# Quantum Computation and Algorithms course

Quantum Computation and Algorithms course

Quantum computing is the flagship quantum technology. Peter Shor's 1994 quantum factoring algorithm kicked off a wave of interest and excitment about quantum computing, and is in large part responsible for the field of quantum information going mainstream. A second wave of even more excitment has been kicked off recently by the near-term prospect of NISQ hardware (noisy, intermediate-scale quantum computers) – i.e. quantum computing hardware that is not obviously useful for anything… but no longer obviously useless either!

However, there is also currently a lot of hype and many unrealistic expectations. The aim of this course is to show you where all the excitment derives from, rigorously and in detail; and to innoculate you against the over-hype.

# How to include material after the letter in newlfm

How to include material after the letter in newlfm

TL;DR: If you want to include additional material after the closing of a `newlfm`

letter, forget the builtin `\restletter`

commands. Add `\newcommand\closelfm{\closlfm\let\closlfm\relax}`

to your `letrinfo.tex`

file, put `\closelfm`

where you want to end the letter (optionally followed by a `\newpage`

), then write the additional material using standard L^{a}T_{e}X markup.

I use the `newlfm`

L^{a}T_{e}X package for all letters I write, personal and professional. It has some nice advantages over the standard `letter`

document class: you can have a database of different sender and recipient addresses, signatures and letterheads; it's easy to create custom letterheads that include images; you can have the correct letterhead and signature selected automatically based on sender or recipient. And many other features I don't make use of.

I have a love-hate relationship with this package, though. It has some quirks/bugs, especially to do with page layout and spacing. And whilst the documentation is extensive, it's not always the clearest. Whenever I need to do anything slightly different to what I've done before with `newlfm`

, getting it to do what I want typically involves hours of effort and frustration the first time around, and I often ending up resorting to ugly kludges to get the letter written without wasting too much time. (This probably says more about my limitations, than those of the package.) Nonetheless, once you have a letter layout configured just the way you want, writing new letters with the same layout with `newlfm`

is a breeze.

I was recently writing a cover letter to accompany the revisions I was submitting to one of my research papers. I wanted to attach the detailed response to the referees' comments after the letter (which contained a lot of mathsso was best written in L^{a}T_{e}X too). The easy thing would have been to write the cover letter using `newlfm`

, write the review response as a separate L^{a}T_{e}X document using some other document class, and stitch the two PDFs together using the pdftk command-line tool (or similar). But having to write these as two separate documents bugged me. I was already writing the cover letter in L^{a}T_{e}X. Why not just write the detailed referee response in the same L^{a}T_{e}X document?

# LaTeX: Cleveref package

LaTeX: Cleveref package The Cleveref package (also available from CTAN) does clever things with cross-references:

- automatic formatting of cross-references based on the type of object referred to (chapter, section, equation, theorem, etc.);
- full control and customisation of the reference format;
- cross-references and page references to multiple items;
- automatic (optional) sorting and compression of multiple cross-references or page references;
- optional output of a sed script that can strip out Cleveref commands and replace them with standard LaTeX, allowing Cleveref to be used e.g. in articles sent to journals or collaborators that don't (yet!) support Cleveref.

A number of other LaTeX packages provide similar features. Some are venerable enough to be documented in *The LaTeX Companion*, or even included in the standard LaTeX distribution itself. But all those that I've come across provide only a subset of the Cleveref features. (See the Introduction to the package documentation for some comparisons.)

# 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.

# Complexity and Computability in Physics course

Complexity and Computability in Physics course Lecture course on quantum complexity theory, encompassing (a brief intro to) computability and complexity theory, complexity theory in quantum physics, and computability theory in physics. Lectured at the 2018 Boulder quantum information summer school.

