Science posts

Blog posts which assume some prior mathematical knowledge are tagged @school, @undergrad or @PhD, to give you a rough idea of what level of expertise is needed to follow the post.

This is just a very rough guideline, to help you restrict the list of blog posts to those that are of most interest to you. Don't let the tag names put you off! If you're at school and keen to read about quantum information or other mathematical topics, by all means go ahead and read a post tagged @PhD. You'll probably understand it better than I would have after my PhD.

Early-Years science: Intro

Early-Years science: Intro 5 May 2021 Whilst I was on paternity leave, I made up simple science experiments and activities to do at home with K      and E     . I still do now.

What I do is very simple.I usually make up science activities on the spur of the moment, using whatever materials we have around the house. Often ideas for experiments come from questions K      or E      ask me, or things we notice whilst out and about. Very often the experiments don't work first time. Then K     , E      and I discuss together why it didn't work, think about ways to adapt it, and try again.

My intention isn't to "teach them science". Knowing by age 4 that plants needs water and light to grow, or that electricity only flows in circuits, is irrelevant. What I want to introduce them to early is how to look at the world scientifically and how to think like a scientist.

Early-Years Science: Buoyancy and Displacement


Investigate how buoyancy and displacement are related.
Notice correlations and connections between different phenomena.

You will need

  • 1 large, waterproof container. Transparent if possible, but any container will do. E.g. a bucket, basin, large mixing bowl, or watertight box. A bath or sink will also work fine.
  • Something boat-like: e.g. a toy boat or a small plastic tub. Plasticine or play-dough to shape into a boat will also work fine.
  • Waterproof marker pen. (A normal felt pen will do if you don't have a waterproof marker.)
  • Small weights, E.g. pebbles or marbles.
  • A balloon.

Early-Years Science: Floating and Sinking


Investigate why things float or sink.
Introduce the idea of formulating and testing hypotheses.

You will need

  • 1 large, waterproof container. E.g. a bucket, basin, or watertight box. A bath or sink will also work fine.
  • A variety of objects of different shapes and sizes, and made out of different materials: stone, metal, plasic, wood. E.g. metal, plastic and wooden spoons of similar sizes; different size spoons; metal and plastic bottle tops; stones; corks; matchsticks.
  • Plasticine or play-dough.
  • A small plastic tub that fits in the big container.

Early-Years Science: Cress-Growing


Investigate what seeds and plants need to grow.
Introduce the basic idea of a controlled trial.

You will need

  • cress seeds
  • 4 trays (plastic food trays are perfect - remember to wash them throughly first!)
  • 2 cardboard boxes (big enough for the trays to fit in)
  • 4 pieces of kitchen roll, or enough cotton wool to line each tray

Toby's ten commandments of being scooped

Toby's ten commandments of being scooped 25 July 2018

[To be read in alongside Luc's ten commandments of authorship.]

Rule 1
Never consult a list of rules regarding being scooped.
Rule 2
Be happy: it implies \(\geq 1\) other person is interested in the problem you solved.
Rule 3
You hear about the other author's paper before it's posted on the arXiv, and are on friendly terms with them. Politely ask if they'd be willing to post to the arXiv simultaneously. This is a win-win: both papers will look more important and get more attention. If they don't agree, go to Rule 6. In either case, cite the other paper and be generous in crediting them.
Rule 4
You agree to post simultaneously, but then they ask for more time. Be generous and give them plenty of extra time. You will feel good. When you post yours, cite the other paper and be generous in crediting them.
Rule 5
You agree to post simultaneously, but then you need more time. Go home, have a beer, tell them you're not going to make the agreed date and let them post first. When you post yours, cite the other paper and be generous in crediting them.
Rule 6
The other results appear on the arXiv before yours, but you can post your results soon enough that it's clearly independent work. Go ahead and post them. You'll still get credit for the work. Cite the other paper and be generous in crediting them.
Rule 7
You're not ready to post soon without stressing about it, but the differences in your results are interesting and you're motivated to work on them. Work on those, post your paper when you have interesting new results. Be happy, smile. You've advanced science a few more steps. Cite the other paper and be generous in crediting them.
Rule 8
Your results are very similar, and the differences aren't interesting enough to you. Get your mind off it—crying, sex, indulging, smoking, hallucinating, and swimming in the North Sea may help. After doing \(\geq 1\) of those activities, look at the differences with fresh eyes. If they look interesting now, go back to Rule 7. If not, let this one go. There are infinitely many new and interesting problems to work on: go and work on one of those.
Rule 9
It happens - do not get stressed about it.

