Technische Universit¨ at M ¨ unchen Approximation Techniques for Special Eigenvalue Problems Konrad Waldherr June 2011 Joint work with Thomas Huckle K. Waldherr: Approximation Techniques for Special Eigenvalue Problems HDA 2011, Bonn, June 2011 1
Technische Universitat Munchen
Approximation Techniques for SpecialEigenvalue Problems
Konrad Waldherr
June 2011
Joint work with Thomas Huckle
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 1
Technische Universitat Munchen
Outline1 Problem Setting: Computation of Ground States
Physical ModelMathematical Model
2 Data-Sparse Representation Formats
3 Matrix Product StatesFormalismComputations with MPSUniqueness of MPSMinimization in terms of MPS
4 Subspace Methods in Terms of MPS
5 Numerical results
6 Conclusions
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 2
Technische Universitat Munchen
Problem: Computation of Ground StatesGiven:
• Physical system: 1D spin chain with N particles
1 2 3 4 5
• Interaction within the system (e.g. nearest-neighbor interaction)
1 2 3 4 5
• External interaction (e.g. exterior magnetic field)
1 2 3 4 5
Goal:• Find ground state, i.e. the state related to the smallest energy of
the system.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 3
Technische Universitat Munchen
Problem: Computation of Ground StatesGiven:
• Physical system: 1D spin chain with N particles
1 2 3 4 5
• Interaction within the system (e.g. nearest-neighbor interaction)
1 2 3 4 5
• External interaction (e.g. exterior magnetic field)
1 2 3 4 5
Goal:• Find ground state, i.e. the state related to the smallest energy of
the system.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 3
Technische Universitat Munchen
Problem: Computation of Ground StatesGiven:
• Physical system: 1D spin chain with N particles
1 2 3 4 5
• Interaction within the system (e.g. nearest-neighbor interaction)
1 2 3 4 5
• External interaction (e.g. exterior magnetic field)
1 2 3 4 5
Goal:• Find ground state, i.e. the state related to the smallest energy of
the system.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 3
Technische Universitat Munchen
Problem: Computation of Ground StatesGiven:
• Physical system: 1D spin chain with N particles
1 2 3 4 5
• Interaction within the system (e.g. nearest-neighbor interaction)
1 2 3 4 5
• External interaction (e.g. exterior magnetic field)
1 2 3 4 5
Goal:• Find ground state, i.e. the state related to the smallest energy of
the system.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 3
Technische Universitat Munchen
Mathematical Model• Any state of the system is represented by a vector x ∈ C2N
.• The physical system can be described by the HamiltonianH ∈ C2N×2N
.
• The Hamiltonian may be formulated as a weighted sum ofKronecker products of Pauli and identity matrices.
• Pauli matrices:
σx =
(0 11 0
), σy =
(0 −ii 0
), σz =
(1 00 −1
).
• Kronecker product:
A⊗B :=
a11 . . . a1n
.... . .
...am1 . . . amn
⊗B =
a11B . . . a1nB...
. . ....
am1B . . . amnB
• Then, the ground state corresponds to the eigenvector related
to the smallest eigenvalue.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 4
Technische Universitat Munchen
Mathematical Model• Any state of the system is represented by a vector x ∈ C2N
.• The physical system can be described by the HamiltonianH ∈ C2N×2N
.• The Hamiltonian may be formulated as a weighted sum of
Kronecker products of Pauli and identity matrices.• Pauli matrices:
σx =
(0 11 0
), σy =
(0 −ii 0
), σz =
(1 00 −1
).
• Kronecker product:
A⊗B :=
a11 . . . a1n
.... . .
...am1 . . . amn
⊗B =
a11B . . . a1nB...
. . ....
am1B . . . amnB
• Then, the ground state corresponds to the eigenvector relatedto the smallest eigenvalue.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 4
Technische Universitat Munchen
Mathematical Model• Any state of the system is represented by a vector x ∈ C2N
.• The physical system can be described by the HamiltonianH ∈ C2N×2N
.• The Hamiltonian may be formulated as a weighted sum of
Kronecker products of Pauli and identity matrices.• Pauli matrices:
σx =
(0 11 0
), σy =
(0 −ii 0
), σz =
(1 00 −1
).
• Kronecker product:
A⊗B :=
a11 . . . a1n
.... . .
...am1 . . . amn
⊗B =
a11B . . . a1nB...
. . ....
