Lectures on Tensor Network States

By Jacob Biamonte (CQT Singapore, Oxford University)

QIC 890/891 Selected Advanced Topics in Quantum Information, The University of Waterloo, Waterloo Ontario, Canada, (2011).
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  • Direct link to this page: http://www.qubit.org/iqc2011
  • Jacob Biamonte Homepage: http://qubit.org/jacob-biamonte
  • My office at IQC is 1011 RAC1 

Synopsis

This lecture series is part of QIC 890/891 held at the University of Waterloo, Spring 2011 and organized by Michele Mosca. We will present our variant of the graphical notation of Penrose, tailored to describe tensor network states, operators and processes, and as a uniting tool for the notation and language of quantum theory, information science and condensed matter physics.

Topics covered: Tensor networks applied to representations of algebras and groups, symmetries, symmetric states, polynomial or algebraic invariants. Hamiltonian models of quantum computation, entanglement spectrum, stabiliser and match gate tensors. Penrose graphical notation and Penrose wire bending duality, tensor networks capturing: map-state duality, channel-state duality and the Choi-Jamiołkowski isomorphism via string diagrams. Factorisation of quantum states into tensor networks.  Extending the quantum circuit model.  Matrix Product States (MPS), Projected Entangled Pair States (PEPS), Multiscale Entanglement Renormalisation Ansatz (MERA), Categorical Tensor Network States (CTNS), Boolean Tensor Network States (BTNS). 

                                 Penrose graphical notation for quantum states

Location and Times

  • Dates: July 5, 7, 12 and 14;  Time: Tuesdays and Thursday, 1:00pm - 2:20pm; Location: RAC1 2009 (IQC main building)
  • Recitations: Friday July 8th and 15th, 1:00 pm in RAC1 2009. 

Lecture Notes

  • Notes from Lecture I, presented July 5th, 2011.  PDF V 1.3
  • Notes from Lecture II, presented July 7th, 2011.  PDF V 1.0
  • Notes from Lecture III, presented July 12th, 2011.  PDF V 1.0
  • Notes from Lecture IV, presented July 14th, 2011.  PDF V 1.0

Lectures on IQC's Youtube Channel  

 

 

  • Credits: lectures filmed and edited by Peter J. Kovacs and Craig Hennessey, trailer and interview orchestrated by Colin Hunter.

Assignments  

  • Assignment 0.  (Warm up) PDF (Problem II in particular is assumed knowledge during the lectures)
  • Assignment I. (Reading assignment)  We will read segments of the 1971 article by Roger Penrose.  Pages 1-7, 11 and 14 and 15.  (pp 221-227, 231, 234 and 235).  
  • These background slides summarize parts of this paper by Penrose. 
  • Assignment II. (Short assignment) PDF Due Thursday, July 7th at the start of class
  • Assignment III. (HWK problem set) PDF Due Tuesday, July 12th at the start of class
  • Assignment IV. (Short assignment) Exercise 62, 66, 67, 69 and 79 from notes III.  Due Thursday, July 14th at the start of class. 
  • Additional problems found in the notes. 
  • email me for solutions! 

Compulsory Background Reading

The graphical language we will use throughout this course and the first tensor network algorithm (for the edge colouring problem, see Assignment III) dates back to Roger Penrose.

  • Reading assignment I. Pages 1-7, 11 and 14 and 15.  (pp 221-227, 231, 234 and 235)

Penrose's work was taken on board by several people.  The earliest and most relevant work to our interests I have found is by Lafont [2, 8, 9].  The following is not assigned reading, but assigned viewing.  It is a good idea to thumb through his paper and get an idea for the sorts of things that are possible in the diagrammatic language.  (Note the link below should let you download his paper).  Lafont's theory was extended, connected with other work and adapted to quantum circuits (in d-dimensions) and tensor networks in [4, 7]. 

  • Reading assignment I-a: Lafont Pages 16 (notice the braid maps S1, S2), Page 18; Page 19, (notice G1 and G2); Page 20 and Page 24.

Optional Resources and Links

Many people find that doing numeric simulations of quantum states, algorithms and process is a great way to learn the modern quantum theory.  The following software package by Ville Bergholm is a popular learning and research tool in the field. 

  • Quantum Information Toolkit http://qit.sourceforge.net/ "Quantum Information Toolkit (QIT) is a free, open source numerical toolkit for various quantum information and computing -related purposes, distributed under GPL. It is available for both MATLAB (version 7.6 or newer) and Python 2.6."
  • Robert Pfeifer who works at PI has recently done a series of lectures entitled, Introduction to Tensor Network Algorithms.  The first lecture covers the so called Matrix Product States (MPS). http://pirsa.org/11060004/
  • Connection to Quantum Gravity.  For those who want to see the general history of how some of the ideas we are using took shape over the years, and also how tensor networks are being used to address problems in quantum gravity, the following paper is a great read.  In fact, the paper even presents a nice overview of some aspects of the work by Penrose mentioned above.
  • John C. Baez and Aaron Lauda, A Prehistory of n-Categorical Physics, preprint 0908.2469, (2009).
  • Related past course by Jacob Biamonte, Stephen Clark, Mark Williamson, and Vlatko Vedral. The Quantum Theory of Information and Computation. Oxford Graduate Course, TT2010.
  • Penrose Graphical Notation for Tensor Network States, talk given at Perimeter Institute for Theoretical Physics Seminar Archive (PIRSA), (2011).  [slides in printer friendly form]

         http://pirsa.org/11080059/ (multiple media formats)

          Click to play

Errata

  • Lecture I.  V1.0: The figure on page 4 illustrating the possible ways to reshape an order-two tensor did not include the transpose of a map in V 1.0.  Changed in V 1.1 at 22:30 July 5th, 2011.  Thanks to Abel Molina for pointing this out.  Please draw the missing addition to the figure if you already printed out version 1.0. V1.2 Corrects a few minor typos in V1.1.   

