Transcript ppt - ECM
Tensor networks and the numerical
study of quantum and classical
systems on infinite lattices
Román Orús
School of Physical Sciences,
The University of Queensland, Brisbane (Australia)
in collaboration with Guifré Vidal and Jacob Jordan
Trobada de Nadal 2006 ECM, December 21st 2006
Outline
0.- Introduction
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
1.- Entanglement renormalization of environment degrees of freedom
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
2.- Contraction of infinite 2-dimensional tensor networks
Critical correlators of the classical Ising model
3.- Outlook
Outline
0.- Introduction
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
1.- Entanglement renormalization of environment degrees of freedom
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
2.- Contraction of infinite 2-dimensional tensor networks
Critical correlators of the classical Ising model
3.- Outlook
Introduction
State of a quantum system of n spins 1/2:
c i i ...i i1,i2 ,...,in
i
1 2
n
2 n coefficients (very inneficient to handle classically)
ci1i2 ... in
A natural ansatz for relevant states of quantum mechanical systems is given in terms of the
contraction of an appropriate tensor network:
Hˆ
hˆ
0 c i i ... i1,i2,...
i, j
i
i, j
1 2
Inspires classical techniques to compute properties of quantum systems which are free
from the sign problem, and which can be implemented in the thermodynamic limit
Matrix Product States (MPS)
…
…
Bonds of dimension
Physical local system of dimension d
[Afflek et al., 1987] [Fannes et al., 1992] [White, 1992] [Ostlund and Rommer, 1995] [Vidal, 2003]
Any quantum state can be represented as an MPS, with large enough .
2
n
For finite systems, the state is represented with nd parameters, instead of d .
Physical observables (e.g. correlators) can be computed in O(poly( )) time.
DMRG
Dynamics
Imaginary-time evolution
Thermal states
Master equations
Great in 1 spatial
dimension because of the logarithmic
scaling of the entaglement entropy [Vidal et al., 2003]
Matrix Product Density Operators (MPDO)
[Verstraete, García-Ripoll, Cirac, 2004]
ˆ
c
j1 , j 2 ,...
i1 ,i2 ,... 1 2
i ,i ,... j1, j2 ,...
{i}{ j}
Purification of local dimension
…
…
Bonds of dimension
Physical local system of dimension d
Any density operator can be represented as an MPDO, with large enough and p
2
2n
For finite systems, the state is represented with 2ndp parameters, instead of d .
Physical observables (e.g. correlators) can be computed in O(poly( )poly(p)) time.
thermal states.
Useful in the computation of 1-dimensional
p
Projected Entangled Pair States (PEPS)
…
[Verstraete and Cirac, 2004]
Physical local system of dimension d
…
…
Bonds of dimension D
…
n
4
For finite systems, the state is represented with ndD parameters, instead of d .
Physical observables (e.g. correlators) can be computed in O(poly(D)) time.
a finite system is an #P-Complete problem
Exact contraction of an arbitrary PEPS for
[N. Schuch et al., 2006].
Successfully applied to variationally compute the ground state of finite quantum systems
in 2 spatial dimensions (up to 11 x 11 sites, [Murg, Verstraete and Cirac, 2006]).
Outline
0.- Introduction
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
1.- Entanglement renormalization of environment degrees of freedom
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
2.- Contraction of infinite 2-dimensional tensor networks
Critical correlators of the classical Ising model
3.- Outlook
Outline
0.- Introduction
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
1.- Entanglement renormalization of environment degrees of freedom
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
2.- Contraction of infinite 2-dimensional tensor networks
Critical correlators of the classical Ising model
3.- Outlook
On thermal states in 1 spatial dimension…
ˆ
c
j1 , j 2 ,...
i1 ,i2 ,... 1 2
i ,i ,... j1, j2 ,...
{i}{ j}
MPDO
MPS-like
…
…
OR
…
…
[Zwolak and Vidal, 2004]
Hˆ / 2
ˆ e
Iˆe
Hˆ / 2
Both ansatzs can be applied to compute thermal states.
However, MPDOs can introduce unphysical correlations
between the environment degrees of freedom
environment
“Unnecessary”
entanglement!
swap
1
Disentanglers on the environment of MPDOs
1
swap
Less expensive
representation
U
Disentangler (renormalization of
correlations flowing across the
environment)
1
This effect is not negligible in the computation of thermal states with MPDOs
Quantum Ising spin chain,
Schmidt coefficients
of the MPS-like
representation
20 h 1.1
BIG!!!