am1B . . . amnB
• Then, the ground state corresponds to the eigenvector related
to the smallest eigenvalue.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 4
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I
+ I ⊗ σz ⊗ σz ⊗ I ⊗ I+ I ⊗ I ⊗ σz ⊗ σz ⊗ I+ I ⊗ I ⊗ I ⊗ σz ⊗ σz+ σx ⊗ I ⊗ I ⊗ I ⊗ I+ I ⊗ σx ⊗ I ⊗ I ⊗ I+ I ⊗ I ⊗ σx ⊗ I ⊗ I+ I ⊗ I ⊗ I ⊗ σx ⊗ I+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I+ I ⊗ σz ⊗ σz ⊗ I ⊗ I
+ I ⊗ I ⊗ σz ⊗ σz ⊗ I+ I ⊗ I ⊗ I ⊗ σz ⊗ σz+ σx ⊗ I ⊗ I ⊗ I ⊗ I+ I ⊗ σx ⊗ I ⊗ I ⊗ I+ I ⊗ I ⊗ σx ⊗ I ⊗ I+ I ⊗ I ⊗ I ⊗ σx ⊗ I+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I+ I ⊗ σz ⊗ σz ⊗ I ⊗ I+ I ⊗ I ⊗ σz ⊗ σz ⊗ I
+ I ⊗ I ⊗ I ⊗ σz ⊗ σz+ σx ⊗ I ⊗ I ⊗ I ⊗ I+ I ⊗ σx ⊗ I ⊗ I ⊗ I+ I ⊗ I ⊗ σx ⊗ I ⊗ I+ I ⊗ I ⊗ I ⊗ σx ⊗ I+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I+ I ⊗ σz ⊗ σz ⊗ I ⊗ I+ I ⊗ I ⊗ σz ⊗ σz ⊗ I+ I ⊗ I ⊗ I ⊗ σz ⊗ σz
+ σx ⊗ I ⊗ I ⊗ I ⊗ I+ I ⊗ σx ⊗ I ⊗ I ⊗ I+ I ⊗ I ⊗ σx ⊗ I ⊗ I+ I ⊗ I ⊗ I ⊗ σx ⊗ I+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I+ I ⊗ σz ⊗ σz ⊗ I ⊗ I+ I ⊗ I ⊗ σz ⊗ σz ⊗ I+ I ⊗ I ⊗ I ⊗ σz ⊗ σz+ σx ⊗ I ⊗ I ⊗ I ⊗ I
+ I ⊗ σx ⊗ I ⊗ I ⊗ I+ I ⊗ I ⊗ σx ⊗ I ⊗ I+ I ⊗ I ⊗ I ⊗ σx ⊗ I+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I+ I ⊗ σz ⊗ σz ⊗ I ⊗ I+ I ⊗ I ⊗ σz ⊗ σz ⊗ I+ I ⊗ I ⊗ I ⊗ σz ⊗ σz+ σx ⊗ I ⊗ I ⊗ I ⊗ I+ I ⊗ σx ⊗ I ⊗ I ⊗ I
+ I ⊗ I ⊗ σx ⊗ I ⊗ I+ I ⊗ I ⊗ I ⊗ σx ⊗ I+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I+ I ⊗ σz ⊗ σz ⊗ I ⊗ I+ I ⊗ I ⊗ σz ⊗ σz ⊗ I+ I ⊗ I ⊗ I ⊗ σz ⊗ σz+ σx ⊗ I ⊗ I ⊗ I ⊗ I+ I ⊗ σx ⊗ I ⊗ I ⊗ I+ I ⊗ I ⊗ σx ⊗ I ⊗ I
+ I ⊗ I ⊗ I ⊗ σx ⊗ I+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I+ I ⊗ σz ⊗ σz ⊗ I ⊗ I+ I ⊗ I ⊗ σz ⊗ σz ⊗ I+ I ⊗ I ⊗ I ⊗ σz ⊗ σz+ σx ⊗ I ⊗ I ⊗ I ⊗ I+ I ⊗ σx ⊗ I ⊗ I ⊗ I+ I ⊗ I ⊗ σx ⊗ I ⊗ I+ I ⊗ I ⊗ I ⊗ σx ⊗ I
+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Example: Ising-type Hamiltonian
1 2 3 4 5
H = σz ⊗ σz ⊗ I ⊗ I ⊗ I+ I ⊗ σz ⊗ σz ⊗ I ⊗ I+ I ⊗ I ⊗ σz ⊗ σz ⊗ I+ I ⊗ I ⊗ I ⊗ σz ⊗ σz+ σx ⊗ I ⊗ I ⊗ I ⊗ I+ I ⊗ σx ⊗ I ⊗ I ⊗ I+ I ⊗ I ⊗ σx ⊗ I ⊗ I+ I ⊗ I ⊗ I ⊗ σx ⊗ I+ I ⊗ I ⊗ I ⊗ I ⊗ σx
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 5
Technische Universitat Munchen
Problem Setting• General formulation of the Hamiltonian H:
H =
M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N , Q(k)i ∈ {id, σx, σy, σz} ⊂ C2×2
• Variational ansatz: Minimization of the Rayleigh quotient:
minx 6=0
xHHx
xHx
• Problems:
• the vector space V = C2N
grows exponentially in N ,• the problem is not generic.
• Idea: Find data-sparse representation formats that allow toovercome the curse of dimensionality.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 6
Technische Universitat Munchen
Problem Setting• General formulation of the Hamiltonian H:
H =
M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N , Q(k)i ∈ {id, σx, σy, σz} ⊂ C2×2
• Variational ansatz: Minimization of the Rayleigh quotient:
minx 6=0
xHHx
xHx
• Problems:
• the vector space V = C2N
grows exponentially in N ,• the problem is not generic.
• Idea: Find data-sparse representation formats that allow toovercome the curse of dimensionality.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 6
Technische Universitat Munchen
Problem Setting• General formulation of the Hamiltonian H:
H =
M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N , Q(k)i ∈ {id, σx, σy, σz} ⊂ C2×2
• Variational ansatz: Minimization of the Rayleigh quotient:
minx 6=0
xHHx
xHx
• Problems:• the vector space V = C2N
grows exponentially in N ,• the problem is not generic.
