# 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
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

## Goals

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

## Goals

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

## Goals

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

[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

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.

## Publications

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

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

## Papers

**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**

*James Watson, Toby Cubitt*
Proceedings 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC), p.764–775 (2022)
arxiv:2107.05060[quant-ph] [57 pages]

**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**

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

**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**

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

**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**

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

**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**

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

**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)