matrix product states for the absolute beginner garnet kin-lic chan princeton university
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Matrix product statesfor the absolute beginner
Garnet Kin-Lic ChanPrinceton 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 positions L spins L 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 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)
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illusion of complexitynature 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)
MERA1D/nD entanglement for gapless systems
different tensor networks reflect geometry of entanglement
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Graphical language
n1 n2 n3
Algebraic form Graphical form
thick line= single tensor
physicalindex
Generalstate
e.g.
spin 1/2
particles
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Graphical language, cont’d
n1 n2 n3
Algebraic form Graphical form
Generaloperator
n1’ n2’ n3’
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Ex: overlap, expectation value
Overlap
Expectationvalue
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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
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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
n1 n2 n3
=
Generalstate
i2
n1
i1
n2 n3
MPS
amplitude is obtained as a product of matrices
1D structureof entanglement
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MPS gaugeMPS are not unique: defined up to gauge on the auxiliary indices
=i j
=insert gauge
matrices
=
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Ex: MPS contraction
Efficient computation: contract in the correct order!
Overlap“d”
“M”
“d”“M”
1 2 3
MPS overlap: total cost
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MPS from general staterecall singular value decomposition (SVD) of matrix
orthogonality conditions
singular values
=i
j =
singular values =i
j
orthogonality conditions
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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
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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
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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
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Typical MPO’s
What is bond dimension as an MPO?
L Rjoins pairs of operators on both sides
MPO bond dimension = 5
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MPO acting on MPS
MPO
MPS
=M1
M2
MPS=M1 x M2
MPO on MPS leads to new MPS with product of bond dimensions
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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.
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MPS: variational compression
original(fixed) MPS
new MPS
1 32
1 32
1 2 3
solve minimization problem
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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
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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
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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
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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
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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.
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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
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
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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 31
compress
2 31
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 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.
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Periodic and infinite MPS
MPS easily extended to PBC and thermodynamic limit
2 31Finite MPS (OBC)
2 31Periodic MPS
2 31Infinite MPS
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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
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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
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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
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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
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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)
MERA1D/nD entanglement for gapless systems
different tensor networks reflect geometry of entanglement
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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)?