It is a long-standing goal to find problems with rigorous quantum advantage. In classical optimization, evidence is dubious. However, when the output is a quantum state, there is (intuitively) more hope. These quantum optimization problems can be formulated as follows: given a local Hamiltonian of $$n$$ qubits

$$ H = \sum_{i=1}^m H_i , , $$

approximate the maximum energy (eigenvalue) of $$H$$. Here, we require the $$H_i$$ act nontrivially on a constant number of qubits and have norm $$\mathcal{O}(1)$$.

A seminal result from [KSV02] showed that approximating the maximum energy to inverse polynomial accuracy in $$n$$ is $$QMA$$-complete. A sequence of works then showed that restricted versions of the Hamiltonian, i.e. to specific choices of $$H_i$$, remain $$QMA$$-complete. Simpler Hamiltonians with hardness guarantees often allow us to tailor algorithms to specific instances of hard problems, thus probing more concretely the approximatability of these problems. This is in direct analogy to the field of classical optimization, where problems such as max-$$k$$-SAT are often used as canonical examples for studying algorithms and complexity.

One such canonical example for quantum optimzation was shown in [PM15] to be $$QMA$$-complete

$$ H = \sum_{(i,j)\in E} \frac{w_{ij}}{2}\left( I_i \otimes I_j - X_i \otimes X_j - Y_i \otimes Y_j - Z_i \otimes Z_j \right) , ,$$

where $$E, w_{ij}\ge 0$$ denote the set of edges in some graph $$G$$ and their respective weights. This problem was formally introduced in [GP19] as the quantum MaxCut (QMC) problem. The name is due to an analogy to the classical MaxCut problem, where a single term on an edge of the graph is a rank-2 projector of the vertices in the edge onto the classical antisymmetric bitstrings

$$H_i = \vert 01\rangle \langle 01\vert + \vert 10 \rangle \langle 10 \vert , .$$

In contrast, quantum MaxCut projects each edge onto the antisymmetric quantum singlet state $$\vert \psi^{-} \rangle = \frac{1}{\sqrt{2}} (\vert 01\rangle - \vert 10\rangle)$$

$$ H_i = \vert \psi^{-} \rangle \langle \psi^{-} \vert , .$$

Much like how $$\vert 01\rangle$$ and $$\vert 10\rangle$$ are the unique antisymmetric computational basis states on two qubits, $$\vert \psi^{-} \rangle$$ is the unique antisymmetric Bell basis state on two qubits. Indeed, one may study Hamiltonians consisting of projectors onto symmetric Bell basis states. This is known as the EPR Hamiltonian [King23].

In this page, you will find descriptions of QMC, EPR, as well as other well-studied quantum optimization problems.

This reference is organized into the following sections:

1. Problems: contains a page for each class of $$2$$-Local Hamiltonians (currently QMC, EPR, and XY) containing

   • Definition: a definition of the problem.
   • Hardness: complexity-theoretic hardness of the problem
   • Algorithms: a table of all known approximation algorithms for the problem with worst-case guarantees, along with a link to the relevant paper and techniques used to derive approximation ratios. Sorted from worst to best.
   • Normalizations and conventions: description of all known normalizations and conventions (coefficient in front of each term in the Hamiltonian) in current literature. Includes names of corresponding problems in statistical physics.
   • Known cases: list of classes of graphs where the maximum energies are known, either analytically or numerically.
   • Classical formulation: a classical description of the problem in terms of the token graphs of $$G$$.
   • Average case: summary of results for algorithms on average-case instances of Hamiltonians, defined over random ensembles of graphs (typically regular graphs).
   • Remarks: assorted remarks and comments about the problem.

2. Techniques: contains a landing page summarizing different analytic techniques used to prove approximation, along with dedicated pages for

   • Lower bounds: list of methods for providing algorithmic lower bounds on the maximum energies of Hamiltonians.
   • Upper bounds: list of methods for providing upper bounds on maximum energies of Hamiltonians.
   • Analysis: summary of techniques for relating lower and upper bounds in order to derive approximation ratios
   • Token Graphs: description of the classical formulation of the Hamiltonians in terms of token graphs.

3. Open questions: a dedicated page for collecting open questions and unproven conjectures.

4. Bibliography: all references used in this website, along with a downloaded .bib file for ease of citation in future work.

This page is inspired by conversations with many people, including Joao Basso, Nathan Ju, Bobak Kiani, Robbie King, Eunou Lee, and Ojas Parekh.

More to add? Email Kunal Marwaha kmarw@uchicago.edu or James Sud jsud@uchicago.edu or open an issue on GitHub.