• Idea: Find data-sparse representation formats that allow toovercome the curse of dimensionality.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 6
Technische Universitat Munchen
Problem Setting• General formulation of the Hamiltonian H:
H =
M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N , Q(k)i ∈ {id, σx, σy, σz} ⊂ C2×2
• Variational ansatz: Minimization of the Rayleigh quotient:
minx 6=0
xHHx
xHx
• Problems:• the vector space V = C2N
grows exponentially in N ,• the problem is not generic.
• Idea: Find data-sparse representation formats that allow toovercome the curse of dimensionality.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 6
Technische Universitat Munchen
Data-sparse representation formats• The data-sparse format has to allow for
1. proper reproduction of the physical setting (e.g.nearest-neighbor interaction),
2. representations that are only polynomial in N ,3. good approximation properties,4. efficient evaluation of the Rayleigh quotient
minx∈U
xHHx
xHx(1)
=⇒ efficient computation of both Hx and yHx.
• Problem setting:• Not: Find a data-sparse format of a given data structure• But: Use the data-sparse format as an ansatz for the
minimization of the Rayleigh quotient (1).
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 7
Technische Universitat Munchen
Data-sparse representation formats• The data-sparse format has to allow for
1. proper reproduction of the physical setting (e.g.nearest-neighbor interaction),
2. representations that are only polynomial in N ,3. good approximation properties,4. efficient evaluation of the Rayleigh quotient
minx∈U
xHHx
xHx(1)
=⇒ efficient computation of both Hx and yHx.• Problem setting:
• Not: Find a data-sparse format of a given data structure• But: Use the data-sparse format as an ansatz for the
minimization of the Rayleigh quotient (1).
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 7
Technische Universitat Munchen
Tensor Decompositions• Any state x ∈ C2N
may be considered as a N th order tensor:
x = (xi1i2...iN )ij=0,1
i1 i2 iN
• Goal: Find convenient tensor decomposition as ansatz for (1)• Classical decomposition formats:
• Tucker decomposition,• Canonical decomposition.
• Physically motivated formats:• Matrix Product States (MPS) for 1D problems,• Projected Entangled Pair States (PEPS) for 2D problems.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 8
Technische Universitat Munchen
Tensor Decompositions• Any state x ∈ C2N
may be considered as a N th order tensor:
x = (xi1i2...iN )ij=0,1
i1 i2 iN
• Goal: Find convenient tensor decomposition as ansatz for (1)
• Classical decomposition formats:• Tucker decomposition,• Canonical decomposition.
• Physically motivated formats:• Matrix Product States (MPS) for 1D problems,• Projected Entangled Pair States (PEPS) for 2D problems.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 8
Technische Universitat Munchen
Tensor Decompositions• Any state x ∈ C2N
may be considered as a N th order tensor:
x = (xi1i2...iN )ij=0,1
i1 i2 iN
• Goal: Find convenient tensor decomposition as ansatz for (1)• Classical decomposition formats:
• Tucker decomposition,• Canonical decomposition.
• Physically motivated formats:• Matrix Product States (MPS) for 1D problems,• Projected Entangled Pair States (PEPS) for 2D problems.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 8
Technische Universitat Munchen
Tensor Decompositions• Any state x ∈ C2N
may be considered as a N th order tensor:
x = (xi1i2...iN )ij=0,1
i1 i2 iN
• Goal: Find convenient tensor decomposition as ansatz for (1)• Classical decomposition formats:
• Tucker decomposition,• Canonical decomposition.
• Physically motivated formats:• Matrix Product States (MPS) for 1D problems,• Projected Entangled Pair States (PEPS) for 2D problems.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 8
Technische Universitat Munchen
Decompositions of a tensor• Tucker decomposition:
Not advantageous in our case of a binary tensor.
• Canonical decomposition (CP):
Allows generalizations in several ways: (see [Huckle 2011])• blockings and mixed blockings,• permutations.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 9
Technische Universitat Munchen
Decompositions of a tensor• Tucker decomposition:
Not advantageous in our case of a binary tensor.• Canonical decomposition (CP):
Allows generalizations in several ways: (see [Huckle 2011])• blockings and mixed blockings,• permutations.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 9
Technische Universitat Munchen
Matrix Product States (MPS)• Consider again the linear 1D spin chain.
b b b b b
site j
• Each site j is related to a matrix pair A(0)j , A
(1)j ∈ CDj×Dj+1
b b b b b
A(ij)j
• For i = (i1, i2, . . . , iN )2, the vector component xi is given by
xi = xi1i2...iN = tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)
b b b b b
ij−1 ij ij+1 ij+2 ij+3
mj−1 mj mj+1 mj+2
(1.X)
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 10
Technische Universitat Munchen
Matrix Product States (MPS)• Consider again the linear 1D spin chain.
b b b b b
site j
• Each site j is related to a matrix pair A(0)j , A
(1)j ∈ CDj×Dj+1
b b b b b
A(ij)j
• For i = (i1, i2, . . . , iN )2, the vector component xi is given by
xi = xi1i2...iN = tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)
b b b b b
ij−1 ij ij+1 ij+2 ij+3
mj−1 mj mj+1 mj+2
(1.X)
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 10
Technische Universitat Munchen
Matrix Product States (MPS)• Consider again the linear 1D spin chain.
b b b b b
site j
• Each site j is related to a matrix pair A(0)j , A
(1)j ∈ CDj×Dj+1
b b b b b
A(ij)j
• For i = (i1, i2, . . . , iN )2, the vector component xi is given by
xi = xi1i2...iN = tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)
b b b b b
ij−1 ij ij+1 ij+2 ij+3
mj−1 mj mj+1 mj+2
(1.X)
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 10
Technische Universitat Munchen
Matrix Product States• Thus, the vector x may be written as
x =
2N∑i=1
xiei =∑
i1,...,ip
tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)︸ ︷︷ ︸
=xi=xi1,...,iN
ei1i2...iN︸ ︷︷ ︸=ei
.
