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Matrix product states for the absolute beginner Garnet Kin-Lic Chan Princeton University

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Brief overview: Why tensor networks? Matrix Product States and Matrix Product Operators Graphical notation Compressing Matrix Product States Energy optimization Time evolution Focus on basic computations and algorithms with MPS not covered: entanglement area laws, RG, topological aspects, symmetries etc. Periodic and infinite MPS

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Quantum mechanics is complex Dirac The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known … the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved. n electron positionsL spinsL particle occupancies Exponential complexity to represent wavefunction

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This view of QM is depressing in general the many-electron wave function Ψ … for a system of N electrons is not a legitimate scientific concept [for large N] Kohn (Nobel lecture, 1998) [The Schrodinger equation] cannot be solved accurately when the number of particles exceeds about 10. No computer existing, or that will ever exist, can break this barrier because it is a catastrophe of dimension... Pines and Laughlin (2000)

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illusion of complexity nature does not explore all possibilities Nature is local: ground-states have low entanglement

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Language for low entanglement states is tensor networks Matrix Product State Tensor Product State (PEPS) 1D entanglement for gapped systems nD entanglement for gapped systems (basis of DMRG - often used in quasi-2D/3D) MERA 1D/nD entanglement for gapless systems different tensor networks reflect geometry of entanglement

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Graphical language n1n1 n2n2 n3n3 Algebraic formGraphical form thick line = single tensor physical index General state e.g.e.g. spin 1/2 particles

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Graphical language, cont’d n1n1 n2n2 n3n3 Algebraic formGraphical form General operator n1’n1’n2’n2’n3’n3’

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Ex: overlap, expectation value Overlap Expectation value

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Low entanglement states What does it mean for a state to have low entanglement? No entanglement local measurements on separated system 1, system 2 can be done independently. Local realism (classical) Entangled state Consider system with two parts, 1 and 2 Low entanglement : small number of terms in the sum

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Matrix product states “bond” or “auxiliary” dimension “M” or “D” or “χ” first and last tensors have one fewer auxiliary index n1n1 n2n2 n3n3 = General state i2i2 n1n1 i1i1 n2n2 n3n3 MPS amplitude is obtained as a product of matrices 1D structure of entanglement

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MPS gauge MPS are not unique: defined up to gauge on the auxiliary indices = ij = insert gauge matrices =

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Ex: MPS contraction Efficient computation: contract in the correct order! Overlap “d” “M” “d” “M” 1 1 2 2 3 3 MPS overlap: total cost

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MPS from general state recall singular value decomposition (SVD) of matrix orthogonality conditions singular values = i j = = i j orthogonality conditions

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MPS from general state, cont’d = 1 1 “n” “m” SVD i j i j = = “n” “m” = SVD = = 2 2 3 3 = 2 2 3 3 1 1 Step 1 Step 2

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Common canonical forms = 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 2 2 different canonical form: absorb singular values into the tensors left canonical = 1 1 1 1 i j = i j all tensors contract to unit matrix from left right canonical all tensors contract to unit matrix from right etc 3 3 3 3 = 2 2 2 2 i j = i j mixed canonical around site 2 (DMRG form) 2 2 3 3 1 1 1 1 1 1 i j 3 3 3 3 = i j “Vidal” form

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Matrix product operators each tensor has a bra and ket physical index n1n1 n2n2 n3n3 n1’n1’n2’n2’n3’n3’ = i2i2 n1n1 i1i1 n2n2 n3n3 n1’n1’n2’n2’n3’n3’ General operator MPO

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Typical MPO’s What is bond dimension as an MPO? LR joins pairs of operators on both sides MPO bond dimension = 5

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MPO acting on MPS MPO MPS = M1M1 M2M2 = M 1 x M 2 MPO on MPS leads to new MPS with product of bond dimensions

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MPS compression: SVD many operations (e.g. MPOxMPS, MPS+MPS) increase bond dim. compression: best approximate MPS with smaller bond dimension. 2 2 3 3 1 1 write MPS in Vidal gauge via SVD’s 2 2 3 3 1 1 M 1 singular values truncate bonds with small singular values 2 2 3 3 1 1 truncated M 2 singular values Each site is compressed independently of new information of other sites: “Local” update: non-optimal.

