From Matrix Product States and Dynamical Mean-Field Theory to Machine Learning Sommerfeld Theory Colloquium, LMU Munich November 9, 2016 F. Alexander Wolf j falexwolf.de Institute of Computational Biology Helmholtz Zentrum M¨unchen Outline • Matrix Product States / Tensor Trains • Dynamical Mean-Field Theory • Machine Learning 2 / 25 \Tensor Trains I": noninteracting bits X1 X2 X3 X4 ... Vector of random variables X 2 f0; 1gL with joint probability mass L 1 −H(x)=T X p(x) = Z e ;H(x) = xn n=1 P −H(x)=T normalized with Z = x e . 3 / 25 \Tensor Trains I": noninteracting bits X1 X2 X3 X4 ... Vector of random variables X 2 f0; 1gL with joint probability mass L 1 −H(x)=T X px = Z e ;H(x) = xn n=1 P −H(x)=T normalized with Z = x e . p has 2L components x 2 f(0; 0; :::; 0); (0; 0; :::; 1);::: g. 3 / 25 \Tensor Trains I": noninteracting bits X1 X2 X3 X4 ... Vector of random variables X 2 f0; 1gL with joint probability mass L 1 −H(x)=T X px = Z e ;H(x) = xn n=1 P −H(x)=T normalized with Z = x e . p has 2L components x 2 f(0; 0; :::; 0); (0; 0; :::; 1);::: g. Note 2100 ' 1030 ' 1015 TB. 3 / 25 \Tensor Trains I": noninteracting bits X1 X2 X3 X4 ... Compute correlations via cov(Xn;Xm) = hXnXmi − hXnihXni, X hXnXmi = xnxm px: x 3 / 25 \Tensor Trains I": noninteracting bits X1 X2 X3 X4 ... Compute correlations via cov(Xn;Xm) = hXnXmi − hXnihXni, X hXnXmi = xnxm px: x . Naive brute force: 2L operations necessary. Monte Carlo: sampling in space of 2L states. 3 / 25 \Tensor Trains I": noninteracting bits X1 X2 X3 X4 ... Better: independent degrees of freedom Xn imply separability PL 1 − n=1 xn=T px = px1;x2;:::;xL = Z e 1 −xn=T = Z ax1 ax2 : : : axL ; axn = e : 3 / 25 \Tensor Trains I": noninteracting bits X1 X2 X3 X4 ... Better: independent degrees of freedom Xn imply separability PL 1 − n=1 xn=T px = px1;x2;:::;xL = Z e 1 −xn=T = Z ax1 ax2 : : : axL ; axn = e : Compute correlations in 2L operations . L 1 X X Y X hX X i = x a x a a n m Z n xn m xm xk xn xm k6=n;m xk = hXnihXmi ::: there are none. 3 / 25 \Tensor Trains II": interacting bits (Ising model) X1 X2 X3 X4 ... L−1 1 −H(x)=T X pex = Z e ;H(x) = − xnxn+1: n=1 4 / 25 \Tensor Trains II": interacting bits (Ising model) X1 X2 X3 X4 ... Two-body interactions imply \almost { separability" X X x1x2=T x2x3=T Z pex = e e ::: x x 4 / 25 \Tensor Trains II": interacting bits (Ising model) X1 X2 X3 X4 ... Two-body interactions imply \almost { separability" X X Z p = A A ::: ex x1x2 x2;x3 x x 4 / 25 \Tensor Trains II": interacting bits (Ising model) X1 X2 X3 X4 ... Two-body interactions imply \almost { separability" X X Z p = A A ::: ex x1x2 x2;x3 x x xnxn+1=T 2×2 = gsumAA : : : ; Axnxn+1 = e ;A 2 R ; where gsum is the grand sum. 4 / 25 \Tensor Trains II": interacting bits (Ising model) X1 X2 X3 X4 ... Two-body interactions imply \almost { separability" X X Z p = A A ::: ex x1x2 x2;x3 x x xnxn+1=T 2×2 = gsumAA : : : ; Axnxn+1 = e ;A 2 R ; where gsum is the grand sum. Compare to non-interacting case X X −xn=T 2 Z px = ax1 ax2 : : : ; axn = e ; a 2 R : x x 4 / 25 \Tensor Trains II": interacting bits (Ising model) X1 X2 X3 X4 ... Compute correlations in 23L operations (L matrix products) n−1 m−1 L−1 1 Y Y Y hX X i = gsum A[k] M A[k] M A[k] n m pe Z k=1 k=n k=m 1 0 where M = 0 −1 4 / 25 \Tensor Trains II": interacting bits (Ising model) X1 X2 X3 X4 ... Compute correlations in 23L operations (L matrix products) n−1 m−1 L−1 1 Y Y Y hX X i = gsum A[k] M A[k] M A[k] n m pe Z k=1 k=n k=m 1 0 where M = 0 −1 . Compare to non-interacting case (2L operations) 1 X X Y X hX X i = x A x A A n m p Z n xn m xm xk xn xm k6=n;m xk 4 / 25 L−2 B 0 = A X Y t xn(2xn+1+xn+2) xnxn+1xn+2 = Bx0 x0 B 0 0 n n+1 xn+1xn+2 2×4 x0 n=1 B 2 R 1 3 3 Tensor Train format . 