• Binary version of the Tensor Train concept. (see [Oseledets andTyrtyshnikov, 2009])
• MPS representation requires 2ND2 instead of 2N entries. (2.X)• Any state can be represented exactly by an MPS (but at the cost
of exponentially growing matrix dimension D),• For approximation problems, D has to scale only polynomially.
(3.X)
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 11
Technische Universitat Munchen
Matrix Product States• Thus, the vector x may be written as
x =
2N∑i=1
xiei =∑
i1,...,ip
tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)︸ ︷︷ ︸
=xi=xi1,...,iN
ei1i2...iN︸ ︷︷ ︸=ei
.
• Binary version of the Tensor Train concept. (see [Oseledets andTyrtyshnikov, 2009])
• MPS representation requires 2ND2 instead of 2N entries. (2.X)
• Any state can be represented exactly by an MPS (but at the costof exponentially growing matrix dimension D),
• For approximation problems, D has to scale only polynomially.(3.X)
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 11
Technische Universitat Munchen
Matrix Product States• Thus, the vector x may be written as
x =
2N∑i=1
xiei =∑
i1,...,ip
tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)︸ ︷︷ ︸
=xi=xi1,...,iN
ei1i2...iN︸ ︷︷ ︸=ei
.
• Binary version of the Tensor Train concept. (see [Oseledets andTyrtyshnikov, 2009])
• MPS representation requires 2ND2 instead of 2N entries. (2.X)• Any state can be represented exactly by an MPS (but at the cost
of exponentially growing matrix dimension D),• For approximation problems, D has to scale only polynomially.
(3.X)
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 11
Technische Universitat Munchen
Matrix Product StatesExample
• Consider the case N = 2 and D = 1:
ψ =
1∑i1,i2=0
(a
(i1)1 a
(i2)2
)︸ ︷︷ ︸
=xi=xi1i2
(ei1 ⊗ ei2)︸ ︷︷ ︸=ei=ei1i2
=
a
(0)1 a
(0)2
a(0)1 a
(1)2
a(1)1 a
(0)2
a(1)1 a
(1)2
!=
x1
x2
x3
x4
• For exact representation:
x1x4 = x2x3
• e1, e2, e3, e4 can be represented,• However, e1 + e4 = (1, 0, 0, 1)T cannot be represented.
• MPS are not closed under addition!
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 12
Technische Universitat Munchen
Matrix Product StatesExample
• Consider the case N = 2 and D = 1:
ψ =
1∑i1,i2=0
(a
(i1)1 a
(i2)2
)︸ ︷︷ ︸
=xi=xi1i2
(ei1 ⊗ ei2)︸ ︷︷ ︸=ei=ei1i2
=
a
(0)1 a
(0)2
a(0)1 a
(1)2
a(1)1 a
(0)2
a(1)1 a
(1)2
!=
x1
x2
x3
x4
• For exact representation:
x1x4 = x2x3
• e1, e2, e3, e4 can be represented,• However, e1 + e4 = (1, 0, 0, 1)T cannot be represented.
• MPS are not closed under addition!
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 12
Technische Universitat Munchen
Matrix Product StatesExample
• Consider the case N = 2 and D = 1:
ψ =
1∑i1,i2=0
(a
(i1)1 a
(i2)2
)︸ ︷︷ ︸
=xi=xi1i2
(ei1 ⊗ ei2)︸ ︷︷ ︸=ei=ei1i2
=
a
(0)1 a
(0)2
a(0)1 a
(1)2
a(1)1 a
(0)2
a(1)1 a
(1)2
!=
x1
x2
x3
x4
• For exact representation:
x1x4 = x2x3
• e1, e2, e3, e4 can be represented,• However, e1 + e4 = (1, 0, 0, 1)T cannot be represented.
• MPS are not closed under addition!
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 12
Technische Universitat Munchen
Matrix Product StatesExample
• Consider the case N = 2 and D = 1:
ψ =
1∑i1,i2=0
(a
(i1)1 a
(i2)2
)︸ ︷︷ ︸
=xi=xi1i2
(ei1 ⊗ ei2)︸ ︷︷ ︸=ei=ei1i2
=
a
(0)1 a
(0)2
a(0)1 a
(1)2
a(1)1 a
(0)2
a(1)1 a
(1)2
!=
x1
x2
x3
x4
• For exact representation:
x1x4 = x2x3
• e1, e2, e3, e4 can be represented,• However, e1 + e4 = (1, 0, 0, 1)T cannot be represented.
• MPS are not closed under addition!
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 12
Technische Universitat Munchen
Computations with MPS• Sum of two MPS vectors
x =∑
i1,...,iN
tr(A(i1)1 A
(i2)2 . . . A
(iN )N )ei1...iN
y =∑
i1,...,iN
tr(B(i1)1 B
(i2)2 . . . B
(iN )N )ei1...iN
x+ y =∑i1,...,iN
tr[(A
(i1)1 0
0 B(i1)1
)(A
(i2)2 0
0 B(i2)2
)︸ ︷︷ ︸
D′×D′
. . .
