Lectured from 2017 as an optional advanced theory course for the

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

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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.
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A possible outline (from which we may deviate to explore other related results) is as follows. We will begin by studying the computational complexity of quantum many-body systems, introducing the necessary complexity-theoretic concepts as briefly as possible in order to move on quickly to their exciting applications to many-body Hamiltonians. An important milestone is Kitaev's proof of QMA-hardness of the ground state energy problem for local Hamiltonians, which we will prove in full.

Kitaev's construction generates systems with highly entangled ground states and polynomially-decaying spectral gap. Which leads us to the second half of the course: a quantum-information-inspired exploration of spectral gaps, correlations, and entanglement in many-body systems. Lieb-Robinson bounds, which limit the speed at which information can propagate in non-relativistic many-body systems, turn out to be surprisingly useful for proving results about their static properties. The culmination of the course will be a full proof of one of Hastings' seminal theorems: spectral gap implies exponential decay of correlations. This will require all of the techniques and mathematical machinery we have developed during the course. If time allows, we will see how these techniques can be extended to prove many other important results relating the spectral gap to entanglement area laws, stability of quantum phases, topological order, and more.

A solid understanding of undergraduate quantum mechanics is assumed. A good grounding in linear algebra, Fourier analysis, and (very basic) functional analysis will be helpful.

The reading material and lecture notes from the Part III Mathematics Quantum Information Theory course are also relevant to this course, and you should familiarise yourself with them. (In particular the introductory notes and the first few chapters of the lecture notes.)

The following books cover all the necessary background (and much more):

- Nielsen, M. and Chuang, I., "Quantum Computation and Quantum Information", Cambridge University Press
- Schumacher, B. and Westmoreland, M. "Quantum Processes, Systems, and Information", Cambridge University Press
- John Preskill's lecture notes on quantum information theory

The Kitaev book covers most of the first half of the course. The rest of the course material is beyond the scope of current text books, but the following references may be helpful:

- Kitaev, A., Shen, A., and Vyalyi M. "Classical and Quantum Computation", American Mathematical Society
- Aharonov, D. and Naveh, T. "Quantum NP - a Survey"
- Gharibian, S., Huang, Y. and Landau, Z. "Hamiltonian Complexity"
- Matt Hastings' Les Houches summer school lecture notes

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