Advanced Quantum Information Theory course

23 June 2017 Lectured from 2017 as an optional advanced theory course for the UCL quantum CDT.

Based on a course I originally lectured from 2013 to 2015 as a Part III Mathematics course at the University of Cambridge.

Lecture notes

  • Notation and terminology
  • Bibliography

Section 1: Hamiltonian Complexity

  • Lecture 1: Whistle-stop introduction to computability and complexity theory
  • Lectures 2: Local Hamiltonians
  • Lectures 3: Kitaev's Theorem
  • Lecture 4: Local clock construction

Section 2: Lieb-Robinson techniques

  • Lecture 5: Many-body quantum physics introduction
  • Lecture 6: Lieb-Robinson bounds
  • Lectures 7-8: Exponential decay of correlations

Quantum Computation and Complexity course

15 July 2016 Lectured at the 2016 Autrans summer school on Stochastic Methods in Quantum Mechanics. The notes are adapted from the first half my Advanced Quantum Information Theory course, with additional material on the basics of computation and complexity theory.

Lecture notes

  • Lectures 1-2: Computation and Complexity
  • Lecture 3: Local Hamiltonians
  • Lectures 3-4: Kitaev's Theorem
  • Lecture 4: Local clock construction

Recommended reading

The Arora-Barak book gives an excellent, modern treatment of the theory of computation and complexity, going far beyond what's covered i

Matrix Product States and PEPS

14 July 2016 Notes from David Perez-Garcia's lecture course on Matrix Product States and PEPS at the 2016 Autrans summer school on Stochastic Methods in Quantum Mechanics.

The slides are courtesy of David. The lecture notes are my handwritten notes from the whiteboard section of David's lectures. All content by David; all mistakes by me!

Lecture notes

  • MPS motivation (slides)
  • MPS lecture notes (handwritten)
  • PEPS and topological order (slides)

(The slides are copyright © 2016 David Perez-Garcia, with all rights reserved. The handwritten notes are copyright ©{

Decoupling Method in Quantum Shannon Theory

19 October 2015 Originally lectured in 2015 as part of the quantum information theory masters course for the UCL quantum CDT.

Lecture Notes

  • Decoupling Method

Recommended reading

Much of the material covered here (and more!) was originally proven in the Mother of All Protocols paper by Abeyesinghe, Devetak, Hayden and Winter.

These notes largely follow Section 10.9 of Preskill's wonderful lecture notes, with a (very) few modifications and additions.

Quantum Mechanics for Mathematicians course

4 November 2011 Lectured in 2011 as the first section of a "Mathematics for Quantum Information" masters course given in the mathematics faculty of the Universidad Complutense de Madrid.

Lecture Notes

  • Section 0: Dirac notation;
    Section 1: The postulates of quantum mechanics
    (lecture 1)
  • Section 2: Combining quantum systems: tensor products
    (lecture 2)
  • Section 3: Non-locality and Bell inequalities
    (lecture 3)
  • Section 4: Ensembles and density operators;
    Section 5: Taking quantum systems apart: reduced states and the partial trace;
    Section 6: A brief introduction to entropy
    (lecture 4)

Quantum Mechanics course

11 May 2010 Lectured from 2007 to 2010 as the second part of the 3rd year mathematics undergraduate "Quantum Mechanics" course at the University of Bristol.

Lecture Notes

  • Section 1: Angular Momentum and Spin
    (lectures 1 and 2)
  • Section 2: Representations of Angular Momentum
    (lectures 3 to 5)
  • Section 3: Orbital Anglular Momentum
    (bonus lecture)
  • Section 4: Measurement
    (lecture 6)
  • Section 5: Multiple Particles and Tensor Products
    (lectures 7 and 8)
  • Section 6: Non-Locality and Bell Inequalities
    (lectures 9 and 1

Classical mechanics and electrodynamics

14 May 2004 I have left up some of the material I prepared for classical mechanics and electrodynamics courses taught by Prof. Weise at the TUM (many, many years ago!) in case it's of use to someone.

Question Sheet Solutions

Given that the question sheets are substantially re-used in subsequent semesters, I've removed the worked solutions that were available here, to help you avoid the temptation to…ahem…short-cut the valuable learning process that struggling to solve the questions provides. (Believe it or not, the question sheets are not some obscure form of torture dreamed up by

Research interests

General interests

  • CP maps (a.k.a. quantum channels)
  • Quantum information theory
  • Many-body physics
  • Complexity theory
  • Hamiltonian complexity
  • Entanglement theory
  • Probability theory
  • Algebraic geometry
  • Learning any other interesting new maths I come across…

That'll do for now.


