Investigate why things float or sink.

Introduce the idea of formulating and testing hypotheses.

- 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.

Investigate what seeds and plants need to grow.

Introduce the basic idea of a controlled trial.

- 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

- 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.

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.

]]>- 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.

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

*Tamara Kohler, Stephen Piddock, Johannes Bausch, Toby Cubitt*
arXiv:2101.12319\[quant-ph\] [22 pages]

*Tamara Kohler, Stephen Piddock, Johannes Bausch and Toby Cubitt*
arXiv:2003.13753\[quant-ph\] [31 pages]

*Laura Clinton, Johannes Bausch and Toby Cubitt*
arXiv:2003.06886\[quant-ph\] [63 pages]

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

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

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

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

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

*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.)

*Tamara Kohler and Toby Cubitt*
J. Stat. Phys. **176**:1 p228–261 (2019)
arxiv:1807.01715\[cond-mat.stat-mech\] [44 pages]

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

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

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

*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\[quant-ph\]

*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".)

*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\[quant-ph\]

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

*Toby Cubitt and Ashley Montanaro*
SIAM J. on Computing, **45**:2, p268–316 (2016) [50 pages]
arXiv:1311.3161\[quant-ph\]

*Gemma de las Cuevas and Toby S. Cubitt*
Science, **351**:6278, p1180-1183 (2016) [47 pages]
arXiv:1406.5955\[cond-mat.stat-mech\]

*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\[quant-ph\]

*Toby S. Cubitt, David Perez-Garcia and Michael M. Wolf*
arXiv:1502.04573\[quant-ph\] (full version, 127 pages)

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

*T. Cubitt, M. Kastoryano, A. Montanaro and K. Temme*
J. Math. Phys. **56**, 102204 (2015) [14 pages]
arXiv:1504.06143\[quant-ph\]

*Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia*
Phys. Rev. A **91**, 040302 (2015)
arXiv:1409.7809\[quant-ph\]

*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\[quant-ph\]

*M. Schwarz, T. S. Cubitt and Frank Verstraete*
arXiv:1311.6474\[quant-ph\]

*Toby Cubitt and Ashley Montanaro*
IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), p120–129 (2014)
arXiv:1311.3161\[quant-ph\]

*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\[quant-ph\]

*Toby S. Cubitt, Angelo Lucia, Spyridon Michalakis, and David Perez-Garcia*
Commun. Math. Phys. **337**, 1275 (2015) [38 pages]
arXiv:1303.4744\[quant-ph\]

*M. Schwarz, K. Temme, F. Verstraete, D. Perez-Garcia and T. S. Cubitt*
Phys. Rev. A, **88**, 032321 (2013) *(Editors' suggestion)*
arXiv:1211.4050\[quant-ph\]

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

*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)

*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)

*T. S. Cubitt, D. Leung, W. Matthews and A. Winter*
IEEE Trans. Inform. Theory **57**:8, 5509–5523 (2011) [15 pages]
arXiv:1003.3195\[quant-ph\]

*Jianxin Chen, Toby S. Cubitt, Aram W. Harrow and Graeme Smith*
IEEE International Symposium on Information Theory (ISIT), p2695–2697 (2010)

*Toby S. Cubitt and Graeme Smith*
IEEE Trans. Inform. Theory **58**:3, 1953–1961 (2012) [9 pages]
arXiv:0912.2737\[quant-ph\]

*T. S. Cubitt, D. Leung, W. Matthews and A. Winter*
Phys. Rev. Lett. **104**, 230503 (2010)
arXiv:0911.5300\[quant-ph\]

*Toby S. Cubitt, Jens Eisert and Michael M. Wolf*
Commun. Math. Phys. **310**, 383–417 (2012) [35 pages]
arXiv:0908.2128\[quant-ph\]

*Toby S. Cubitt, Jianxin Chen and Aram W. Harrow*
IEEE Trans. Inform. Theory **57**:12, 8114–8126 (2011) [8 pages]
arXiv:0906.2547\[quant-ph\]

*Joonwoo Bae, Toby S. Cubitt and Antonio Acín*
Phys. Rev. A **79**, 032304 (2009)
arXiv:0806.1606\[quant-ph\]

*Toby S. Cubitt, Mary Beth Ruskai and Graeme Smith*
J. Math. Phys. **49**, 102104 (2008) [27 pages]
arXiv:0802.1460\[quant-ph\]

*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\[quant-ph\]

*M. M. Wolf, J. Eisert, T. S. Cubitt and J.I. Cirac*
Phys. Rev. Lett. **101**, 150402 (2008)
arXiv:0711.3172\[quant-ph\]

*Toby S. Cubitt, Ashley Montanaro and Andreas Winter*
J. Math. Phys. **49**, 022107 (2008)
arXiv:0706.0705\[quant-ph\]

*T. S. Cubitt and J.I. Cirac*
Phys. Rev. Lett. **100**, 180406 (2008)
arXiv:quant-ph/0701053

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

*T. S. Cubitt, F. Verstraete and J.I. Cirac*
Phys. Rev. A **71**, 052308 (2005) [12 pages]
arXiv:quant-ph/0404179

*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)