Matrix Product States and Tensor Network States Norbert Schuch Max-Planck-Institute of Quantum Optics, Munich Overview ● Quantum many-body systems are all around! condensed matter quantum chemistr high-ener! ph sics ● "an exhibit complex quantum correlations #$multipartite entan!lement% & rich and uncon'entional physics, but difficult to understand! ● Quantum information and ntan!lement Theory: Toolbox to characterize and utili*e entanglement #im$ Study strongly correlated quantum many-body systems from the perspective of quantum information % entanglement theory& ntan!lement structure of quantum many-body states Quantum many-body systems ● +ide range of quantum man -bod (QM,% s stems exists ● Our focus: spin models #$qudits% on lattices) local interactions 5 t pically transl. invariant ● Realized in man s stems) locali*ed d.f electrons half-/lled band quantum simulators, e1!. optical lattices ● 0xpecially interested in the !round state , i.e., the lo2est eigen'ector (It is the “most quantum” state, and it also carries relevant information about excitations.) Mean-field theory ● In man cases, entan!lement in QM, s stems is ne!li!ible ● S stem can be studied with product state ansat( )mean 'eld t"eory* ● Consequence of )mono!amy of entan!lement* (& de Finetti theorem% ● ,eha'ior fully characterized b a sin!le spin – – a local property #order parameter) & +andau theory of phases ● Beha'ior insensitive to boundary conditions, topolo! , ... xotic phases and topological order ● S stems exist which cannot be described by mean 'eld theory de!eneracy depends on s stem supports !lobal properties exotic excitations #3an ons4% 5 e1!1 Kitae';s “8oric Code41 & impossible within mean-/eld ansat* & orderin! in entan!lement & 8o understand these s stems: need to capture their entan!lement! ● 9seful as quantum memories and for topolo!ical quantum computin! T"e physical corner of Hilbert space ● <o2 can we describe entan!led QMB states= ● general state of spins) exponentially lar!e Hilbert space! ● but then again ..1 has only parameters & ground state must live in a small “physical corner” of ,ilbert space! ● Is there a “nice” wa to describe states in the p"ysical corner= & use entanglement structure! ntanglement ● Consider bipartition of QMB s stem into A and B Schmidt decomposition ( ON,% ● Schmidt coe.cients characterize bipartite entanglement more disorder & more entan!lement ● Measure of entanglement: ntan!lement entropy ntanglement structure$ The area law ● <o2 much is a region of a QMB s stem entangled with the rest= ● entan!lement entropy of a region scales as boundary #'s. 'olume% )area law” for entan!lement #for <amiltonians 2ith a spectral !ap; but approx. true even 2ithout gap% ● Interpretation) entanglement is distributed locally One dimension: Matrix Product States #n ansatz for states with an area law ... ● each site composed of t2o auxiliary particles #3'irtual particles4% formin! max. entangled bonds #D) “bond dimension4% ● apply linear map (“pro@ector”) ⇒ ● satis/es area law b construction ● state characterized b → parameters ● family of states: enlarged by increasing /ormulation in terms of Matrix Products matrices ● iterate this for the whole state ) )Matrix Product State” 0MPS) #or for open boundaries% /ormulation in terms of Tensor Networks ● Tensor Network notation) ● Matrix Product States can be written as 111 with )Tensor Network States* Completeness of MPS ● MPS form a complete family – e'ery state can be written as an MPS: ● Can be understood in terms of teleportin! using the entangled bonds #pproximation by MPS ● 3eneral MPS with possibly 'er lar!e bond dimension ● Schmidt decomposition across some cut) ● Pro@ect onto lar!est Schmidt values ) & error ● Rapidly decayin! #A bounded entrop %: total error ● .cient approximation of states 2ith area law (and thus ground states% Matrix Product States can e.ciently approximate states with an area law, and !round states of 0!apped1 one-dimensional ,amiltonians& 2omputin! properties of MPS ● Biven an MPS , can we compute exp. values for local ? O )transfer operator* ● computing = multiplication of matrices & computation time ● OBC scalin!: [and if done properly, e'en (P,"% and #O,"%D The transfer operator ● consider translational invariant s stem) ● spectrum of transfer operator go'erns scalin! of correlations #a% largest eigen'alue unique) exponential decay of correlations #b% largest eigen'alue degenerate: long-range correlations ● uniqueness of purification: contains all non-local information about state ● is Choi matrix of quantum channel Numerical optimization of MPS ● MPS approximate ground states e(cientl ● expectation values can be computed e(ciently ● can we e(ciently 'nd t"e which minimi(es = ● 'arious methods) - EM-B) optimi(e sequentially & iterate - gradient methods: optimi*e all simultaneously - hybrid methods ... is quadratic in each & each step can be done e(ciently ● "ard instances exist #NP-hard), but methods practically conver!e very well ● provably workin! poly-time method exists MPS form the basis for powerful variational met"ods for the simulation of one-dimensional spin chains xample: The A6+T state – a rotationally invariant model : pro@ector onto the spin-K representation of singlet #3 4% L )#6+T state* C>Geck, Kenned , Hieb & 8asaki, '87] ● Resulting state is invariant under #$spin rotation% by construction) ● Can construct states w. s mmetries b encoding symmetries locally The A6+T Hamiltonian ● consider 9 sites of #6+T model M sites ha'e spin impossible8 ● and L is a (frustration free% !round state of #frustration free = it minimized each indi'iduall % )parent ,amiltonian* ● inherits spin-rotation symmetry of state by construction (specificall , % ● One can pro'e) - is the unique !round state of - has a spectral !ap abo'e the ground state Parent Hamiltonians ● > parent Hamiltonian can be constructed for an MPS: lives in –dimensional space possible boundary conditions choose s.th. → doesn;t ha'e full rank ● "onstruct parent ,amiltonian , ● "an pro'e) - has unique !round state - has a spectral !ap abo'e the ground state ● This + abilit of MPS to approximate ground states of general <amiltonians & MPS form right frame2ork to study p"ysics of :; QMB systems MPS and symmetries ● Symmetries in MPS can alwa s be encoded locally L L … L and con'ersel L ● Symmetries are inherited by t"e parent ,amiltonian! /ractionali(ation ● "onsider A:LT model on chain with open boundaries ● all choices of boundaries are !round states of parent Hamiltonian & zero energy )ed!e excitations* with spin ● )fractionali(ation” of ph sical spin into at the boundary & impossible in mean-/eld theory & non-trivial )topolo!ical* p"ase #3<aldane phase4% ● can pro'e: cannot smoothly connect MPS 2ith integer and half-integer spin at edge & inequivalent phases! MPS encode p"ysical symmetries locally5 and can be used to model p"ysical systems and study t"eir di<erent non-trivial phases& Two dimensions$ Projected Entangled Pair States Two dimensions: Pro=ected Entan!led Pair States ● Natural generalization of MPS to two dimensions) bond dimension Tensor Network Notation$ Pro=ected ntan!led Pair States 0PEPS) ● approximate !round states of local <amiltonians 2ell ● P0PS form a complete family 2ith accurac parameter 1 ● P0PS can also be de/ned on other lattices, in three and more dimensions, e'en on any !rap" ● symmetries 9;$ Symmetries and parent Hamiltonians ● can be encoded locally in ho2e'er, a $ ● #but there are partial results% 2e can also de/ne !eneral characteri(ation ... ... ... ● entanglement again, a full characterization of ... ... parent ,amiltonians #and again, there are partial results% ... ... of in'erse direction is degrees of freedom) ... ... ... ... ... $ ... ... ... !round space and spectral !ap ... ... still missin! ... ... ... ... 5 is missing … 2omputational complexity of P PS ● expectation values in P0PS #e1g. correlation functions%) transfer operator ● resembles 1E situation, but ... 5 exact contraction is a hard problem (more precisely, #P-hard% ● approximation met"ods necessary – e1!1 b a!ain using MPS Pro=ected ntan!led Pair States 0PEPS) approximate two-dimensional systems faithfully, can be used for numerical simulations5 and allow to locally encode the p"ysics of 9; systems& Tensor Networks and Topological Order T"e Toric Code model ● Toric 2ode: ground state = superposition of all loop patterns 111 ● <amiltonian: #i) vertex term → enforce closed loops #ii% plaquette term → 'x phase when Pipping plaquette ● degenerate !round states: labeled b parity of loops around torus ● non-tri'ial excitations: #i) broken strings (come in pairs% #ii) 2rong relative phase (also in pairs% Tensor networks for topological states ● Tensor network for Toric Code) N N N N N ... ● 8oric Code tensor has symmetry ($e'en parit %) ● +hat are consequences of such an entanglement symmetry in a P0PS= ntan!lement symmetry and pulling throug" ● S mmetry can be rephrased as “pullin!-through condition*) A $ A $ ● pulling-through condition L Strin!s can be freely moved8 ● Strings are invisible locally #e.g. to Hamiltonian% ● Note: Beneralization of “pullin!-through condition* allo2s to characterize all known 0non-chiral1 topolo!ical phases Topological ground space manifold ● Torus: closed strings ield di<erent !round states ● de!eneracy depends on topolo!y #genus%: Topological order8 & local characteri(ation of topological order & parametri*ation of !round space manifold based on symmetry of single tensor & !ives us the tools to explicitly construct > study !round states & works for s