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


Session 1 (June 1st): 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 8th): 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 15th): 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 22nd): 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 29th): 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 6th): 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 13th): 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 20th): 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 27th): 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 3rd): 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:

[] 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. (linknote 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)