You can find a (possibly not-quite-up-to-date) list of my publications on this web site, including copies of the papers, as well as the slides from some of my talks. For a more up-to-date list, try the arXiv.


You can also find all of my papers on the arXiv (which is sometimes more up-to-date than this list).

Published Papers

  1. The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension Johannes Bausch, Toby Cubitt and Maris Ozols Annales Henri Poincaré (to appear) [68 pages] arXiv:1605.01718[quant-ph]
  2. Fundamental limitations in the purifications of tensor networks G. De las Cuevas, T. S. Cubitt, J.I. Cirac, M. M. Wolf and D. Perez-Garcia J. Math. Phys. 57, 071902 (2016) [8 pages] arXiv:1512.05709[quant-ph]
  3. The Complexity of Divisibility Johannes Bausch and Toby S. Cubitt J. Linear Alg. 504, p64–107 (2016) [50 pages] arXiv:1411.7380[math.PR]
  4. Complexity Classification of Local Hamiltonian Problems Toby Cubitt and Ashley Montanaro SIAM J. on Computing, 45:2, p268–316 (2016) [50 pages] arXiv:1311.3161[quant-ph]
  5. Simple Universal Models Capture all Classical Spin Physics Gemma de las Cuevas and Toby S. Cubitt Science, 351:6278, p1180-1183 (2016) [47 pages] arXiv:1406.5955[cond-mat.stat-mech]
  6. Area law for fixed points of rapidly mixing dissipative quantum systems F. G. S. L. Brandao, T. S. Cubitt, A. Lucia, S. Michalakis and D. Perez-Garcia J. Math. Phys. 56, 102202 (2015) [17 pages] arXiv:1505.02776[quant-ph]
  7. Undecidability of the Spectral Gap Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf Nature, 528, p207–211, (2015) arXiv:1502.04135[quant-ph] (short version) arXiv:1502.04573[quant-ph] (full version, 143 pages)
  8. Quantum reverse hypercontractivity T. Cubitt, M. Kastoryano, A. Montanaro and K. Temme J. Math. Phys. 56, 102204 (2015) [14 pages] arXiv:1504.06143[quant-ph]
  9. Rapid Mixing and Stability of Quantum Dissipative Systems Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia Phys. Rev. A 91, 040302 (2015) arXiv:1409.7809[quant-ph]
  10. Unbounded Number of Channel Uses may be Required to Detect Quantum Capacity D. Elkouss, S. Strelchuck, W. Matthews, M. Ozols, D. Perez-Garcia and T. S. Cubitt Nature Communications 6, 7739 (2015) [11 pages] arXiv:1408.5115[quant-ph]
  11. Complexity Classification of Local Hamiltonian Problems Toby Cubitt and Ashley Montanaro IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), p120–129 (2014) arXiv:1311.3161[quant-ph]
  12. Bounds on Entanglement Assisted Source-Channel Coding via the Lovász Theta Number and its Variants Toby Cubitt, Laura Mancinska, David Roberson, Simone Severini, Dan Stahlke and Andreas Winter IEEE Trans. Inform. Theory 60, 7330 (2014) [15 pages] arXiv:1310.7120[quant-ph]
  13. Stability of local quantum dissipative systems Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia Commun. Math. Phys. 337, 1275 (2015) [38 pages] arXiv:1303.4744[quant-ph]
  14. Preparing Topological PEPS on a Quantum Computer M. Schwarz, K. Temme, F. Verstraete, D. Perez-Garcia and T. S. Cubitt Phys. Rev. A, 88, 032321 (2013) (Editors' suggestion) arXiv:1211.4050[quant-ph]
  15. Entanglement can Completely Defeat Quantum Noise Jianxin Chen, Toby S. Cubitt, Aram W. Harrow and Graeme Smith Phys. Rev. Lett. 107, 250504 (2011) (Editor's suggestion) arXiv:1109.0540[quant-ph] (highlighted in APS Physics article)
  16. Extracting Dynamical Equations from Experimental Data is NP-Hard Toby S. Cubitt, Jens Eisert and Michael M. Wolf Phys. Rev. Lett. 108, 120503 (2012) (Editor's suggestion) arXiv:1005.0005[quant-ph] (highlighted in Science NOW article and in APS Physics article)
  17. Zero-Error Channel Capacity and Simulation Assisted by Non-Local Correlations T. S. Cubitt, D. Leung, W. Matthews and A. Winter IEEE Trans. Inform. Theory 57:8, 5509–5523 (2011) [15 pages] arXiv:1003.3195[quant-ph]
  18. Super-duper-activation of the zero-error quantum capacity Jianxin Chen, Toby S. Cubitt, Aram W. Harrow and Graeme Smith IEEE International Symposium on Information Theory (ISIT), p2695–2697 (2010)
  19. An Extreme Form of Superactivation for Quantum Zero-Error Capacities Toby S. Cubitt and Graeme Smith IEEE Trans. Inform. Theory 58:3, 1953–1961 (2012) [9 pages] arXiv:0912.2737[quant-ph]
  20. Improving Zero-Error Classical Communication with Entanglement T. S. Cubitt, D. Leung, W. Matthews and A. Winter Phys. Rev. Lett. 104, 230503 (2010) arXiv:0911.5300[quant-ph]
  21. The Complexity of Relating Quantum Channels to Master Equations Toby S. Cubitt, Jens Eisert and Michael M. Wolf Commun. Math. Phys. 310, 383–417 (2012) [35 pages] arXiv:0908.2128[quant-ph]
  22. Superactivation of the Asymptotic Zero-Error Classical Capacity of a Quantum Channel Toby S. Cubitt, Jianxin Chen and Aram W. Harrow IEEE Trans. Inform. Theory 57:12, 8114–8126 (2011) [8 pages] arXiv:0906.2547[quant-ph]
  23. Non-Secret Correlations can be Used to Distribute Secrecy Joonwoo Bae, Toby S. Cubitt and Antonio Acín Phys. Rev. A 79, 032304 (2009) arXiv:0806.1606[quant-ph]
  24. The Structure of Degradable Quantum Channels Toby S. Cubitt, Mary Beth Ruskai and Graeme Smith J. Math. Phys. 49, 102104 (2008) [27 pages] arXiv:0802.1460[quant-ph]
  25. Counterexamples to Additivity of Minimum Output p-Rényi Entropy for p close to 0 Toby S. Cubitt, Aram W. Harrow, Debbie Leung, Ashley Montanaro and Andreas Winter Commun. Math. Phys. 284, 281–290 (2008) [9 pages] arXiv:0712.3628[quant-ph]
  26. Assessing non-Markovian Dynamics M. M. Wolf, J. Eisert, T. S. Cubitt and J.I. Cirac Phys. Rev. Lett. 101, 150402 (2008) arXiv:0711.3172[quant-ph]
  27. On the Dimension of Subspaces with Bounded Schmidt Rank Toby S. Cubitt, Ashley Montanaro and Andreas Winter J. Math. Phys. 49, 022107 (2008) arXiv:0706.0705[quant-ph]
  28. Engineering Correlation and Entanglement Dynamics in Spin Systems T. S. Cubitt and J.I. Cirac Phys. Rev. Lett. 100, 180406 (2008) arXiv:quant-ph/0701053
  29. Entanglement Flow in Multipartite Systems T. S. Cubitt, F. Verstraete and J.I. Cirac Phys. Rev. A 71, 052308 (2005) [12 pages] arXiv:quant-ph/0404179
  30. Separable States can be Used to Distribute Entanglement T. S. Cubitt, F. Verstraete, W. Dür, J.I. Cirac Phys. Rev. Lett. 91, 037902 (2003) arXiv:quant-ph/0302168 (highlighted in Science NOW article)