(A
(iN )N 0
0 B(iN )N
)]ei1...iN
• Summing MPS vectors leads to an augmentation of the matrixsizes D → D′.
• Use compression techniques to keep the size of the MPSmatrices constant.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 13
Technische Universitat Munchen
Computations with MPS• Sum of two MPS vectors
x =∑
i1,...,iN
tr(A(i1)1 A
(i2)2 . . . A
(iN )N )ei1...iN
y =∑
i1,...,iN
tr(B(i1)1 B
(i2)2 . . . B
(iN )N )ei1...iN
x+ y =∑i1,...,iN
tr[(A
(i1)1 0
0 B(i1)1
)(A
(i2)2 0
0 B(i2)2
)︸ ︷︷ ︸
D′×D′
. . .
(A
(iN )N 0
0 B(iN )N
)]ei1...iN
• Summing MPS vectors leads to an augmentation of the matrixsizes D → D′.
• Use compression techniques to keep the size of the MPSmatrices constant.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 13
Technische Universitat Munchen
Computations with MPS• Sum of two MPS vectors
x =∑
i1,...,iN
tr(A(i1)1 A
(i2)2 . . . A
(iN )N )ei1...iN
y =∑
i1,...,iN
tr(B(i1)1 B
(i2)2 . . . B
(iN )N )ei1...iN
x+ y =∑i1,...,iN
tr[(A
(i1)1 0
0 B(i1)1
)(A
(i2)2 0
0 B(i2)2
)︸ ︷︷ ︸
D′×D′
. . .
(A
(iN )N 0
0 B(iN )N
)]ei1...iN
• Summing MPS vectors leads to an augmentation of the matrixsizes D → D′.
• Use compression techniques to keep the size of the MPSmatrices constant.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 13
Technische Universitat Munchen
Computations with MPS• Inner products
xHy =∑
i1,...,iN
xi1,...,iN︷ ︸︸ ︷(tr(A
(i1)1 · · · A(iN )
N
))·
yi1,...,iN︷ ︸︸ ︷(tr(B
(i1)1 · · ·B(iN )
N
))=∑ij
∑kj
a(i1)1;k1,k2
· · · a(iN )N ;kN ,k1
∑mj
b(i1)1;m1,m2
· · · b(iN )N ;mN ,m1
Find an efficient ordering of the summations!• Matrix × vector
Hx =
(M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N
) ∑i1,...,iN
tr(A(i1)1 . . . A
(iN )N )ei1...iN
=
M∑k=1
y(k)MPS
=⇒ 4. XK. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 14
Technische Universitat Munchen
Uniqueness of MPS• The MPS formalism is not unique:∑
i1,...,iN
tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)ei1,...,iN
=∑
i1,...,iN
tr(
(A(i1)1 M−1
2 )(M2A(i2)2 M−1
3 ) · · · (MNA(iN )N )
)ei1,...,iN
• Replace the matrices A(0)j , A
(1)j by parts of unitary matrices using
the SVD: (A
(0)j
A(1)j
)=
(U
(0)j
U(1)j
)ΣjVj .
• Multiply the ΣjVj parts on the right neighbour.• Replace all A matrices by U matrices, up to one.• This remaining factor is considered to be optimized.• Leads to better convergence properties and numerical stability.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 15
Technische Universitat Munchen
Uniqueness of MPS• The MPS formalism is not unique:∑
i1,...,iN
tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)ei1,...,iN
=∑
i1,...,iN
tr(
(A(i1)1 M−1
2 )(M2A(i2)2 M−1
3 ) · · · (MNA(iN )N )
)ei1,...,iN
• Replace the matrices A(0)j , A
(1)j by parts of unitary matrices using
the SVD: (A
(0)j
A(1)j
)=
(U
(0)j
U(1)j
)ΣjVj .
• Multiply the ΣjVj parts on the right neighbour.• Replace all A matrices by U matrices, up to one.• This remaining factor is considered to be optimized.• Leads to better convergence properties and numerical stability.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 15
Technische Universitat Munchen
Uniqueness of MPS• The MPS formalism is not unique:∑
i1,...,iN
tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)ei1,...,iN
=∑
i1,...,iN
tr(
(A(i1)1 M−1
2 )(M2A(i2)2 M−1
3 ) · · · (MNA(iN )N )
)ei1,...,iN
• Replace the matrices A(0)j , A
(1)j by parts of unitary matrices using
the SVD: (A
(0)j
A(1)j
)=
(U
(0)j
U(1)j
)ΣjVj .
• Multiply the ΣjVj parts on the right neighbour.
• Replace all A matrices by U matrices, up to one.• This remaining factor is considered to be optimized.• Leads to better convergence properties and numerical stability.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 15
Technische Universitat Munchen
Uniqueness of MPS• The MPS formalism is not unique:∑
i1,...,iN
tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)ei1,...,iN
=∑
i1,...,iN
tr(
(A(i1)1 M−1
2 )(M2A(i2)2 M−1
3 ) · · · (MNA(iN )N )
)ei1,...,iN
• Replace the matrices A(0)j , A
(1)j by parts of unitary matrices using
the SVD: (A
(0)j
A(1)j
)=
(U
(0)j
U(1)j
)ΣjVj .
• Multiply the ΣjVj parts on the right neighbour.• Replace all A matrices by U matrices, up to one.• This remaining factor is considered to be optimized.
