**Syllabus and program for the 2022 PHYS
500.001 Adv. Sem. on “Tensor Network Methods”**

**Session 1 (June 1 ^{st}): Course overview and
introduction to tensor networks**

Course overview and intro to code-along track. Introduction
to tensor networks and tensor network notation. Singular value decomposition.
Examples of the applications of tensor networks in physics and in quantum
information.

Reference: [Bridgeman2017, chapters 0-2]

**Session 2 (June 8 ^{th}): Matrix product states (MPS)**

MPS motivation and structure. Normalization, canonical forms,
and gauge degrees of freedom. MPS decomposition algorithm. Open vs periodic
boundary conditions. MPS notation. Examples of MPS.

Main references: [Schollwöck2013, chapter 16.2],
[Bridgeman2017, chapter 3], and [Vidal2003]. See also [Schollwöck2011, chapter
4] and [Perez-Garcia2007].

**Session 3 (June 15 ^{th}): Matrix product operators
(MPOs) and Hamiltonians**

MPO structure. Application of MPOs to MPS. Bond contraction
order. MPO decomposition algorithm. MPS overlaps. Adding MPOs to MPOs. Ground
state search.

References: [Schollwöck2013, chapter 16.3, 16.6, 16.9],
[Schollwöck2011, chapter 5, 6], and [Perez-Garcia2007].

**Session 4 (June 22 ^{nd}): MPS bond dimensions, area
law, and bond dimension compression**

Bond dimension in MPS – relation with entanglement. Area vs
volume laws. MPS correlation lengths. SVD compression. Variational compression.
Bond dimension SVD compression algorithm.

References: [Schollwöck2013, chapter 16.4], [Schollwöck2011,
chapter 4.5], [Paeckel2019, chapter 2.6], and [Eisert2010].

**Session 5 (June 29 ^{th}): Time evolution of MPS**

Time-dependent MPS. Time-evolving block decimation.
Time-dependent variational principle. Schrödinger picture vs Heisenberg picture
(light cones vs contraction). Ground state search using imaginary time
evolution.

Main references: [Schollwöck2013, chapters 16.5, 16.8],
[Schollwöck2011, chapters 7, 8.1], and [Paeckel2019]. See also [Daley2004].

**Session 6 (July 6 ^{th}): Open quantum system dynamics
using tensor networks**

Simulation methods for open quantum systems. Quantum
trajectories for MPS. Lindblad master equation for MPOs. Matrix product density
operators. Measurement-induced
entanglement phase transitions.

Main references: [Weimer2021, sections I-III], [Cheng2021],
and [Czischek2021]. See also [Werner2015], [Choi2020], and [Skinner2019].

**Session 7 (July 13 ^{th}): The Density Matrix
Renormalization Group (DMRG)**

The idea behind DMRG. Infinite DMRG (iDMRG). Finite DMRG.
DMRG for non-equilibrium systems and time-dependent DMRG. DMRG vs
imaginary-time MPS ground state search. Applications of DMRG.

Main references: [Schollwöck2011, chapters 2, 3, 9, 10] and
[Schollwöck2005]. See also [De Chiara2009] and [Hallberg2006].

**Session 8 (July 20 ^{th}): Projected Entangled Pair
States (PEPS)**

PEPS construction in 1D. Extension to higher dimensions.
Properties of PEPS and examples. Injective PEPS and parent Hamiltonians.
Comparison with MPS.

Main references: [Bridgeman2017, chapters 3.1, 6],
[Verstraete2006], and [Verstraete2008, chapter 6]. See also [Schuch2010].

**Session 9 (July 27 ^{th}): Machine learning using
tensor networks**

Restricted Boltzmann machine and tensor network states correspondence.
Implications of correspondence. Machine learning topological states, spatial
geometry, quantum state tomography.

Main references: [Chen2018], [Deng2017], [Torlai2018], and
[You2018]. See also [Gao2017], [Glasser2018], and [Huang2021].

**Session 10 (August 3 ^{rd}): Quantum error correction
using tensor networks**

Application of tensor network methods in quantum error
correction. Maximum likelihood decoding in surface codes using MPS.

Main reference: [Bravyi2014]. See also [Pastawski2015].

**The following two electives were not
included in the ten course sessions and have been left here as reference for
the curious reader:**

**Elective 2: Classification of gapped phases in 1D**

Quantum phases. Injective MPS. Parent Hamiltonian.
Classification of phases in 1D using MPS. Classification of phases in 2D using
PEPS.

Main references: [Bridgeman2017, chapter 4] and [Schuch2011].
See also [Chen2011].

**Elective 4: Multiscale Entanglement Renormalization Ansatz
(MERA)**

MERA state construction. Properties of MERA. Application to
ground states of gapless Hamiltonians. Comparison with MPS, PEPS.

References: [Bridgeman2017, chapter 7] and [Vidal2010].

**Suggested supplementary resources: **

[TensorNetwork.org] Wiki-style
website focused on tensor networks. A good resource that also provides
references!

[Schollwöck2021] Ulrich Schollwöck’s online lecture on MPS.

[Cirac2021] Rev. Mod. Phys. Paper on MPS and PEPS by Cirac,
Perez-Garcia, Schuch, and Verstraete.

[Schuch2014] Talk by Schuch on topological order in PEPS.

[Biamonte2020] An exhaustive look at tensor network methods.

[Bañuls2022] A short but detailed review of tensor network
methods.