  1. Universal Quantum Hamiltonians Toby Cubitt, Ashley Montanaro and Stephen Piddock arXiv:1701.05182[quant-ph]
  2. Comment on "On the uncomputability of the spectral gap" Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf arXiv:1603.00825[quant-ph]
  3. Universal Refocusing of Systematic Quantum Noise Imdad S. B. Sardharwalla, Toby S. Cubitt, Aram W. Harrow and Noah Linden arXiv:1602.07963[quant-ph]
  4. Size-Driven Quantum Phase Transitions Johannes Bausch, Toby S. Cubitt, Angelo Lucia, David Perez-Garcia and Michael M. Wolf arXiv:1512.05687[quant-ph]
  5. Undecidability of the Spectral Gap (full version) Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf arXiv:1502.04573[quant-ph]
  6. An Information-Theoretic Proof of the Constructive Commutative Quantum Lovász Local Lemma M. Schwarz, T. S. Cubitt and Frank Verstraete arXiv:1311.6474[quant-ph]
  7. Are Problems in Quantum Information Theory (Un)decidable? Michael M. Wolf, Toby S. Cubitt and David Perez-Garcia arXiv:1111.5425[quant-ph]
  8. Entanglement in the Stabilizer Formalism David Fattal, Toby S. Cubitt, Yoshihisa Yamamoto, Sergey Bravyi and Isaac L. Chuang arXiv:quant-ph/0406168