y = ( ) ( ) 0 i 1 0, σ i 0 z =, I = 0 1 ( ) QCCC 2013, Prien, October 20,

43 Symmetries of Spin- 1 2 Hamiltonians H = σ x = M k=1 α k Q (k) 1 Q (k) N, Q(k) i {I, σ x, σ y, σ z } C 2 2 ( ) 0 1, σ 1 0 y = ( ) ( ) 0 i 1 0, σ i 0 z =, I = 0 1 σ x, σ z and I are symmetric, σ y /i is skew-symmetric, ( ) QCCC 2013, Prien, October 20,

44 Symmetries of Spin- 1 2 Hamiltonians H = σ x = M k=1 α k Q (k) 1 Q (k) N, Q(k) i {I, σ x, σ y, σ z } C 2 2 ( ) 0 1, σ 1 0 y = ( ) ( ) 0 i 1 0, σ i 0 z =, I = 0 1 ( ) σ x, σ z and I are symmetric, σ y /i is skew-symmetric, σ x, σ y /i and I are persymmetric, σ z is skew-persymmetric, QCCC 2013, Prien, October 20,

45 Symmetries of Spin- 1 2 Hamiltonians H = σ x = M k=1 α k Q (k) 1 Q (k) N, Q(k) i {I, σ x, σ y, σ z } C 2 2 ( ) 0 1, σ 1 0 y = ( ) ( ) 0 i 1 0, σ i 0 z =, I = 0 1 ( ) σ x, σ z and I are symmetric, σ y /i is skew-symmetric, σ x, σ y /i and I are persymmetric, σ z is skew-persymmetric, Kronecker product of two skew-symmetric matrices is symmetric, Kronecker product of two skew-persymmetric matrices is persymmetric. QCCC 2013, Prien, October 20,

46 Symmetries of Spin- 1 2 Hamiltonians H = σ x = M k=1 α k Q (k) 1 Q (k) N, Q(k) i {I, σ x, σ y, σ z } C 2 2 ( ) 0 1, σ 1 0 y = ( ) ( ) 0 i 1 0, σ i 0 z =, I = 0 1 ( ) σ x, σ z and I are symmetric, σ y /i is skew-symmetric, σ x, σ y /i and I are persymmetric, σ z is skew-persymmetric, Kronecker product of two skew-symmetric matrices is symmetric, Kronecker product of two skew-persymmetric matrices is persymmetric. Symmetry of Spin Hamiltonians Hamiltonians of typical physical models (Ising-ZZ, various Heisenberg models) are real-valued and symmetric persymmetric. QCCC 2013, Prien, October 20,

47 Matrix Symmetries and Eigenvectors Eigenvectors of Symmetric Persymmetric Matrices ([CB76]) A symmetric persymmetric matrix A can be orthogonally transformed to a block diagonal matrix: ( ) B C A = C T = 1 ( ) ( ) ( ) I I B + JC 0 I J. JBJ 2 J J 0 B JC I J Hence, the eigenvectors of A are of the form ( ) ( ) vi vi or Jv i Jv i. QCCC 2013, Prien, October 20,

48 Matrix Symmetries and Eigenvectors Eigenvectors of Symmetric Persymmetric Matrices ([CB76]) A symmetric persymmetric matrix A can be orthogonally transformed to a block diagonal matrix: ( ) B C A = C T = 1 ( ) ( ) ( ) I I B + JC 0 I J. JBJ 2 J J 0 B JC I J Hence, the eigenvectors of A are of the form ( ) ( ) vi vi or Jv i Jv i. Ground States of Spin Hamiltonians Ground states of typical physical models (Ising-ZZ, various Heisenberg models) are either symmetric or skew-symmetric. QCCC 2013, Prien, October 20,

49 MPS Representation of Symmetric Vectors x i1,i 2,i 3,...,i N 1,i N = x i1,i 2,i 3,...,i N 1,i N Jx = x (BF) Flip-operator i := 1 i. Example (N=3): x = ( a b c d d c b a ) T Theorem: MPS Representation of the Vector Symmetry ([Huckle13]) 1 If all MPS matrix pairs (A (0) j, A (1) j ) are connected via A (ij) j with involutions U j (Uj 2 symmetric (BF). = U j A (īj) j U j+1, j = 1,..., N, (1) = I), the represented vector is 2 Any symmetric vector (BF) can be represented by an MPS fulfilling relations (1). Reduction factor: 2 QCCC 2013, Prien, October 20,

