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

Matrix product statesfor the absolute beginner

Garnet Kin-Lic ChanPrinceton University

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

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 positions L spins L particle occupancies

Exponential complexity to represent wavefunction

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 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)

illusion of complexitynature does not explore all possibilities

Nature is local: ground-states have low entanglement

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)

MERA1D/nD entanglement for gapless systems

different tensor networks reflect geometry of entanglement

Graphical language

n1 n2 n3

Algebraic form Graphical form

thick line= single tensor

physicalindex

Generalstate

e.g.

spin 1/2

particles

Graphical language, cont’d

n1 n2 n3

Algebraic form Graphical form

Generaloperator

n1’ n2’ n3’

Ex: overlap, expectation value

Overlap

Expectationvalue

Low entanglement statesWhat does it mean for a state to have low entanglement?

No entanglement

local measurements on separated system 1, system 2can 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

Matrix product states

“bond” or “auxiliary” dimension “M” or “D” or “χ”

first and last tensors have one fewer auxiliary index

n1 n2 n3

=

Generalstate

i2

n1

i1

n2 n3

MPS

amplitude is obtained as a product of matrices

1D structureof entanglement

MPS gaugeMPS are not unique: defined up to gauge on the auxiliary indices

=i j

=insert gauge

matrices

=

Ex: MPS contraction

Efficient computation: contract in the correct order!

Overlap“d”

“M”

“d”“M”

1 2 3

MPS overlap: total cost

MPS from general staterecall singular value decomposition (SVD) of matrix

orthogonality conditions

singular values

=i

j =

singular values =i

j

orthogonality conditions

MPS from general state, cont’d

= 1

“n” “m”

SVD

i

j

i

j

=

=

“n” “m”

=SVD

=

=2 3

= 2 31

Step 1

Step 2

Common canonical forms

= 2 31

2 31

2 31

2

2

different canonical form: absorb singular values into the tensors

left canonical

=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=

2

2i

j =i

j etc

mixed canonicalaround site 2(DMRG form) 2 31

1

1 i

j3

3=

i

j

“Vidal” form

Matrix product operators

each tensor has a bra and ket physical index

n1 n2 n3

n1’ n2’ n3’

= i2

n1

i1

n2 n3

n1’ n2’ n3’

Generaloperator

MPO

Typical MPO’s

What is bond dimension as an MPO?

L Rjoins pairs of operators on both sides

MPO bond dimension = 5

MPO acting on MPS

MPO

MPS

=M1

M2

MPS=M1 x M2

MPO on MPS leads to new MPS with product of bond dimensions

MPS compression: SVDmany operations (e.g. MPOxMPS, MPS+MPS) increase bond dim.

compression: best approximate MPS with smaller bond dimension.

2 31

write MPS in Vidal gauge via SVD’s

2 31

M1 singular values

truncatebonds withsmall singular values

2 31

truncated M2 singular values

Each site is compressedindependently of new information of other sites:“Local” update: non-optimal.

MPS: variational compression

original(fixed) MPS

new MPS

1 32

1 32

1 2 3

solve minimization problem

Gradient algorithmTo minimize quantity, follow its gradient until it vanishes

1 2 32

linear in 2

=1 3

2 1 32

1 32

quadratic in 2

=1 3

1 32

x 2

2

Gradient step

Sweep algorithm (DMRG style)

1 2 3bilinear in 2 1 32

1 32

1 3

1 2 3

consider 2 as vector

where1 3

1 32

1 32where

1 3

1 3

Sweep algorithm cont’d

1 2 3

Minimization performed site by site by solving

1 2 3

1 2 3

1 31 3

1 3M b

2 332

32

M b

1 21 2

1 2M b

use updated tensors from previous step

Sweep and mixed canonical form

in mixed canonical form

1 2 32 332

32Mb

mixed canonical around site 1

1 2 3 1 31 3

1 3M b

mixed canonical around site 2note: updated singular values

change canonical form

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 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.

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

DMRG energy minimization

2

1 3

1 32

1 3

1 32

use mixed canonical form around site 2

unit matrix

eigenvalue problem foreach site, in mixed canonical form

where

1 3

1 3

DMRG “superblock” Hamiltonian

Time evolution

real time evolution

imaginary time evolution: replace i by 1.

projects onto ground-state at long times.

General time-evolution

2 31

compress

2 31

repeat

Short range H: Trotter form

evolution on pairs of bonds

Even-odd evolution

time evolution can be broken up into even and odd bonds

Time-evolving block decimationeven/odd evolution easy to combine with SVD compression: TEBD

SVDincrease of bond dimensionof unconnected bonds: SVD compression can be done independently on each bond.

Periodic and infinite MPS

MPS easily extended to PBC and thermodynamic limit

2 31Finite MPS (OBC)

2 31Periodic MPS

2 31Infinite MPS

Infinite TEBD

local algorithms such as TEBD easy to extend to infinite MPS

A B A B A B

Unit cell = 2 site infinite MPS

i-TEBD cont’d

A B

even bond evolution

+ compression updated

not updated

Step 1

Step 2

B A

odd bond evolution

+ compressionupdated

not updated

Repeat

A B

B A

SymmetriesGiven global symmetry group, local site basis can be labelledby 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

MPS and symmetrybond indices can be labelled by same symmetry labels as physical sites

labelled by integere.g. particle number symmetry

MPS: well defined Abelian symmetry, each tensor fulfils rule

Choice of convention:

tensor with no arrowsleaving gives totalstate quantum number

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

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)

MERA1D/nD entanglement for gapless systems

different tensor networks reflect geometry of entanglement

Questions

1. What is the dimension of MPS (M1) + MPS (M2)?

2. How would we graphically represent the DM of an MPS, (tracing out sites n3 to nL?)

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)?