## Lecture videos and notes

- Lecture 1, Computation and Complexity video and lecture notes
- Lecture 2: Complexity in Physics video and lecture notes
- Lecture 3: Computability in Physics video and lecture notes

# Advanced Quantum Information Theory course

Advanced Quantum Information Theory course

Quantum information theory is neither wholly physics (though it's mostly about quantum mechanics), nor wholly mathematics (though it mostly proves rigorous mathematical results), nor wholly computer science (though it's mostly about storing, processing, or transmitting information). Over the last two decades, it has developed into a rich mathematical theory of information in quantum mechanical systems, that draws on all three of these disciplines. More recently, this has been turned on its head: quantum information is beginning to be *used* to attack deep problems in physics, computer science, and mathematics.

The aim of this course is to select one or two advanced topics in quantum information theory, close to the cutting edge of research, and cover them in some depth and rigour.

This time around, I will focus on quantum information in many-body systems. What do these two topics have to do with each other? Quantum computation aims to engineer complex many-body systems to process information in ways that would not be possible classically. Many-body physics aims to understand the complex behaviour of naturally-occurring many-body systems. In a sense, they are two sides of the same coin. Quantum information theory is now used both to prove important results in many-body physics, and to construct many-body models that exhibit very unusual physics, providing counterexamples to long-standing beliefs in condensed matter theory. This is now one of the fastest-developing areas of the field.

# Paternity leave: reflections

When K~~ ~~ was still at the theory stage, S~~ ~~ and I spent inordinate amounts of time discussing how we wanted to organise things once she arrived. One of the few things we spent almost no time discussing was what to do about parental leave. It just seemed obvious that we'd split it 50/50, and each take 6 months.It didn't occur to us we were doing anything unusual. If we thought about it at all, I guess it just seemed fair to give each of us an equal share of time with K~~ ~~, and also to tryto share any career impact it might have.

Once I started my six months leave, I didn't need to look up the statistics to quickly realise how few men were taking any parental leave at all. It was a rare baby singing, music, swimming or soft-play class where I wasn't the only man present. I never found that an issue. To their credit, all the mothers I met made me feel as accepted and welcome as any other parent of a young baby, and I made some really good friends. There's honour among thieves…and solidarity amongst sleep-deprived parents of any gender!

So how do I feel about it having done it? Two things struck me particularly forcefully over the six months.

# 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.

# Emacs: Data structure packages

Emacs: Data structure packages These packages provide basic (and not so basic) data structures. They are all relatively stable, though bug-fixes and new features are added occasionally. (Latest update: August 2017).

In recent versions of Emacs (>=24.1), you can install all the non-obsolete packages from within Emacs itself, via GNU ELPA. Use `M-x list-packages`

and take it from there. This is the preferred installation method. (Occasionally, the ELPA version might lag slightly behind the latest version available here.)

# Dear Diane

[Letter sent to my local MP, Diane Abbott. Do the same! Keep up the pressure on MPs to represent your views.]

Dear Diane,

When I first moved to Hackney, I was proud to tell people I had you as my MP. As one of the few voices on the Labour backbenches consistently voting according to conscience, defying the party whip when it was at odds with your principles and your constituents' interests, you stood out from the crowd of political apparatchiks toting the party line. On issues ranging from the Iraq war, to defending the NHS from privatisation, to resisting the incoming tide of government mass surveillance, your voting record aligned even more closely with my views than the overall Labour party line. Though I've been a lifelong Labour voter, I was even happier to be a Diane Abbot voter.

You campaigned for remaining in the EU. Your constituents voted overwhelmingly remain, the joint-second highest remain vote (with Lambeth) after Gibraltar. You know that opposing a hard Brexit and fighting to keep the UK in the common market is in the best interests of your constituents, not just economically but also socially. You know that, for 40 years, the strongest bulwark against dismantling of social protections, civil liberties, and workers' rights in the UK has been European legislation. You know that fighting to retain as much of that as possible is fighting to prevent Theresa May's race to the very bottom in pandering to Trump, legitimising corporate tax evasion, liberating corporations to exploit employees, and dismantling and privatising the services that provide a safety net to so many in the UK.

