Presentation on theme: "Matrix product states for the absolute beginner"— Presentation transcript:
1 Matrix product states for the absolute beginner Garnet Kin-Lic ChanPrinceton University----- Meeting Notes (12/12/14 12:49) -----b
2 Brief overview: Why tensor networks? Graphical notationMatrix Product States and Matrix Product OperatorsCompressing Matrix Product StatesEnergy optimizationTime evolutionPeriodic and infinite MPSFocus on basic computations and algorithms with MPSnot covered: entanglement area laws, RG, topological aspects, symmetries etc.
3 Quantum mechanics is complex DiracThe 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.So: ideas in quantum chemistry. Well let’s start at the very beginning, which is to recognize that quantum mechanics is the foundation of chemistry, which was realized byPaul Dirac, when he wrote down his equation and concluded that the fundamental laws for all of chemistry were known.The only problem, as recognized by Dirac also, is that the direct solution of quantum mechanics appears too complicated. Why is this?Well, the fundamental object in quantum mechanics is the wavefunction, and this is a joint or simultaneous probability amplitude for all the electrons.n electron positionsL spinsL particle occupanciesExponential complexity to represent wavefunction
4 This view of QM is depressing [The Schrodinger equation] cannot be solved accurately whenthe number of particles exceeds about 10. No computer existing,or that will ever exist, can break this barrier because it is acatastrophe of dimension ...Pines and Laughlin (2000)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)
5 illusion of complexity nature does not explore all possibilitiesNature is local: ground-states have low entanglement
6 Language for low entanglement states is tensor networks different tensor networks reflect geometry of entanglementMatrix Product State1D entanglement for gapped systems(basis of DMRG - often used in quasi-2D/3D)MERA1D/nD entanglement for gapless systemsTensor Product State (PEPS)nD entanglement for gapped systems
7 Graphical language n1 n2 n3 Algebraic form Graphical form spin 1/2 e.g.particlesAlgebraic formGraphical formn1n2n3physicalindexGeneralstatethick line= single tensor
10 Low entanglement states What does it mean for a state to have low entanglement?Consider system with two parts, 1 and 2No entanglementlocal measurements on separated system 1, system 2can be done independently. Local realism (classical)Entangled stateLow entanglement : small number of terms in the sum
11 “bond” or “auxiliary” dimension Matrix product statesfirst and last tensors have one fewer auxiliary index1D structureof entanglement“bond” or “auxiliary” dimension“M” or “D” or “χ”amplitude is obtained as a product of matricesGeneralstatei2n1i1n2n3MPSn1n2n3=
12 MPS gaugeMPS are not unique: defined up to gauge on the auxiliary indices=ijinsert gaugematrices==
13 Ex: MPS contractionOverlap“d”“M”Efficient computation: contract in the correct order!123“d”“M”MPS overlap: total cost
14 = MPS from general state = = recall singular value decomposition (SVD) of matrixsingular valuesorthogonality conditionsi==ji=singular valuesjorthogonality conditions
15 MPS from general state, cont’d iStep 1“n”“m”=jSVD=i1=jStep 2=“m”“n”SVD=23==231
16 Common canonical forms “Vidal” form23=1different canonical form: absorb singular values into the tensorsall tensors contract to unit matrix from leftleft canonical2311iji==2etcj2right canonicalall tensors contract to unit matrix from right23ii3=2=etc1jj32mixed canonicalaround site 2(DMRG form)231iji3=1j3
17 Matrix product operators each tensor has a bra and ket physical indexn1n2n3i2n1i1n2n3n1’n2’n3’Generaloperator=MPOn1’n2’n3’
18 Typical MPO’s What is bond dimension as an MPO? L R joins pairs of operators on both sidesMPO bond dimension = 5
19 = = MPO acting on MPS M2 MPO M1 MPS MPS M1 x M2 MPO on MPS leads to new MPS with product of bond dimensions
20 MPS compression: SVDmany operations (e.g. MPOxMPS, MPS+MPS) increase bond dim.compression: best approximate MPS with smaller bond dimension.write MPS in Vidal gauge via SVD’s231123truncatebonds withsmall singularvaluesM1 singular valuesEach site is compressedindependently of newinformation of other sites:“Local” update: non-optimal.231truncated M2 singular values
24 Sweep algorithm cont’d Minimization performed site by site by solving3223Mb123M13b13123use updated tensors from previous stepM12b12123
25 Sweep and mixed canonical form in mixed canonical formM3223b123mixed canonical around site 1change canonical formM13b13123mixed canonical around site 2note: updated singular values
26 SVD vs. variational compression variational algorithms – optimization for each site depends onall other sites. Uses “full environment”SVD compression: “local update”. Not as robust, but cheap!MPS: full environment / local update same computationalscaling, only differ by number of iterations.General tensor networks (e.g. PEPS): full environment may be expensive to compute or need further approximations.
27 Energy optimization bilinear form: similar to compression problem commonly used algorithmsDMRG: variational sweep with full environmentimag. TEBD: local update, imag. time evolution + SVD compression
28 DMRG energy minimization 2use mixed canonical form around site 2132132unit matrixwhere13eigenvalue problem foreach site, in mixed canonical formDMRG “superblock”Hamiltonian
29 Time evolution real time evolution imaginary time evolution: replace i by 1.projects onto ground-state at long times.
31 Short range H: Trotter form evolution on pairs of bonds
32 Even-odd evolutiontime evolution can be broken up into even and odd bonds
33 Time-evolving block decimation even/odd evolution easy to combine with SVD compression: TEBDincrease of bond dimensionof unconnected bonds:SVD compression can be doneindependently on each bond.SVD
34 Periodic and infinite MPS MPS easily extended to PBC and thermodynamic limitFinite MPS (OBC)231231Periodic MPS231Infinite MPS
35 Unit cell = 2 site infinite MPS Infinite TEBDlocal algorithms such as TEBD easy to extend to infinite MPSUnit cell = 2 site infinite MPSABABAB
36 i-TEBD cont’d A B even bond evolution + compression updated not updatedStep 1Step 2odd bondRepeat
37 SymmetriesGiven global symmetry group, local site basis can be labelledby irreps of group – quantum numbersU(1) – site basis labelled by integer n (particle number)n=0, 1, 2 etc...SU(2) symmetry – site basis labelled by j, m (spin quanta)Total state associated with good quantum numbers
38 MPS and symmetrybond indices can be labelled by same symmetry labels as physical sitese.g. particle number symmetrylabelled by integerMPS: well defined Abelian symmetry, each tensor fulfils ruletensor with no arrowsleaving gives totalstate quantum numberChoice of convention:
39 Brief overview: Why tensor networks? Graphical notationMatrix Product States and Matrix Product OperatorsCompressing Matrix Product StatesEnergy optimizationTime evolutionPeriodic and infinite MPSFocus on basic computations and algorithms with MPSnot covered: entanglement area laws, RG, topological aspects, symmetries etc.
40 Language for low entanglement states is tensor networks different tensor networks reflect geometry of entanglementMatrix Product State1D entanglement for gapped systems(basis of DMRG - often used in quasi-2D/3D)MERA1D/nD entanglement for gapless systemsTensor Product State (PEPS)nD entanglement for gapped systems
41 Questions What is the dimension of MPS (M1) + MPS (M2)? How would we graphically represent the DM of an MPS, (tracing out sites n3 to nL?)What is the dimension of the MPO of an electronic Hamiltonian with general quartic interactions?What happens when we use an MPS to represent a 2D system?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?How would we alter the discussion of symmetry for non-Abelian symmetry e.g. SU(2)?