Stochastic Methods in Quantum Mechanics. The notes are adapted from the first half my Advanced Quantum Information Theory course.

Lectured at the 2016 Autrans summer school on- Lectures 1-2: Computation and Complexity
- Lecture 3: Local Hamiltonians
- Lectures 3-4: Kitaev's Theorem
- Lecture 4: Local clock construction

The lecture notes sheets are copyright ©Toby Cubitt, and are licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

The Arora-Barak book gives an excellent, modern treatment of the theory of computation and complexity, going far beyond what's covered in this short course. The proof of Kitaev's theorem closely follows the original from the Kitaev-Schen-Vyalyi book. The other references are review papers on Hamiltonian complexity, which may also be of interest.

- Arora and Barak, "Complexity Theory: A Modern Approach, Cambridge University Press
- 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"

The following is a selective and incomplete list of links to the arXiv versions of papers that proved key results in Hamiltonian Complexity post-Kitaev. (These are the papers I mentioned in the brief survey at the very end of the lecture course.)

Proves QMA-completeness of the k-local Hamiltonian problem for \(k=3\).

Proves QMA-completeness of the k-local Hamiltonian problem for \(k=2\). Introduces the perturbation gadget technique.

Proves QMA-completeness of the k-local Hamiltonian problem for nearest-neighbour interactions (\(k=2\)) between qubits on a 2D square lattice. (Interactions are *not* translationally-invariant). Developes stronger perturbation gadget techniques.

- D. Aharonov, D. Gottesman, S. Irani and Julia Kempe, "The power of quantum systems on a line" (2007)

Proves QMA-completeness of the k-local Hamiltonian problem for nearest-neighbour interactions (\(k=2\)) bewteen qu\(d\)its on a line, for \(d=13\). (Interactions are *not* translationally-invariant.) Later improved to \(d=8\) by Nagaj et al.

Proves QMA_{EXP}-completeness of the local Hamiltonian problem for *translationally-invariant*, nearest-neighbour interactions (\(k=2\)) between qu\(d\)its on a line, with a *fixed* Hamiltonian and \(d\approx 10^6\). (The only remaining parameter in the problem is the length of the chain!)

Proves the Gottesman-Irani result for \(d=42\).

- TC, D. Perez-Garcia and M. Wolf, "Undecidability of the Spectral Gap (full version)" and (short version)" (2015)

A computability theory rather than a complexity theory result. Proves undecidability of the spectral gap problem for translationally-invariant, nearest-neighbour interactions between qu\(d\)its on a 2D square lattice in the thermodynamic limit, with \(d\approx 10^100\) (or maybe a bit smaller). Makes key use of ideas from the Gottesman-Irani result, amongst (many) other ingredients.

Proves QMA-completeness of the 2-local Hamiltonian problem for qubit Hamiltonians containing only XZ, X and Z interactions (amongst other similar results).

Proves a complete complexity classification for the k-local Hamiltonian problem with 2-qubit interactions, according to the type of interactions allowed. (A quantum analogue of Schaeffer's dichotomy theorem for boolean constraint satisfaction problems.)

Tightens the Cubitt-Montanaro classification by proving one of the four classes in the classification is equal to stoqMA (a highly non-trivial improvement!), amongst other results.

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