Complexity and Computability in Physics course

21 July 2018 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


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

Recommended reading

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 in the course closely follows the original from the Kitaev-Schen-Vyalyi book. The other references in this list are review papers on Hamiltonian complexity, which may also be of interest.


The following is a selective and incomplete list of links to the arXiv versions of papers that proved key results in Hamiltonian Complexity and Computability theory post-Kitaev.

QMA-completeness with stronger locality conditions, and related results

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.

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 QMAEXP-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 \(d=42\).

QMA-completeness with restricted types of interaction, and related results

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.

Undecidability in physics

Shows that questions about the dynamics of a particle bouncing around a 3D box with linear and parabolic reflectors is undecidable.

Contains all the technical details and proofs for the above result, and more.

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.

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