2 (2 + 4 )L operations L−2 xnxn+1xn+2=T X X Y Axnxn+1xn+2 = e Z px = Ax x x n n+1 n+2 2×2×2 x x n=1 A 2 R \Tensor Trains III": long-range interacting bit chain Wolf (2015) X1 X2 X3 X4 ... L−2 1 −H(x)=T X px = Z e ;H(x) = − xnxn+1xn+2 n=1 5 / 25 L−2 B 0 = A X Y t xn(2xn+1+xn+2) xnxn+1xn+2 = Bx0 x0 B 0 0 n n+1 xn+1xn+2 2×4 x0 n=1 B 2 R 1 3 3 Tensor Train format . 2 (2 + 4 )L operations \Tensor Trains III": long-range interacting bit chain Wolf (2015) X1 X2 X3 X4 ... L−2 1 −H(x)=T X px = Z e ;H(x) = − xnxn+1xn+2 n=1 L−2 xnxn+1xn+2=T X X Y Axnxn+1xn+2 = e Z px = Ax x x n n+1 n+2 2×2×2 x x n=1 A 2 R 5 / 25 1 3 3 Tensor Train format . 2 (2 + 4 )L operations \Tensor Trains III": long-range interacting bit chain Wolf (2015) X1 X2 X3 X4 ... L−2 1 −H(x)=T X px = Z e ;H(x) = − xnxn+1xn+2 n=1 L−2 xnxn+1xn+2=T X X Y Axnxn+1xn+2 = e Z px = Ax x x n n+1 n+2 2×2×2 x x n=1 A 2 R L−2 B 0 = A X Y t xn(2xn+1+xn+2) xnxn+1xn+2 = Bx0 x0 B 0 0 n n+1 xn+1xn+2 2×4 x0 n=1 B 2 R 5 / 25 \Tensor Trains III": long-range interacting bit chain Wolf (2015) X1 X2 X3 X4 ... L−2 1 −H(x)=T X px = Z e ;H(x) = − xnxn+1xn+2 n=1 L−2 xnxn+1xn+2=T X X Y Axnxn+1xn+2 = e Z px = Ax x x n n+1 n+2 2×2×2 x x n=1 A 2 R L−2 B 0 = A X Y t xn(2xn+1+xn+2) xnxn+1xn+2 = Bx0 x0 B 0 0 n n+1 xn+1xn+2 2×4 x0 n=1 B 2 R 1 3 3 Tensor Train format . 2 (2 + 4 )L operations 5 / 25 If px = p(x) does not couple all index components xn among each other, there is a low rank Tensor Train representation. This reduces computational cost in summations over p(x) from exponential to linear in system size. • What about quantum mechanics? \Tensor Trains" in Statistical Mechanics • Write probability mass function L p : f0; 1; :::; dg ! R; d; L 2 N; as vector dL px = p(x); p 2 R ; which is indexed and parametrized by x 2 f0; 1; :::; dgL. 6 / 25 This reduces computational cost in summations over p(x) from exponential to linear in system size. • What about quantum mechanics? \Tensor Trains" in Statistical Mechanics • Write probability mass function L p : f0; 1; :::; dg ! R; d; L 2 N; as vector dL px = p(x); p 2 R ; which is indexed and parametrized by x 2 f0; 1; :::; dgL. If px = p(x) does not couple all index components xn among each other, there is a low rank Tensor Train representation. 6 / 25 • What about quantum mechanics? \Tensor Trains" in Statistical Mechanics • Write probability mass function L p : f0; 1; :::; dg ! R; d; L 2 N; as vector dL px = p(x); p 2 R ; which is indexed and parametrized by x 2 f0; 1; :::; dgL. If px = p(x) does not couple all index components xn among each other, there is a low rank Tensor Train representation. This reduces computational cost in summations over p(x) from exponential to linear in system size. 6 / 25 \Tensor Trains" in Statistical Mechanics • Write probability mass function L p : f0; 1; :::; dg ! R; d; L 2 N; as vector dL px = p(x); p 2 R ; which is indexed and parametrized by x 2 f0; 1; :::; dgL. If px = p(x) does not couple all index components xn among each other, there is a low rank Tensor Train representation. This reduces computational cost in summations over p(x) from exponential to linear in system size. • What about quantum mechanics? 6 / 25 We now consider quantum many-body states X j i = cxjxi; x where jxi = jx1i ⊗ jx2i ⊗ · · · ⊗ jxLi = jx1x2 : : : xLi is a tensor product of single-particle basis states jxii. For example jxii 2 fj "ii; j #iig • But, do we know anything about how the vector of coefficients c = (cx) couples its components, so that the tensor train format is meaningful? Statistical Mechanics vs.