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![Page 1: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/1.jpg)
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
![Page 2: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/2.jpg)
Outline
0.- Introduction
1.- Entanglement renormalization of environment degrees of freedom
2.- Contraction of infinite 2-dimensional tensor networks
3.- Outlook
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
Critical correlators of the classical Ising model
![Page 3: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/3.jpg)
Outline
0.- Introduction
1.- Entanglement renormalization of environment degrees of freedom
2.- Contraction of infinite 2-dimensional tensor networks
3.- Outlook
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
Critical correlators of the classical Ising model
![Page 4: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/4.jpg)
€
ψ = c i1i2 ...ini{ }
∑ i1,i2,...,in
n2
State of a quantum system of n spins 1/2:
coefficients (very inneficient to handle classically) niiic ...21
Introduction
A natural ansatz for relevant states of quantum mechanical systems is given in terms of the contraction of an appropriate tensor network:
€
ˆ H = ˆ h i, j
<i, j>
∑
€
ψ0 = c i1i2 ...
i{ }
∑ i1,i2,...
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
![Page 5: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/5.jpg)
Matrix Product States (MPS)
[Afflek et al., 1987] [Fannes et al., 1992] [White, 1992] [Ostlund and Rommer, 1995] [Vidal, 2003]
Physical local system of dimension Bonds of dimension χ
€
d
For finite systems, the state is represented with parameters, instead of .
€
ndχ 2
€
dn
Any quantum state can be represented as an MPS, with large enough .
Physical observables (e.g. correlators) can be computed in time.
€
O( poly(χ ))€
χ
Great in 1 spatial dimension because of the logarithmic scaling of the entaglement entropy [Vidal et al., 2003]
DMRGDynamics
Imaginary-time evolutionThermal states
Master equations
… …
![Page 6: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/6.jpg)
Matrix Product Density Operators (MPDO)
€
ˆ ρ = c i1 ,i2 ,...j1 , j2 ,... i1,i2,... j1, j2,...
{i}{ j}
∑
Physical local system of dimension Bonds of dimension χ
€
d
… …Purification of local dimension
€
p
For finite systems, the state is represented with parameters, instead of .
€
2ndpχ 2
€
d2n
Any density operator can be represented as an MPDO, with large enough and
Physical observables (e.g. correlators) can be computed in time.
€
O( poly(χ )poly( p))€
χ
€
p
Useful in the computation of 1-dimensional thermal states.
[Verstraete, García-Ripoll, Cirac, 2004]
![Page 7: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/7.jpg)
Projected Entangled Pair States (PEPS)
Physical local system of dimension
€
d
Bonds of dimension
€
D
For finite systems, the state is represented with parameters, instead of .
€
ndD4
€
dn
Physical observables (e.g. correlators) can be computed in time.
€
O( poly(D))
Exact contraction of an arbitrary PEPS for a finite system is an #P-Complete problem [N. Schuch et al., 2006].
Successfully applied to variationally compute the ground state of finite quantum systemsin 2 spatial dimensions (up to 11 x 11 sites, [Murg, Verstraete and Cirac, 2006]).
… …
……
[Verstraete and Cirac, 2004]
![Page 8: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/8.jpg)
Outline
0.- Introduction
1.- Entanglement renormalization of environment degrees of freedom
2.- Contraction of infinite 2-dimensional tensor networks
3.- Outlook
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
Critical correlators of the classical Ising model
![Page 9: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/9.jpg)
Outline
0.- Introduction
1.- Entanglement renormalization of environment degrees of freedom
2.- Contraction of infinite 2-dimensional tensor networks
3.- Outlook
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
Critical correlators of the classical Ising model
![Page 10: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/10.jpg)
On thermal states in 1 spatial dimension…
€
ˆ ρ = c i1 ,i2 ,...j1 , j2 ,... i1,i2,... j1, j2,...
{i}{ j}
∑
OR … …
MPDO
€
ˆ ρ ∝ e−β ˆ H / 2 ˆ I e−β ˆ H / 2Both ansatzs can be applied to compute thermal states. However, MPDOs can introduce unphysical correlations between the environment degrees of freedom
environment
swap
€
χ >1
“Unnecessary”entanglement!
… …
MPS-like
[Zwolak and Vidal, 2004]
![Page 11: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/11.jpg)
Disentanglers on the environment of MPDOs
swap
€
χ >1
U
Disentangler (renormalization of correlations flowing across the environment)
This effect is not negligible in the computation of thermal states with MPDOs
€
χ =1
Less expensive representation
![Page 12: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/12.jpg)
Quantum Ising spin chain,
€
β =20
€
h =1.1
Schmidt coefficientsof the MPS-like representation
BIG!!!