Credits

It is my pleasure to acknowledge great conversations and meaningful collaborations with several researchers which have influenced the course material.  I work with some of them at the Centre for Quantum Technologies.  In alphabetical order, a partial list includes

Suggested Additional Reading

The following list of papers (in no particular order) influenced the lecture content and/or are papers which I think others will find interesting.  More citations can be found in the lecture notes.  The list here is designed to have easy access by providing links, when possible. 

  1. Roger Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications, Academic Press (1971).
  2. Y. Lafont, Penrose diagrams and 2-dimensional rewriting, Applications of Categories in Computer Science, London Mathematical Society Lecture Note Series 177, p. 191-201, Cambridge University Press (1992).
  3. Jacob Biamonte, Lectures on Tensor Network States, QIC 890/891 Selected Advanced Topics in Quantum Information, The University of Waterloo, Waterloo Ontario, Canada, (2011), http://www.qubit.org/iqc2011.
  4. Ville Bergholm and Jacob Biamonte, Categorical Quantum Circuits, Journal of Physics A: Mathematical and Theoretical, Vol. 44, No. 17, pages 25304, (2011), http://arxiv.org/abs/1010.4840.
  5. S. J. Denny, J. D. Biamonte, D. Jaksch and S. R. Clark, Algebraically contractible topological tensor network states, Journal of Physics A: Mathematical and Theoretical, (2011), http://arxiv.org/abs/1108.0888.
  6. Roger Penrose, The theory of quantized directions, unpublished (1967).
  7. J. D. Biamonte, S. R. Clark, and D. Jaksch, Categorical tensor network states, to appear AIP Advances (2011), http://arxiv.org/abs/1012.0531.
  8. Y. Lafont, Towards an Algebraic Theory of Boolean Circuits, Journal of Pure and Applied Algebra 184 (2-3), p. 257-310, Elsevier (2003). (link)
  9. Y. Lafont, Equational reasoning with 2-dimensional diagrams, in Term Rewriting, Lecture Notes in Computer Science 909, p. 170-195, Springer-Verlag (1995). (link)
  10. F. Verstraete, V. Murg, and J. I. Cirac. Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. Advances in Physics, 57(2):143–224, 2008.http://arxiv.org/abs/0907.2796
  11. J. I. Cirac and F. Verstraete. Renormalization and tensor product states in spin chainsand lattices. J. Phys. A Math. Gen., 42:4004, 2009. http://arxiv.org/abs/0910.1130
  12. Samson Abramsky, Temperley-Lieb algebra: from knot theory to logic and computation via quantum mechanics, Mathematics of Quantum Computing and Technology, Taylor and Francis, 515–558 (2007). [link]
  13. G. Evenbly and G. Vidal, Tensor network states and geometry, ArXiv e-prints, arXiv:1106.1082, (2011). http://arxiv.org/abs/1106.1082
  14. Joachim Kock. Frobenius algebras and 2-d topological quantum field theories. Cambridge University Press, (2003). (well written introduction to modern algebra)
  15. Ross Duncan and Simon Perdrix, Rewriting Measurement-Based Quantum Computations with Generalised Flow,
    Lecture Notes in Computer Science, 6199/2010:285-296, (2010).
  16. John C. Baez and Aaron Lauda, A Prehistory of n-Categorical Physics, preprint 0908.2469, (2009).
  17. John Baez and Jacob Biamonte, Baez-Biamonte Network Theory Project, nLab (2011).
  18. T. H. Johnson, S. R. Clark, D. Jaksch, Dynamical simulations of classical stochastic systems using matrix product states, Phys. Rev. E 82, 036702 (2010). http://arxiv.org/abs/1006.2639
  19. Roger Penrose, The Road to Reality, Jonathan Cape Ltd, (2004).
  20. T. H. Johnson, J. D. Biamonte, S. R. Clark and D. Jaksch, Verifier Tensor Networks, preprint, (2011).
  21. Samson Abramsky, Bob Coecke, Categorical quantum mechanics, Chapter in the Handbook of Quantum Logic and Quantum Structures vol II, Elsevier, (2008).  [link]
  22. P Selinger, A survey of graphical languages for monoidal categories, arXiv:0908.3347 (2009).
  23. Markus Grassl, Describing Entanglement Using Invariant Theory, IQOQI Group Seminar, (2007). (transparencies).
  24. Markus Grassl, Entanglement and Invariant Theory, Quantum Computation and Information Seminar, UC Berkeley, (2002), (transparencies).

How to Draw Penrose Tensor Diagrams

The tensor network diagrams in the course notes were created in SVG format using Inkscape which is a standard vector graphics software package, chosen because it is free on Linux.  Any other standard graphics package should work fine to create tensor diagrams like the ones in the notes. 

  • ZIP file containing SVG source for all of the figures from the lecture notes.  (tensor-diagrams.ZIP)

 

 

Markus Grassl,
"Describing Entanglement Using Invariant Theory",
IQOQI Group Seminar, 17.10.2007.
Transparencies of the talk.