# Quantum Computation and Complexity course

Quantum Computation and Complexity course 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.

## Lecture notes

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

# Matrix Product States and PEPS

Matrix Product States and PEPS 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 © 2016 Toby Cubitt, and are licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.)

# Paternity leave: statistics

## Lies, damn lies, and…

A couple of months ago, the statistic that only 1% of men had taken up shared parental leave was splashed all over the British media. (Shared Parental Leave was introduced in the UK in 2015, and essentially allows parents to share 12 months of leave however they like. Taking it consecutively, simultaneously, alternating blocks of leave between both parents, or a mixture of the above are all permitted.)

My Family Care, the company that carried out the survey on which this statistic was based, apparetly asked Human Resources directors at 200 businesses what percentage of men in their company had taken shared parental leave in the year since it was introduced. But, as Radio's 4's excellent More or Less programme pointed out, they forgot to ask what percentage of those men were actually elligible for parental leave in the first place! Most of them won't have had children at all in the last year. Some of them won't even have any children!! Ooops.

# Paternity leave: reactions

Paternity leave: reactions
I've been on paternity leave since January, taking care of K~~ ~~ full time. The reactions I've had from people when they discover I'm on paternity leave for half a year have been entirely positive. But some of the comments I've had in response have been interesting. I've collected the ones that stuck in my mind, together with my thoughts on them.

(If you know me personally, and think you recognise something you've said, you don't! These aren't direct quotes. I've paraphrased things that have been said to me multiple times by many different people.)

# Decoupling Method in Quantum Shannon Theory

Decoupling Method in Quantum Shannon Theory Originally lectured in 2015 as part of the quantum information theory masters course for the UCL quantum CDT.

## Lecture Notes

## 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.

# Emacs: Miscelaneous packages

Emacs: Miscelaneous packages These packages provide miscelaneous features I needed at some point. So I coded them. Currently, they're all to do with displaying useful information in the mode line.

`show-point-mode`

displays the current value of the point in the mode line. I primarily find it useful when debugging Elisp code that uses overlays and markers.

`wc-mode`

displays output similar to the Unix `wc`

command in the mode line, i.e. the character count, word count and line count for the current buffer. (I primarily find this Useful when writing grant applications with character or word limits. Though I'm sure it's useful for other more productive activities, too…)

- show-point-mode.el (version 0.3)
- wc-mode.el (version 0.3)

# Matlab code

Matlab code I've collected here various functions, routines, and other bits of Matlab, Octave and Mathematica code organized by topic, that might save someone, somewhere, from re-inventing the wheel. Some of them are so simple it would probably be quicker to re-code them than find this page, but since you're here anyway…

Comments within the code should be enough to figure out what they do and how to use them (try `help <function>`

from within Matlab or Octave). No guarantee they work as advertised, but I use them myself so I do correct bugs when I come across them. The Matlab code *should* run under both Octave and Matlab.

All the Matlab, Octave and Mathematica code linked from this page is released under the GPL license, version 2 or later.

If you make use of this code in your research, consider including a citation to this web page in any resulting publication. Not only is it fair to give credit when you've made use of other people's work, it is important for scientific reproducability to document any code you used to help produce your results. Also, if you found this code useful, then others probably will too! Citing this page helps others find it.

I leave it to your judgement whether you feel your results made sufficient use of this code to warrant a citation; I do not insist on it. But consider whether using this code has been as helpful to your research as the least useful paper you are citing. (Typically, this sets the bar very low!). If so, you should probably include a citation to this web page.

# Emacs: Undo Tree package

Emacs: Undo Tree package
Emacs has a powerful undo system. Unlike the standard undo/redo system in most software, it allows you to recover *any* past state of a buffer (whereas the standard undo/redo system can lose past states as soon as you redo). However, this power comes at a price: many people find Emacs' undo system confusing and difficult to use, spawning a number of packages that replace it with the less powerful but more intuitive undo/redo system. (See the Emacs Wiki.)