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MPS: variational compression original (fixed) MPS new MPS 1 1 3 3 2 2 1 1 3 3 2 2 1 1 2 2 3 3 solve minimization problem

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Gradient algorithm To minimize quantity, follow its gradient until it vanishes 1 1 2 2 3 3 2 2 linear in 2 2 = 1 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 2 2 quadratic in 2 2 = 1 1 3 3 1 1 3 3 2 2 x 2 2 2 Gradient step

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Sweep algorithm (DMRG style) 1 1 2 2 3 3 bilinear in 2 2 1 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 1 1 2 2 3 3 consider 2 2 as vector where 1 1 3 3 1 1 3 3 2 2 1 1 3 3 2 2 1 1 3 3 1 1 3 3

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Sweep algorithm cont’d 1 1 2 2 3 3 Minimization performed site by site by solving 1 1 2 2 3 3 1 1 2 2 3 3 1 1 3 3 1 1 3 3 1 1 3 3 Mb 2 2 3 3 3 3 2 2 3 3 2 2 Mb 1 1 2 2 1 1 2 2 1 1 2 2 Mb use updated tensors from previous step

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Sweep and mixed canonical form in mixed canonical form 1 1 2 2 3 3 2 2 3 3 3 3 2 2 3 3 2 2 M b mixed canonical around site 1 1 1 2 2 3 3 1 1 3 3 1 1 3 3 1 1 3 3 Mb mixed canonical around site 2 note: updated singular values change canonical form

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SVD vs. variational compression variational algorithms – optimization for each site depends on all other sites. Uses “full environment” SVD compression: “local update”. Not as robust, but cheap! MPS: full environment / local update same computational scaling, only differ by number of iterations. General tensor networks (e.g. PEPS): full environment may be expensive to compute or need further approximations.

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Energy optimization bilinear form: similar to compression problem commonly used algorithms DMRG: variational sweep with full environment imag. TEBD: local update, imag. time evolution + SVD compression

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DMRG energy minimization 2 2 1 1 3 3 1 1 3 3 2 2 1 1 3 3 1 1 3 3 2 2 use mixed canonical form around site 2 unit matrix eigenvalue problem for each site, in mixed canonical form where 1 1 3 3 1 1 3 3 DMRG “superblock” Hamiltonian

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Time evolution real time evolution imaginary time evolution: replace i by 1. projects onto ground-state at long times.

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General time-evolution 2 2 3 3 1 1 compress 2 2 3 3 1 1 repeat

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Short range H: Trotter form evolution on pairs of bonds

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Even-odd evolution time evolution can be broken up into even and odd bonds

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Time-evolving block decimation even/odd evolution easy to combine with SVD compression: TEBD SVD increase of bond dimension of unconnected bonds: SVD compression can be done independently on each bond.

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Periodic and infinite MPS MPS easily extended to PBC and thermodynamic limit 2 2 3 3 1 1 Finite MPS (OBC) 2 2 3 3 1 1 Periodic MPS 2 2 3 3 1 1 Infinite MPS

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Infinite TEBD local algorithms such as TEBD easy to extend to infinite MPS A A B B A A B B A A B B Unit cell = 2 site infinite MPS

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i-TEBD cont’d A A B B even bond evolution + compression updated not updated Step 1 Step 2 B B A A odd bond evolution + compression updated not updated Repeat A A B B B B A A

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Symmetries Given global symmetry group, local site basis can be labelled by irreps of group – quantum numbers U(1) – site basis labelled by integer n (particle number) SU(2) symmetry – site basis labelled by j, m (spin quanta) n=0, 1, 2 etc... Total state associated with good quantum numbers

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MPS and symmetry bond indices can be labelled by same symmetry labels as physical sites labelled by integer e.g. particle number symmetry MPS: well defined Abelian symmetry, each tensor fulfils rule Choice of convention: tensor with no arrows leaving gives total state quantum number

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Brief overview: Why tensor networks? Matrix Product States and Matrix Product Operators Graphical notation Compressing Matrix Product States Energy optimization Time evolution Focus on basic computations and algorithms with MPS not covered: entanglement area laws, RG, topological aspects, symmetries etc. Periodic and infinite MPS

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Language for low entanglement states is tensor networks Matrix Product State Tensor Product State (PEPS) 1D entanglement for gapped systems nD entanglement for gapped systems (basis of DMRG - often used in quasi-2D/3D) MERA 1D/nD entanglement for gapless systems different tensor networks reflect geometry of entanglement

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Questions 1.What is the dimension of MPS (M 1 ) + MPS (M 2 )? 2.How would we graphically represent the DM of an MPS, (tracing out sites n 3 to n L ?) 3.What is the dimension of the MPO of an electronic Hamiltonian with general quartic interactions? 4.What happens when we use an MPS to represent a 2D system? 5.What happens to the bond-dimension of an MPS as we evolve it in time? Do we expect the MPS to be compressible? How about for imaginary time evolution? 6.How would we alter the discussion of symmetry for non- Abelian symmetry e.g. SU(2)?

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