**Bibliography:** (all references have also been
uploaded and sorted by topic in
this OneDrive folder)

[Bañuls2022] Bañuls, “Tensor Network Algorithms: a Route
Map”. arXiv:2205.10345. (link)

[Biamonte2020] Biamonte, “Lectures on Quantum Tensor Networks
– a pathway to modern diagrammatic reasoning”. arXiv:1912.10049v2. (link)

[Bridgeman2017] Bridgeman and Chubb, “Hand-waving and
Interpretive Dance: An Introductory Course on Tensor Networks”.
arXiv:1603.03039. (link)

[Chen2011] Chen et al., “Complete classification of
one-dimensional gapped quantum phases in interacting spin systems”. Phys. Rev.
B **84**, 235128 (2011). (link)

[Chen2018] Chen et al., “Equivalence of restricted Boltzmann
machines and tensor network states”. Phys. Rev. B **97**, 085104 (2018). (link)

[Cheng2021] Cheng et al., “Simulating noisy quantum circuits
with matrix product density operators”. Phys. Rev. Research **3**, 023005
(2021). (link)

[Choi2020] Choi et al., “Quantum Error Correction in
Scrambling Dynamics and Measurement-Induced Phase Transition”. Phys. Rev. Lett.
125, 030505 (2020). (link)

[Cirac2021] Cirac et al., “Matrix product states and
projected entangled pair states: Concepts, symmetries, theorems”. Rev. Mod.
Phys. **93**, 045003 (2021). (link)

[Czischek2021] Czischek et al., “Simulating a measurement-induced phase transition for trapped ion
circuits”. Phys. Rev. A **104**, 062405 (2021). (link)

[Daley2004] Daley et al., “Time-dependent density-matrix
renormalization-group using adaptive effective Hilbert spaces”. J. Stat. Mech
(2004) P04005. (link)

[De Chiara2009] De Chiara et al., “Density Matrix
Renormalization Group for Dummies”. J. Comput. Theor. Nanosci. 5, 1277-1288
(2008). (arXiv link)

[Deng2017] Deng et al., “Machine learning topological
states”. Phys. Rev. B **96**, 195145 (2017). (link)

[Eisert2010] Eisert et al., “Area laws for the entanglement
entropy”. Rev. Mod. Phys. **82**, 277 (2010). (link)

[Gao2017] Gao and Duan, “Efficient representation of quantum
many-body states with deep neural networks”. Nat. Commun. **8**, 662 (2017).
(link)

[Glasser2018] Glasser et al., “Neural-Network Quantum States,
String-Bond States, and Chiral Topological States”. Phys. Rev. X **8**,
011006 (2018). (link)

[Hallberg2006] Hallberg, “New trends in density matrix
renormalization”. Advances in Physics **55**, 477 (2006). (link, arXiv link)

[Huang2021] Huang and Moore, “Neural Network Representation
of Tensor Network and Chiral States”. Phys. Rev. Lett. **127**, 170601
(2021). (link)

[Paeckel2019] Paeckel et al., “Time-evolution methods for
matrix-product states”. Annals of Physics **411**, 167998 (2019). (link)

[Pastawski2015] Pastawski et al., “Holographic quantum
error-correcting codes: toy models for the bulk/boundary correspondence”. J.
High Energ. Phys. **2015**, 149 (2015). (link)

[Perez-Garcia2007] Perez-Garcia, Verstraete, Wolf, and Cirac,
“Matrix product state representations”. Quantum Inf. Comput. **7**, 401
(2007). (arXiv link)

[Schollwöck2005] Schollwöck, “The density-matrix
renormalization group”. Rev. Mod. Phys. **77**, 259 (2005). (link)

[Schollwöck2011] Schollwöck, “The density-matrix
renormalization group in the age of matrix product states”. arXiv:1008.3477v2 (link)

[Schollwöck2013] Pavarini, Koch, and Schollwöck, “Lecture
Notes of the Autumn School Correlated Electrons 2013”, chapter 16. (link
– **note that we only want chapter 16**)

[Schollwöck2021] Schollwöck, “Introduction to MPS”. Recorded
talk at the International Symposium on Correlated Electrons 2021. (link)

[Schuch2010] Schuch et al., “PEPS as ground states:
Degeneracy and topology”. Annals of Physics **325**, 2153 (2010). (link)

[Schuch2011] Schuch et al., “Classifying quantum phases using
matrix product states and projected entangled pair states”. Phys. Rev. B **84**,
165139 (2011). (link)

[Schuch2014] Schuch, “Topological Order in Projected
Entangled Pair States”. Recorded talk at the Simons Institute for the Theory of
Computing. (link, alternate link)

[Skinner2019] Skinner et al., “Measurement-Induced Phase
Transitions in the Dynamics of Entanglement”. Phys. Rev. X **9**, 031009
(2019). (link)

[Torlai2018] Torlai et al., “Neural-network quantum state
tomography”. Nature Phys. **14**, 447 (2018). (link)

[Verstraete2006] Verstraete et al., “Criticality, the Area
Law, and the Computational Power of Projected Entangled Pair States”. Phys.
Rev. Lett. **96**, 220601 (2006). (link)

[Verstraete2008] Verstraete et al., “Matrix Product States,
Projected Entangled Pair States, and variational renormalization group methods
for quantum spin systems”. Adv. Phys. **57**, 143 (2008). (arXiv link)

[Vidal2003] Vidal, “Efficient Classical Simulation of
Slightly Entangled Quantum Computations”. Phys. Rev. Lett. **91**, 147902
(2003). (link)

[Vidal2010] Vidal, “Entanglement Renormalization: an
introduction”. arXiv:0912.1651v2 (link)

[Weimer2021] Weimer et al., “Simulation methods for open
quantum many-body systems”. Rev. Mod. Phys. **93**, 015008 (2021). (link)

[Werner2015] Werner et al., “A positive tensor network
approach for simulating open quantum many-body systems”. Phys. Rev. Lett. **116**,
237201 (2016). (link)

[You2018] You et al., “Machine learning spatial geometry from
entanglement features”. Phys. Rev. B **97**, 045153 (2018). (link)