• Leads to better convergence properties and numerical stability.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 15
Technische Universitat Munchen
Uniqueness of MPS• The MPS formalism is not unique:∑
i1,...,iN
tr(A
(i1)1 ·A(i2)
2 · · ·A(iN )N
)ei1,...,iN
=∑
i1,...,iN
tr(
(A(i1)1 M−1
2 )(M2A(i2)2 M−1
3 ) · · · (MNA(iN )N )
)ei1,...,iN
• Replace the matrices A(0)j , A
(1)j by parts of unitary matrices using
the SVD: (A
(0)j
A(1)j
)=
(U
(0)j
U(1)j
)ΣjVj .
• Multiply the ΣjVj parts on the right neighbour.• Replace all A matrices by U matrices, up to one.• This remaining factor is considered to be optimized.• Leads to better convergence properties and numerical stability.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 15
Technische Universitat Munchen
Computation of the Ground StateHow to calculate the smallest eigenvalue of the Hamiltonian H?
1. Variational ansatz:
minxMPS
xHMPSHxMPS
xHMPSxMPS
• Find optimal matrix pairs A(ij)j for all sites j
• Use Alternating Least Squares: Keep all matrix pairs up toone fixed and optimize the remaining one
2. New idea: Use subspace iteration
minx∈Um
xHHx
xHx
• Find optimal solution in the subspace Um
• Um is defined by MPS vectors
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 16
Technische Universitat Munchen
Computation of the Ground StateHow to calculate the smallest eigenvalue of the Hamiltonian H?
1. Variational ansatz:
minxMPS
xHMPSHxMPS
xHMPSxMPS
• Find optimal matrix pairs A(ij)j for all sites j
• Use Alternating Least Squares: Keep all matrix pairs up toone fixed and optimize the remaining one
2. New idea: Use subspace iteration
minx∈Um
xHHx
xHx
• Find optimal solution in the subspace Um
• Um is defined by MPS vectors
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 16
Technische Universitat Munchen
Minimization Algorithm in terms of MPS• Start with initial guesses for the A(ij)
j matrices.
• Replace A(ij)j by U (ij)
j up to A(i1)1 .
• Consider all U matrices as fixed and optimize X1 = A1:
minX1
(∑ij
tr(X
(i1)1 U
(i2)2 · · ·U(iN )
N
)ei
)H
H
(∑ij
tr(X
(i1)1 U
(i2)2 · · ·U(iN )
N
)ei
)(∑
ij
tr(X
(i1)1 U
(i2)2 · · ·U(iN )
N
)ei
)H(∑ij
tr(X
(i1)1 U
(i2)2 · · ·U(iN )
N
)ei
)
= minX1
X1HR1X1
X1HX1
with effective Hamiltonian R1
• =⇒ dense eigenvalue problem for the (2D2 × 2D2) matrix R1.• Orthogonalize the solution and continue with the right neighbor.
Modifications of this algorithm lead to the DMRG method.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 17
Technische Universitat Munchen
Subspace iteration for the smallest eigenvalue• Idea: Minimize the Rayleigh quotient xHHx
xHx over a subspace (!)Um := span{q1, . . . , qm}.
• Let Vm = (q1, . . . , qm) ∈ C2N×m
• The minimization over Um yields
minx∈Um\{0}
xHHx
xHx
x=Vmy= min
y∈Cm\{0}
(Vmy)HH(Vmy)
(Vmy)HVmy
= miny∈Cm\{0}
yH(V HmHVm
)y
yH (V HmVm) y
= miny∈Cm\{0}
yHAy
yHBy,
a generalized eigenvalue problem Ay = λminBy of size m×m.• x = Vmy or restart with q1 = Vmy
• How to choose an appropriate subspace Um?
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 18
Technische Universitat Munchen
Subspace iteration for the smallest eigenvalue• Idea: Minimize the Rayleigh quotient xHHx
xHx over a subspace (!)Um := span{q1, . . . , qm}.
• Let Vm = (q1, . . . , qm) ∈ C2N×m
• The minimization over Um yields
minx∈Um\{0}
xHHx
xHx
x=Vmy= min
y∈Cm\{0}
(Vmy)HH(Vmy)
(Vmy)HVmy
= miny∈Cm\{0}
yH(V HmHVm
)y
yH (V HmVm) y
= miny∈Cm\{0}
yHAy
yHBy,
a generalized eigenvalue problem Ay = λminBy of size m×m.
• x = Vmy or restart with q1 = Vmy
• How to choose an appropriate subspace Um?
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 18
Technische Universitat Munchen
Subspace iteration for the smallest eigenvalue• Idea: Minimize the Rayleigh quotient xHHx
xHx over a subspace (!)Um := span{q1, . . . , qm}.
• Let Vm = (q1, . . . , qm) ∈ C2N×m
• The minimization over Um yields
minx∈Um\{0}
xHHx
xHx
x=Vmy= min
y∈Cm\{0}
(Vmy)HH(Vmy)
(Vmy)HVmy
= miny∈Cm\{0}
yH(V HmHVm
)y
yH (V HmVm) y
= miny∈Cm\{0}
yHAy
yHBy,
a generalized eigenvalue problem Ay = λminBy of size m×m.• x = Vmy or restart with q1 = Vmy
• How to choose an appropriate subspace Um?