Truths about proofs and groups

Truths about proofs and groups 9 February 2018

A while back, a PhD student in our group asked me whether the sum over all elements of a stabilizer is a projector. If you know what this means, you're probably (a) a quantum information theorist (in which case, stop reading here) and (b) already know the answer and how to prove it (unlike me, who'd forgotten both).

The answer is no doubt well-known to anyone who works on quantum stabilizer codes, and we could have just googled for the result. It seemed like a nice, self-contained mathematical question, though. So rather than googling, we tried to figure it out for ourselves at the blackboard.

If you just want to see the simple final answer, skip to the end. But then you'll miss all the fun and the main point of this post. The way we came up with the solution makes for a nice toy example of the convoluted, messy and inelegant process by which mathematical results are really proven. Before they get polished up into the simple, elegant, pristine proofs "from The Book" that are all you ever get to see in textbooks and research papers. The unspoken (or at least unpublished) reality is that elegant proofs invariably emerge after following numerous blind alleys, unjustified intuitive leaps, and inelegant, round-the-houses arguments. All of which get simplified away in time for publication. (Or maybe that's just my proofs!)

Instead of just explaining the elegant final answer, I'm going to explain the inelegant process we went through to reach it.

Research interests

General interests

  • Quantum information theory
  • Many-body physics
  • Complexity theory
  • Hamiltonian complexity
  • Hamiltonian simulation
  • CP maps (a.k.a. quantum channels)
  • 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 with links to the papers, as well as the slides from some of my talks. For a more up-to-date list, try the arXiv.


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


A mathematical framework for quantum Hamiltonian simulation and duality

Harriet Apel and Toby Cubitt arXiv:2208.11941[quant-ph] [39 pages]

Towards near-term quantum simulation of materials

Laura Clinton, Toby Cubitt, Brian Flynn, Filippo Maria Gambetta, Joel Klassen, Ashley Montanaro, Stephen Piddock, Raul A. Santos and Evan Sheridan arXiv:2205.15256[quant-ph] [94 pages]

Phase Estimation of Local Hamiltonians on NISQ Hardware

Laura Clinton, Johannes Bausch, Joel Klassen, Toby Cubitt arXiv:2110.13584[quant-ph]

Undecidability of the Spectral Gap (full version)

Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf Forum of Mathematics Pi, 10:e14 (2022) arXiv:1502.04573[quant-ph] [127 pages]

Computational Complexity of the Ground State Energy Density Problem

Holographic duality between local Hamiltonians from random tensor networks

Harriet Apel, Tamara Kohler, Toby Cubitt J. High Energy Phys. 2022:52 (2022) arXiv:2105.12067[quant-ph] [38 pages]

A Note on the Second Spectral Gap Incompleteness Theorem

Toby S. Cubitt arXiv:2105.09854[quant-ph]

Fitting quantum noise models to tomography data

Emilio Onorati, Tamara Kohler, Toby Cubitt arXiv:2103.17243[quant-ph] [56 pages]

Uncomputably Complex Renormalisation Group Flows

James D. Watson, Emilio Onorati, Toby S. Cubitt arXiv:2102.05145[quant-ph] [88 pages]

General conditions for universality of Quantum Hamiltonians

Tamara Kohler, Stephen Piddock, Johannes Bausch, Toby Cubitt PRX Quantum 3:010308 (2022) arXiv:2101.12319[quant-ph] [22 pages]

Translationally-Invariant Universal Quantum Hamiltonians in 1D

Tamara Kohler, Stephen Piddock, Johannes Bausch and Toby Cubitt Annales Henri Poincaré, 23, p.223–254 (2021) arXiv:2003.13753[quant-ph] [31 pages]