Both the loss of data with standard undo/redo, and the confusion of Emacs' undo, stem from trying to treat undo history as a linear sequence of changes. It's not. The `undo-tree-mode`

provided by this package replaces Emacs' undo system with a system that treats undo history as what it is: a branching tree of changes. This simple idea allows the more intuitive behaviour of the standard undo/redo system to be combined with the power of never losing any history. An added side bonus is that undo history can in some cases be stored more efficiently, allowing more changes to accumulate before Emacs starts discarding history.

It gets better. You don't have to imagine the undo tree, because `undo-tree-mode`

includes an undo-tree visualizer which draws it for you, and lets you browse around the undo history.

The only downside to this more advanced yet simpler undo system is that it was inspired by Vim. But, after all, most successful religions steal the best ideas from their competitors!

# Emacs: Auto-Overlays package

Emacs: Auto-Overlays package The Auto-Overlays package allows you to define overlays that are created (and updated and destroyed) automatically when text in a buffer matches a regular expression.

Various classes of automatic overlay are provided, to make it easy to define matches for different text regions: words, lines, regions enclosed by start and end tags, or regions enclosed by delimiters. You can also define your own custom classes.

The overlays are updated just before any buffer modification. The built in overlay classes only update as much as is necessary to ensure that overlays covering the point are consistent. Therefore the overlays at the point are guaranteed to be correct before any buffer modification takes place there, but updating the overlays is fast and usually causes no noticeable delay.

# Emacs: Completion User Interface package

Emacs: Completion User Interface package The Completion User Interface package is a library that implements user-interfaces for in-buffer completion.

Typically, in packages providing some kind of text completion, a large amount of code deals with providing the user interface rather than finding good completions. The goal of Completion-UI is to be the swiss-army knife of in-buffer completion user-interfaces; a library which any completion package can use to provide an in-buffer completion user-interface, thereby freeing completion package writers to concentrate on the task of finding the completions in the first place.

In fact, Completion-UI is even better than a swiss-army knife, because it's also extensible: it's easyto add new completion user-interfaces and hook them into Completion-UI. The new interface will then be available to any completion package that uses the Completion-UI library, *without* needing to make any changes to that package.

Various standard completion user-interfaces and commands are provided "out of the box". These can be separately enabled, disabled and tweaked by the Emacs user via the usual bewildering array of customization variables.

# Emacs: Predictive Completion package

Emacs: Predictive Completion package The Emacs Predictive Completion package adds a new minor-mode to the GNU Emacs editor. When enabled, predictive mode exploits the redundancy inherent in languages in order to complete words you are typing before you've finished typing them (somewhat like the IntelliSense feature in some IDEs). It is highly customisable, and works happily alongside other Emacs major modes. See the documentation for more details.

Predictive mode only works under GNU Emacs, not under XEmacs. It may be possible to get it to work under XEmacs with a modicum of work. (At the very least, the overlay compatibility package would be required.) If you're interested in attempting this, then I'm happy to answer questions, but I have no plans to do it myself.

# LaTeX: Quantum package

LaTeX: Quantum package The Quantum package defines a number of commands and short-hands useful when writing about quantum mechanics, and quantum information theory in particular.

There is no separate documentation; read the package source to find out what commands it provides.

- Quantum package

# LaTeX: Authord package

LaTeX: Authord package Gives a complete solution to the problem of precedence in scientific pubication, in a way that Don Knuth would surely approve of.

- Authord package
- Authord documentation

# Quantum Mechanics for Mathematicians course

Quantum Mechanics for Mathematicians course 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

Quantum Mechanics course 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 10)

# Why I use TMDA

Why I use TMDA Mine is a sad and familiar story. I was drowning in a deluge of spam (a.k.a. junk email), and it had become such a problem that email was fast becoming useless for me. Having to sort through and delete hundreds of spam emails per day was bad enough. Worse was the increasing frequency with which I was accidentally deleting legitimate email along with the spam.