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 18
Technische Universitat Munchen
Subspace iteration for the smallest eigenvalue• Idea: Minimize the Rayleigh quotient xHHx
xHx over a subspace (!)Um := span{q1, . . . , qm}.
• Let Vm = (q1, . . . , qm) ∈ C2N×m
• The minimization over Um yields
minx∈Um\{0}
xHHx
xHx
x=Vmy= min
y∈Cm\{0}
(Vmy)HH(Vmy)
(Vmy)HVmy
= miny∈Cm\{0}
yH(V HmHVm
)y
yH (V HmVm) y
= miny∈Cm\{0}
yHAy
yHBy,
a generalized eigenvalue problem Ay = λminBy of size m×m.• x = Vmy or restart with q1 = Vmy
• How to choose an appropriate subspace Um?
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 18
Technische Universitat Munchen
Subspace iteration for the smallest eigenvalue• Choose the Krylov space Km(H, b) as subspace:
Um = Km(H, b) := span{b,Hb, . . . ,Hm−1b}
• This ansatz yields
A = V HmHVm =
bHHb bHH2b . . . bHHmbbHH2b bHH3b . . . bHHm+1b
......
. . ....
bHHmb bHHm+1b . . . bHH2m−1b
B = V HmVm =
bHb bHHb . . . bHHm−1bbHHb bHH3b . . . bHHmb
......
. . ....
bHHm−1b bHHmb . . . bHH2m−2b
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 19
Technische Universitat Munchen
Subspace iteration for the smallest eigenvalue• Choose the Krylov space Km(H, b) as subspace:
Um = Km(H, b) := span{b,Hb, . . . ,Hm−1b}
• This ansatz yields
A = V HmHVm =
bHHb bHH2b . . . bHHmbbHH2b bHH3b . . . bHHm+1b
......
. . ....
bHHmb bHHm+1b . . . bHH2m−1b
B = V HmVm =
bHb bHHb . . . bHHm−1bbHHb bHH3b . . . bHHmb
......
. . ....
bHHm−1b bHHmb . . . bHH2m−2b
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 19
Technische Universitat Munchen
Subspace iteration in terms of MPS• b = bMPS andKm(H, bMPS) = span{bMPS , HbMPS , . . . ,H
m−1bMPS}:
• In principle the same proceeding, BUT
• Computation of the matrix vector products:
HjbMPS =
(M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N
)j
bMPS
=
M∑k1=1
· · ·M∑
kj=1
b(k1,...,kj)MPS
Number of summands grows dramatically!• Application of the back-transformation:
Vmymin =m∑j=1
ymin,jHj−1bMPS
does not lie in the MPS space.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 20
Technische Universitat Munchen
Subspace iteration in terms of MPS• b = bMPS andKm(H, bMPS) = span{bMPS , HbMPS , . . . ,H
m−1bMPS}:• In principle the same proceeding, BUT
• Computation of the matrix vector products:
HjbMPS =
(M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N
)j
bMPS
=M∑
k1=1
· · ·M∑
kj=1
b(k1,...,kj)MPS
Number of summands grows dramatically!• Application of the back-transformation:
Vmymin =m∑j=1
ymin,jHj−1bMPS
does not lie in the MPS space.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 20
Technische Universitat Munchen
Subspace iteration in terms of MPS• b = bMPS andKm(H, bMPS) = span{bMPS , HbMPS , . . . ,H
m−1bMPS}:• In principle the same proceeding, BUT
• Computation of the matrix vector products:
HjbMPS =
(M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N
)j
bMPS
=
M∑k1=1
· · ·M∑
kj=1
b(k1,...,kj)MPS
Number of summands grows dramatically!
• Application of the back-transformation:
Vmymin =m∑j=1
ymin,jHj−1bMPS
does not lie in the MPS space.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 20
Technische Universitat Munchen
Subspace iteration in terms of MPS• b = bMPS andKm(H, bMPS) = span{bMPS , HbMPS , . . . ,H
m−1bMPS}:• In principle the same proceeding, BUT
• Computation of the matrix vector products:
HjbMPS =
(M∑k=1
αkQ(k)1 ⊗ · · · ⊗Q(k)
N
)j
bMPS
=
M∑k1=1
· · ·M∑
kj=1
b(k1,...,kj)MPS
Number of summands grows dramatically!• Application of the back-transformation:
Vmymin =
m∑j=1
ymin,jHj−1bMPS
does not lie in the MPS space.K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 20
Technische Universitat Munchen
Subspace iteration in terms of MPS• Solution: Application of a projection P onto the MPS space:
P :
M∑k=1
y(k)MPS 7→ argminxMPS
∥∥∥∥∥M∑k=1
y(k)MPS − xMPS
∥∥∥∥∥
• Solve the optimization problem
minA
∥∥∥∥∥∥∑
i1,...,iN
tr
(M∑k=1
(B(i1;k)1 · · ·B(iN ;k)
N )−A(i1)1 · · ·A(iN )
N
)ei1...iN
∥∥∥∥∥∥• Two possible projection methods:
• SVD-based truncation• ALS-based optimization
• Use ALS to find optimal matrix pair A(0)r , A
(1)r
• Forcing the derivative to be zero leads to a linear system.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 21
Technische Universitat Munchen
Subspace iteration in terms of MPS• Solution: Application of a projection P onto the MPS space:
P :
M∑k=1
y(k)MPS 7→ argminxMPS
∥∥∥∥∥M∑k=1
y(k)MPS − xMPS
∥∥∥∥∥• Solve the optimization problem
minA
∥∥∥∥∥∥∑
i1,...,iN
tr
(M∑k=1
(B(i1;k)1 · · ·B(iN ;k)
N )−A(i1)1 · · ·A(iN )
N
)ei1...iN
∥∥∥∥∥∥
• Two possible projection methods:• SVD-based truncation• ALS-based optimization
• Use ALS to find optimal matrix pair A(0)r , A
(1)r
• Forcing the derivative to be zero leads to a linear system.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 21