![Page 13: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/13.jpg)
Simulating master equations with MPDOs
€
d ˆ ρ
dt= L[ ˆ ρ ] = −i H, ˆ ρ [ ] + 2Aμ
ˆ ρ Aμ+ − Aμ Aμ
+ ˆ ρ − Aμ+Aμ
ˆ ρ ( )μ >0
∑
€
L = Lr,r+1
r
∑
€
ˆ ρ (t + dt) ≅ ⊗r odd
edtLr ,r+1
( ) ⊗r even
edtLr ,r+1
( ) ˆ ρ (t)
€
(edtLr ,r+1 )[ ˆ ρ (t)] = Mμ r ,r+1ˆ ρ (t)
μ r ,r+1
∑ Mμ r ,r+1
+Kraus operators
W
M
It is possible to introduce “disentangling isometries” acting in the environment
subspace that truncate the proliferation of indices at
each step
BUT…M
M
M
M
M
M
M
M
M
M
M
M
M
M
Proliferation of indices makes “naive” simulation not feasible
… …
![Page 14: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/14.jpg)
Quantum Ising spin chain with amplitude damping,
€
h =1.1
€
γ=0.1
€
ˆ ρ (t = 0) = + +( )∞⊗
![Page 15: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/15.jpg)
Quantum Ising spin chain with amplitude damping,
€
h =1.1
€
γ=0.1
€
ˆ ρ (t = 0) ∝ e−β ˆ H
€
β =20 with and without partial disentanglement
![Page 16: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/16.jpg)
Outline
0.- Introduction
1.- Entanglement renormalization of environment degrees of freedom
2.- Contraction of infinite 2-dimensional tensor networks
3.- Outlook
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
Critical correlators of the classical Ising model
![Page 17: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/17.jpg)
Outline
0.- Introduction
1.- Entanglement renormalization of environment degrees of freedom
2.- Contraction of infinite 2-dimensional tensor networks
3.- Outlook
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
Critical correlators of the classical Ising model
![Page 18: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/18.jpg)
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
€
D
€
D
€
D
… …
……
€
D2
![Page 19: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/19.jpg)
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.
![Page 20: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/20.jpg)
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.
![Page 21: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/21.jpg)
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
€
χ
![Page 22: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/22.jpg)
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
![Page 23: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/23.jpg)
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
![Page 24: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/24.jpg)
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
![Page 25: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/25.jpg)
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
… …
![Page 26: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/26.jpg)
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
r
€
σ z
€
σ z
r
€
D
€
D
€
D
€
D
€
D
€
D
€
D
€
D
€
σ z
![Page 27: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/27.jpg)
An example: classical Ising model at criticality
€
H = − σ i
<i, j>
∑ σ j
€
βC =1
2ln 1+ 2( )
€
σ iσ i+r β C≈
1
r1
4
It is possible to build a quantum PEPS such that the expected values correspond to those of
the classical ensemble
€
C(r) = ψ βˆ σ i
z ˆ σ i+rz ψ β = σ iσ i+r β
€
β =βC − 0.1
€
χ =20€
χ =30
exact
Very good agreement up to ~100 sites with modest computational effort!€
logC(r)
€
log(r)
![Page 28: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/28.jpg)
Outline
0.- Introduction
1.- Entanglement renormalization of environment degrees of freedom
2.- Contraction of infinite 2-dimensional tensor networks
3.- Outlook
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
Critical correlators of the classical Ising model
![Page 29: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/29.jpg)
Outline
0.- Introduction
1.- Entanglement renormalization of environment degrees of freedom
2.- Contraction of infinite 2-dimensional tensor networks
3.- Outlook
Matrix Product States (MPS)
Matrix Product Density Operators (MPDO)
Projected Entangled Pair States (PEPS)
Infinite 1-dimensional thermal states and disentanglers
Infinite 1-dimensional master equations
Critical correlators of the classical Ising model
![Page 30: Tensor networks and the numerical study of quantum and classical systems on infinite lattices Román Orús School of Physical Sciences, The University of.](https://reader035.fdocuments.in/reader035/viewer/2022062803/56649f385503460f94c5481d/html5/thumbnails/30.jpg)
Outlook
Question: why tensor networks are good for you?
Answer: because, potentially, you can apply them to study…
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, topological effects, finite-temperature systems, systems away 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
Soon application to compute the ground state properties and dynamics of infinite quantum many-
body systems in 2 spatial dimensions
in collaboration with G. Vidal, J. Jordan, F. Verstraete and I. Cirac