Hamiltonian Simulation Algorithms for Near-Term Quantum Hardware

Laura Clinton, Johannes Bausch and Toby Cubitt Nature Communications 12(1), 4929 (2021) arXiv:2003.06886[quant-ph] [63 pages]

Compact fermion to qubit mappings

Charles Derby, Joel Klassen, Johannes Bausch, Toby Cubitt Phys. Rev. B 104:035118 (2021) Combines arXiv:2003.06939[quant-ph] and arXiv:2003.07125[quant-ph]

Mitigating Errors in Local Fermionic Encodings

Johannes Bausch, Toby Cubitt, Charles Derby and Joel Klassen arXiv:2003.07125[quant-ph] [23 pages]

Uncomputability of Phase Diagrams

Johannes Bausch, Toby S. Cubitt and James D. Watson Nature Communications 12(1), 1, (2021) arXiv:1910.01631[quant-ph] [64 pages]

Toy Models of Holographic Duality between local Hamiltonians

Tamara Kohler and Toby Cubitt J. High Energy Phys. 2019:17 (2019) arXiv:1810.08992[hep-th] [62 pages]

History-state Hamiltonians are critical

Carlos E. González-Guillén and Toby S. Cubitt arXiv:1810.06528[quant-ph] [35 pages]

Undecidability of the Spectral Gap in One Dimension

Johannes Bausch, Toby Cubitt, Angelo Lucia and David Perez-Garcia Phys. Rev. X 10, 031038 (2020) arxiv:1810.01858[quant-ph] [54 pages]

The Unsolvable Problem

Toby Cubitt, David Perez-Garcia and Michael Wolf Scientific American, Volume 319, Issue 4, October 2018 (Cover article; selected for 2018 "Scientific American Articles of the Year" special issue, and for 2019 "Best Writing on Mathematics" Princeton University Press anthology.)

Translationally invariant universal classical Hamiltonians

Universal Quantum Hamiltonians

Toby Cubitt, Ashley Montanaro and Stephen Piddock Proc. Natl. Acad. Sci. 115:38 p9497–9502 (2018) arXiv:1701.05182[quant-ph] [82 pages]

Comment on "On the uncomputability of the spectral gap"

Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf arXiv:1603.00825[quant-ph]

Universal Refocusing of Systematic Quantum Noise

Imdad S. B. Sardharwalla, Toby S. Cubitt, Aram W. Harrow and Noah Linden arXiv:1602.07963[quant-ph]

Size-Driven Quantum Phase Transitions

Johannes Bausch, Toby S. Cubitt, Angelo Lucia, David Perez-Garcia and Michael M. Wolf Proc. Natl. Acad. Sci. 115:1, p19–23 (2018) [18 pages] arXiv:1512.05687[quant-ph]

The Complexity of Translationally-Invariant Spin Chains with Low Local Dimension

Johannes Bausch, Toby Cubitt and Maris Ozols Annales Henri Poincaré, 18:11, p3449–3513 (2017) [63 pages] arXiv:1605.01718[quant-ph] (Winner of 2018 AHP-Birkhauser Prize for "most remarkable paper published in Annales Henri Poincaré".)

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]

The Complexity of Divisibility

Johannes Bausch and Toby S. Cubitt J. Linear Alg. 504, p64–107 (2016) [50 pages] arXiv:1411.7380[math.PR]

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]

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]

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]

Undecidability of the Spectral Gap (short version)

Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf Nature, 528, p207–211, (2015) arXiv:1502.04135[quant-ph] (short version)

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]

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]

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]

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]

Complexity Classification of Local Hamiltonian Problems

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]

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]

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]

Are Problems in Quantum Information Theory (Un)decidable?

Michael M. Wolf, Toby S. Cubitt and David Perez-Garcia arXiv:1111.5425[quant-ph]

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)

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)

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]

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)

An Extreme Form of Superactivation for Quantum Zero-Error Capacities

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]

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]

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]

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]

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]

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]

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]

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]

Engineering Correlation and Entanglement Dynamics in Spin Systems

Entanglement in the Stabilizer Formalism

David Fattal, Toby S. Cubitt, Yoshihisa Yamamoto, Sergey Bravyi and Isaac L. Chuang arXiv:quant-ph/0406168

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

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)