50 MPS Representation of Symmetric Vectors Theorem: Uniqueness Result for Symmetric Vectors ([Huckle13]) Assume that the MPS matrices (over K) are related by A (1) j = U j 1 A (0) j V j, with square matrices V j and U j of appropriate size (j = 1,..., N). If any choice of matrices A (0) j results in the symmetry of the represented vector x, i.e., Jx = x, then U j and V j are up to a scalar factor involutions for all j: Uj 2 = u ji, Vj 2 = v ji. Furthermore, U j = c j V j, j = 1,..., N with constants c j. This theorem gives an argument to use involutions as heuristic. An involution has as eigenvalues ±1 = choose U j = J. QCCC 2013, Prien, October 20,

51 Exploiting Symmetries in Computations Impose symmetry conditions into variational ground-state search Suppose that the desired ground-state vector x is symmetric, i.e., Jx = x Therefore, 0 x H (I J) x = x H (x Jx) = 0. }{{} =0 λ min yh Hy y H y yh Hy y H y + ρyh (I J) y y H y = yh (H + ρ(i J)) y y H y. Apply ground-state algorithm (e.g., DMRG) to modified Hamiltonian H(ρ) := H + ρ(i J) for some ρ > 0. QCCC 2013, Prien, October 20,

52 Numerical Results: Number of Iterations Table : Heisenberg-XXX model of N = 100 spins: Number of DMRG sweeps until convergence. ρ D QCCC 2013, Prien, October 20,

53 Numerical Studies: Convergence λ1 λ1 / λ D = 5: ρ = 0 ρ = 1 D = 10: ρ = 0 ρ = 1 D = 15: ρ = 0 ρ = outer ALS iteration D = 20: ρ = 0 ρ = 1 Figure : Heisenberg-XXX model of N = 100 spins: Approximation error. QCCC 2013, Prien, October 20,

54 Conclusions Simulation of quantum systems relies on standard problems of numerical linear algebra Hilbert space grows exponentially in the physical system size QCCC 2013, Prien, October 20,

55 Conclusions Simulation of quantum systems relies on standard problems of numerical linear algebra Hilbert space grows exponentially in the physical system size Non-generic behavior of quantum systems Nature of the underlying system translates into structured objects of linear and multilinear algebra (e.g., symmetries) Numerical treatment requires efficient algorithms acting on problem-adapted data structures NMR computation: problem reduction and algorithmic improvement Projection on MPS manifold may improve the solution Symmetry-adapted ground-state search improves variational MPS methods QCCC 2013, Prien, October 20,

56 Conclusions Simulation of quantum systems relies on standard problems of numerical linear algebra Hilbert space grows exponentially in the physical system size Non-generic behavior of quantum systems Nature of the underlying system translates into structured objects of linear and multilinear algebra (e.g., symmetries) Numerical treatment requires efficient algorithms acting on problem-adapted data structures NMR computation: problem reduction and algorithmic improvement Projection on MPS manifold may improve the solution Symmetry-adapted ground-state search improves variational MPS methods QCCC 2013, Prien, October 20,

57 Conclusions Simulation of quantum systems relies on standard problems of numerical linear algebra Hilbert space grows exponentially in the physical system size Non-generic behavior of quantum systems Nature of the underlying system translates into structured objects of linear and multilinear algebra (e.g., symmetries) Numerical treatment requires efficient algorithms acting on problem-adapted data structures NMR computation: problem reduction and algorithmic improvement Projection on MPS manifold may improve the solution Symmetry-adapted ground-state search improves variational MPS methods Thank you for your attention. QCCC 2013, Prien, October 20,

58 T. Auckenthaler, M. Bader, T. Huckle, A. Spörl, and K. Waldherr Matrix exponentials and parallel prefix computation in a quantum control problem. Parallel Comput., T. Huckle, and K. Waldherr Subspace iteration methods in terms of matrix product states. Proc. Appl. Math. Mech., T. Huckle, K. Waldherr, and T. Schulte-Herbrüggen Computations in quantum tensor networks. Lin. Alg. Appl., T. Huckle, K. Waldherr, and T. Schulte-Herbrüggen Exploiting matrix symmetries and physical symmetries in matrix product states and tensor trains. Linear Multilinear A., QCCC 2013, Prien, October 20,