Entanglement and complexity of many-body wavefunctions

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1 Enanglemen and complexiy of many-body wavefuncions Frank Versraee, Universiy of Vienna Norber Schuch, Calech Ignacio Cirac, Max Planck Insiue for Quanum Opics Tobias Osborne, Univ. Hannover

2 Overview Compuaional complexiy and quanum compuaion Wha consequences would a soluion o he following problems have for complexiy heory N-represenabiliy Efficien descripion of he Universal Densiy Funcional Theoreical resuls and variaional ansazes for wavefuncions of gapped laice Hamilonians Reduced densiy marices and area laws Marix Produc Saes The variaional principle for quanum field heories Coninuous marix produc saes A quanum Gross-Piaevski equaion

3 Quanum Compuaion How o exploi he superposiion principle o perform compuaions much faser han on a classical compuer Basic building block: qubi = -level quanum sysem Can be spin degree of freedom of an elecron (quanum dos), almos degenerae energy levels in aoms (ion raps), inernal degrees of freedom of neural aoms (Bose-Hubbard model in opical laice), Wha is needed: Time-dependen conrol of he - and -body erms in he Hamilonian (very much non-equilibrium sysem!) Wha is he boleneck: Decoherence: leaking of he quanum informaion o he environmen However: heorem on quanum faul olerance

4 Quanum Algorihms Shor s algorihm: polynomial algorihm for facoring an N-digi number Imporan applicaions in field of crypography Quanum Simulaion: given a many-body quanum Hamilonian, can we use a quanum compuer o. find ground / hermal / excied saes?. calculae ime-dependen non-equilibrium properies? Answers: () is rivial, () is much harder bu can eiher be solved using Adiabaic evoluion from a known ground sae (Farhi e al, 00) Quanum Meropolis algorihm: sample in he eigenbasis (no sign problem!) (K. Temme e al., 09)

5 Compuaional Complexiy Traveling Salesman, 3-SAT, ground sae of spin glass Ground saes of quanum spin glasses NP Primaliy Tesing, any algorihm ha makes sense P QMA BQP Facoring, Quanum Meropolis P: class of problems ha can be solved efficienly using classical compuer BQP: class of problems ha can be solved efficienly using quanum compuer NP: class of problems whose soluion can be checked efficienly using classical compuer QMA: class of problems whose soluion can be checked efficienly using quanum compuer

6 Quanum Spin Glasses: he Feynman-Kiaev Hamilonian Quanum analogue consrucion o he 3-SAT consrucion of Cook and Levin: encode soluion o he problem ino he ground sae of a Hamilonian: is i frusraed or unfrusraed? Is a generic Hamilonian on spin ½ sysems wih a mos 5-body erms Deermining ground sae energy of such a Hamilonian allows o solve ANY problem in he class QMA (and hence NP)! T T H H U U I H H H z k final N k z k ini T 0 0

7 Reducions o more realisic Hamilonians Using h order perurbaion heory ( Gadges ), his problem can be reduced o Finding ground saes of generic -body Hamilonian on qubis (wih long-range ineracions) (Kempe, Kiaev, 04) -body Hamilonians on a square -D laice wih neares neighbour ineracions (Oliveria and Terhal 06) Heisenberg model on a square -D laice wih n.n ineracions and sie-dependen magneic fields Hubbard model on a square laice and sie-dependen magneic fields Hence finding ground sae energy of such a sysem is QMA-hard

8 Example of a gadge: from Heisenberg ineracion + magneic field o he XY-model by second order perurbaion heory ) / ( Z Z Z Z Z Y Y X X Z Z YY X X H 3 3 YY X X H eff

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10 Nex sep: reducing he problem o he many-elecron Schrodinger equaion wih a given exernal poenial: Can we define an exernal poenial such ha he low-energy subspace is described by he Hubbard model in a magneic field? Sep : model non-ineracing par as sum of -D Kronig-Penney models which have band of bound saes Sep : rea Coulomb + local magneic field as a perurbaion (noe: his is only possible by working on a FINITE sysem and choosing he laice parameer as a funcion of he oal number of paricles such ha he long-range effecs of he Coulomb force can be ignored) Effec: local Hubbard ineracion erm + magneic field

11 Summary: here exis a choice of a exernal poenial + spaial dependen magneic field such ha he resuling low-energy secor is well described by he Hubbard model, Heisenberg model,, quanum spin glass+ correcions ha can be made arbirarily small I is herefore QMA-hard o find ground saes for elecrons ineracing via Coulomb in such an exernal poenial!

12 Connecion o DFT: Use consrain search formulaion of (spin!) DFT Schuch, FV Na. Phys. 09 If we impose ha he local densiy arises from an N-body densiy marix (no necessarily pure!), hen he universal funcional is convex; his means ha he problem is a convex problem Hence, if here were an efficien (black-box) descripion of his universal funcional, we can easily find he minimum of his funcion (convex programming) Hence: efficien descripion of he universal spin densiy funcional would imply ha we can solve QMA-hard problems Would imply QMA=NP=P! (= million dollar Clay quesion!) Similar (bu much simpler) proof possible for -represenabiliy (Liu, Chrisandl, FV 06).

13 Example ha we gave is cerainly academic, bu neverheless shows ha i is an inracable problem o find he universal funcional! Remarks abou compuaional complexiy Cavea: final Hamilonian is no very physical (srenghs of local poenial and magneic fields have o scale wih he sysem size and be unable a every laice sie) Compuaional complexiy has o do wih he wors-case scenario Could sill be ha we can approximae soluion well enough (alhough: PCP heorem) Why would we like o simulae quanum spin glasses if naure iself does no hermalize in such a siuaion? Does no say anyhing abou he complexiy of simulaing e.g. a ranslaional invarian sysem, we canno even define NP or QMA, as here are no enough parameers