Technische Universitat Munchen
Subspace iteration in terms of MPS• Solution: Application of a projection P onto the MPS space:
P :
M∑k=1
y(k)MPS 7→ argminxMPS
∥∥∥∥∥M∑k=1
y(k)MPS − xMPS
∥∥∥∥∥• Solve the optimization problem
minA
∥∥∥∥∥∥∑
i1,...,iN
tr
(M∑k=1
(B(i1;k)1 · · ·B(iN ;k)
N )−A(i1)1 · · ·A(iN )
N
)ei1...iN
∥∥∥∥∥∥• Two possible projection methods:
• SVD-based truncation• ALS-based optimization
• Use ALS to find optimal matrix pair A(0)r , A
(1)r
• Forcing the derivative to be zero leads to a linear system.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 21
Technische Universitat Munchen
Subspace iteration in terms of MPS• Solution: Application of a projection P onto the MPS space:
P :
M∑k=1
y(k)MPS 7→ argminxMPS
∥∥∥∥∥M∑k=1
y(k)MPS − xMPS
∥∥∥∥∥• Solve the optimization problem
minA
∥∥∥∥∥∥∑
i1,...,iN
tr
(M∑k=1
(B(i1;k)1 · · ·B(iN ;k)
N )−A(i1)1 · · ·A(iN )
N
)ei1...iN
∥∥∥∥∥∥• Two possible projection methods:
• SVD-based truncation• ALS-based optimization
• Use ALS to find optimal matrix pair A(0)r , A
(1)r
• Forcing the derivative to be zero leads to a linear system.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 21
Technische Universitat Munchen
Numerical resultsIsing-type Hamiltonian, N = 10, dim = 2
1 5 10 15 20 25 30 35 40 45 5010
−6
10−4
10−2
100
102
Number of iterations
App
roxi
mat
ion
erro
r
Full vectorMPS, D=1MPS, D=2MPS, D=3MPS, D=4
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 22
Technische Universitat Munchen
Numerical resultsIsing-type Hamiltonian, N = 10, dim = 3
1 5 10 15 20 25 30 35 40 45 5010
−6
10−4
10−2
100
102
Number of iterations
App
roxi
mat
ion
erro
r
Full vectorMPS, D=1MPS, D=2MPS, D=3MPS, D=4
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 23
Technische Universitat Munchen
Numerical resultsIsing-type Hamiltonian, N = 10, dim = 4
5 10 15 20 25 30 35 40 45 50110
−6
10−4
10−2
100
102
Number of iterations
App
roxi
mat
ion
erro
r
Full vectorMPS, D=1MPS, D=2MPS, D=3MPS, D=4
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 24
Technische Universitat Munchen
Numerical resultsIsing-type Hamiltonian, N = 25, dim = 5
1 5 10 1510
−4
10−3
10−2
10−1
100
101
102
Number of iterations
App
roxi
mat
ion
erro
r
MPS, D=2MPS, D=3MPS, D=4MPS, D=5
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 25
Technische Universitat Munchen
Conclusions• Matrix Product States
• data-sparse representation format• represent ground states faithfully• DMRG in terms of MPS
• Subspace methods in terms of MPS• more flexibility (choice of subspace dimension)• approximation not only of the lowest eigenvalue• projection required to keep the number of summands limited• projection can even improve the approximation• convergence to DMRG solution• convergence properties of subspace iteration and
approximation properties of MPS
Thank you for your attention.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 26
Technische Universitat Munchen
Conclusions• Matrix Product States
• data-sparse representation format• represent ground states faithfully• DMRG in terms of MPS
• Subspace methods in terms of MPS• more flexibility (choice of subspace dimension)• approximation not only of the lowest eigenvalue• projection required to keep the number of summands limited• projection can even improve the approximation• convergence to DMRG solution• convergence properties of subspace iteration and
approximation properties of MPS
Thank you for your attention.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 26
Technische Universitat Munchen
Conclusions• Matrix Product States
• data-sparse representation format• represent ground states faithfully• DMRG in terms of MPS
• Subspace methods in terms of MPS• more flexibility (choice of subspace dimension)• approximation not only of the lowest eigenvalue• projection required to keep the number of summands limited• projection can even improve the approximation• convergence to DMRG solution• convergence properties of subspace iteration and
approximation properties of MPS
Thank you for your attention.
K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 26
Technische Universitat Munchen
Conclusions• Matrix Product States
• data-sparse representation format• represent ground states faithfully• DMRG in terms of MPS
• Subspace methods in terms of MPS• more flexibility (choice of subspace dimension)• approximation not only of the lowest eigenvalue• projection required to keep the number of summands limited• projection can even improve the approximation• convergence to DMRG solution• convergence properties of subspace iteration and
approximation properties of MPS
Thank you for your attention.K. Waldherr: Approximation Techniques for Special Eigenvalue Problems
HDA 2011, Bonn, June 2011 26