Simulating master equations with MPDOs
ˆ
d
ˆ ] iH,
ˆ 2A
ˆ A A A
ˆ A A
ˆ
L[
dt
0
ˆ (t dt) e dtL
L Lr,r 1
r odd
r
(e
…
dtLr,r1
ˆ (t)]
)[
M
M
M
M
M
M
M
M
M
…
r,r1
ˆ (t)M
W
BUT…
M
Proliferation of indices makes “naive”
simulation not feasible
M
ˆ (t)
e dtLr,r1
r even
M
M
M
M
r,r1
M
r,r1
Kraus operators
r,r1
It is possible to introduce
“disentangling isometries”
acting in the environment
subspace that truncate the
proliferation of indices at
each step
Quantum Ising spin chain with amplitude damping,
ˆ (t 0)
h 1.1 0.1
Quantum Ising spin chain with amplitude damping,
ˆ
ˆ ( t 0) e H
20
with and without partial disentanglement
h 1.1 0.1
Outline
0.- Introduction
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
1.- Entanglement renormalization of environment degrees of freedom
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
2.- Contraction of infinite 2-dimensional tensor networks
Critical correlators of the classical Ising model
3.- Outlook
Outline
0.- Introduction
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
1.- Entanglement renormalization of environment degrees of freedom
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
2.- Contraction of infinite 2-dimensional tensor networks
Critical correlators of the classical Ising model
3.- Outlook
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an #P-Complete problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac (2004).
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
D
D
D
d
D
…
…
D
D2
D
…
D
…
D
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an NP-hard problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
…
…
…
…
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an NP-hard problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an NP-hard problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
Boundary MPS with bond
dimension
Action of non-unitary
gates on an infinite
MPS
Can be efficiently computed, taking
care of orthonormalization issues
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an NP-hard problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
Iterate until a fixed point is found
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an NP-hard problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
Iterate until a fixed point is found
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an NP-hard problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
Iterate until a fixed point is found
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an NP-hard problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
Once there is convergence, contract it from the other side
and compute e.g. correlators on the diagonal with the obtained MPS
…
…
The difficult problem of a PEPS…
In order to compute expected values of observables, one must necessarily contract the PEPS
tensor network, and this is an NP-hard problem in general.
For finite systems, there is a variational technique to efficiently approximate such a
contraction (up to lattices of 11 x 11 spins) due to Verstraete and Cirac.
We have developed a technique to contract the whole PEPS tensor network in the
thermodynamic limit for translationally-invariant systems.
Once there is convergence, contract it from the other side
and compute e.g. correlators on the diagonal with the obtained MPS
D
D
D
D
D
D
z
D
r
z
D
r
z
An example: classical Ising model at criticality
H i j
i, j
1
C ln 1 2
2
i ir
C
1
r
1
4
It is possible to build a quantum PEPS such that the expected values correspond to those of
the classical ensemble
z z
ˆ i
ˆ ir i ir
C(r)
C 0.1
log C(r)
exact
Very good agreement up to ~100 sites
with modest computational effort!
30
log( r)
20
Outline
0.- Introduction
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
1.- Entanglement renormalization of environment degrees of freedom
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
2.- Contraction of infinite 2-dimensional tensor networks
Critical correlators of the classical Ising model
3.- Outlook
Outline
0.- Introduction
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
1.- Entanglement renormalization of environment degrees of freedom
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
2.- Contraction of infinite 2-dimensional tensor networks
Critical correlators of the classical Ising model
3.- Outlook
Outlook
Question: why tensor networks are good for you?
Answer: because, potentially, you can apply them to study…
Soon application to compute the ground state
properties and dynamics of infinite quantum manybody
systems
in 2 spatial
dimensions
strongly-correlated quantum
many-body
systems
in 1, 2, and
more spatial dimensions, in the finite
case and in the thermodynamic limit, Hubbard models, high-Tc superconductivity, frustrated lattices,
in collaboration
with G.
Vidal, J.systems
Jordan,away
F. Verstraete
and I. Cirac
topological effects,
finite-temperature
systems,
from equilibrium,
master equations
and dissipative systems, classical statistical models, quantum field theories on infinite lattices, at
finite temperature and away from equilibrium, effects of boundary conditions, RG transformations,
computational complexity of physical systems, etc