There are various ways to fight this deluge of spam. The most common is to use a filter that tries to recognise and delete the spam (or, more usually, move it to a spam box for later perusal). This is quite effective. A small amount of spam will not be recognised as such (false-negatives), and will end up in your inbox anyway. But the amount of spam will usually be cut down to a manageable amount, rendering email usable again.

The problem with the filtering approach is the false-positives: legitimate mail that gets mis-identified as spam. Even if it's moved to a spam box rather than deleted, when you're searching through hundreds of spam emails you're almost certain to miss the one or two legitimate mails hiding amongst them, and you'll delete them along with the spam. (At least, that's what I found myself doing.)

A second approach is to use techniques such as domain blocking, real-time blacklists, and other methods of blocking whole groups of addresses known to send spam. But this is really just a variation on filtering (filters usually take the sender's address into account when deciding whether an email is spam or not, as well as the body).

I didn't want to run the risk of someone sending me an email, me deleting it accidentally, and them never knowing that I didn't received it. So I chose to use a third approach: white-list plus challenge/response (plus a number of other features of the impressive TMDA system).

# Classical mechanics and electrodynamics

Classical mechanics and electrodynamics 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 bitter and twisted physics professors).

If anyone involved in teaching the courses is interested in obtaining the solutions, drop me an email. I have scanned copies for about half the mechanics question sheets and all the electrodynamics question sheets.

# Emailing me

## Email address

I can be reached by email at toby@dr-qubit.org, associated with this PGP key:

4096R/0xA96F4A674DC39B79 BB74 FB42 4C64 4CB7 3571 39AA A96F 4A67 4DC3 9B79

As of 2 August 2013, I transitioned from an old 1024-bit DSA key to this new 4096-bit RSA key. I will be signing all software releases with the new key. Please also use the new key for all correspondence. See the transition statement to certify the transition, and for more details.

Note that I use FLOSS spam-reduction software called TMDA to protect my addresses from junk-mail.

If you've never exchanged any email with me previously, you'll receive a message asking you to verify your email address. By simply replying to the message (literally just hit "Reply" then "Send"), your original message will be delivered. You'll only have to confirm your address once ever. All subsequent email from that address will be delivered directly.

# Toby 'qubit' Cubitt

## Who am I? (a brief Curriculum Vitae)

I'm a nationality-confused European, born and raised in Luxembourg but technically British.

I went to the European school in Luxembourg, graduating with the European Baccalaureate in 1998. From there, I hopped across the Channel to Churchill College, Cambridge, studying physics under the Natural Sciences Tripos at the University of Cambridge.

After graduating in 2002, I decided to see what the other end of Europe was like, and moved to the Max Planck Institute for Quantum Optics just outside Munich, Germany to do a PhD in quantum information theory under the wonderful Ignacio Cirac.

# 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

**Complete Toy Models of Holographic Duality**

*Tamara Kohler, Toby Cubitt*
arXiv:1810.08992[hep-th] [62 pages]

**History-state Hamiltonians are critical**

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

**Undecidability of the Spectral Gap in One Dimension**

*Johannes Bausch, Toby Cubitt, Angelo Lucia, David Perez-Garcia*
arxiv:1810.01858[quant-ph] [54 pages]

**The Unsolvable Problem**

*Toby Cubitt, David Perez-Garcia, Michael Wolf*
Scientific American, Volume 319, Issue 4, October 2018
(Cover article; selected for 2018 "Scientific American Articles of the Year" special issue.)

**Translationally invariant universal classical Hamiltonians**

*Tamara Kohler and Toby Cubitt*
J. Stat. Phys. (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** (full version)

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

**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)

**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)