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Phase diagrams chart material properties with respect to one or more externalor internal parameters such as pressure or magnetisation; as such, they play afundamental role in many theoretical and applied fields of science. In thiswork, we prove that provided the phase of the Hamiltonian at a finite sizereflects the phase in the thermodynamic limit, approximating the criticalboundary in its phase diagram to constant precision is$P^{QMA_{EXP}}$-complete. This holds even for translationally-invariant nearestneighbour couplings, and even if the system's phase diagram is promised to havea single critical boundary delineating two phases. For the simpler case of asingle parameter, the same problem remains $QMA_{EXP}$-hard. Our results extendthe study of quantum phases to systems with more realistic phase diagrams thanpreviously studied. Furthermore, our findings place complexity-theoreticconstraints on the effectiveness of (computational or analytic) methods basedon finite size criteria, similar in spirit to the Knabe bound, for the task ofextrapolating the properties (e.g. gapped/gapless) of a system from finite-sizeobservations to the thermodynamic limit.

The Complexity of Approximating Critical Points of Quantum Phase Transitions