## Many bounded versions of undecidable problems are NP-hard

Andreas Klingler, Mirte van der Eyden, Sebastian Stengele, Tobias Reinhart, Gemma de las Cuevas

SciPost Phys. 14, 173 (2023) · published 29 June 2023

- doi: 10.21468/SciPostPhys.14.6.173
- Submissions/Reports

### Abstract

Several physically inspired problems have been proven undecidable; examples are the spectral gap problem and the membership problem for quantum correlations. Most of these results rely on reductions from a handful of undecidable problems, such as the halting problem, the tiling problem, the Post correspondence problem or the matrix mortality problem. All these problems have a common property: they have an NP-hard bounded version. This work establishes a relation between undecidable unbounded problems and their bounded NP-hard versions. Specifically, we show that NP-hardness of a bounded version follows easily from the reduction of the unbounded problems. This leads to new and simpler proofs of the NP-hardness of bounded version of the Post correspondence problem, the matrix mortality problem, the positivity of matrix product operators, the reachability problem, the tiling problem, and the ground state energy problem. This work sheds light on the intractability of problems in theoretical physics and on the computational consequences of bounding a parameter.

### Authors / Affiliation: mappings to Contributors and Organizations

See all Organizations.-
^{1}Andreas Klingler, -
^{1}Mirte van der Eyden, -
^{1}Sebastian Stengele, -
^{1}Tobias Reinhart, -
^{1}Gemma de las Cuevas

^{1}Institut für Theoretische Physik / Institute for Theoretical Physics, University of Innsbruck [ITP]

- Austrian Science Fund (FWF) (through Organization: Fonds zur Förderung der wissenschaftlichen Forschung / FWF Austrian Science Fund [FWF])
- Österreichischen Akademie der Wissenschaften (through Organization: Österreichische Akademie der Wissenschaften / Austrian Academy of Sciences [ÖAW])