Open Questions

APS Conjectures

[APS25] conjecture the following upper bounds on the maximum eigenvalues of \(2\)-Local Hamiltonians for all weighted graphs

\[\lambda_{max} (H^{QMC}(G)) \le W(G) + M(G)\] \[\lambda_{max} (H^{EPR}(G)) \le W(G) + M(G)\] \[\lambda_{max} (H^{XY}(G)) \le W(G) + \tfrac{M(G)}{2}.\]

Supporting numerical evidence is given in [APS25] and [TZ25b]. The conjecture could be related to Brouwer’s Conjecture.

Approximability of XY

Are there any hardness of approximability results for XY analagous to the \(0.956\) barrier for QMC from [HNPTW22]?

EPR

Is finding the maximum energy of the EPR Hamiltonian in \(P\)? Is it \(StoqMA\)-complete?