Undecidability and Intractability in Theoretical Physics

T. N. Palmer

Klaas Landsman

Andreas Klingler, Mirte van der Eyden, Sebastian Stengele +2 more

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.

Doug Doucette

This paper provides a self-contained conceptual overview of one of the most striking results in contemporary quantum many-body physics: the undecidability of the spectral gap problem. The spectral gap—the energy difference between a system's ground state and its first excited state—is a fundamental property that distinguishes insulating from conducting behavior and marks quantum phase transitions. In 2015, Cubitt, Pérez-García, and Wolf proved that deciding whether a translationally invariant quantum spin system with finite-range interactions is gapped or gapless is algorithmically undecidable. This result was extended to one-dimensional systems by Bausch and colleagues in 2020, overturning the intuition that lower-dimensional systems are inherently more tractable. The proof architecture rests on two main ingredients: aperiodic tilings and history-state constructions. Aperiodic tilings encode undecidable combinatorial properties into the spatial structure of the Hamiltonian, while history states embed the evolution of a Turing machine into the ground state. By combining these elements, the authors construct explicit Hamiltonians whose spectral gap depends directly on whether a simulated Turing machine halts—a problem known to be undecidable. Thus, no algorithm can classify every instance correctly. The paper also distinguishes two forms of undecidability: algorithmic undecidability, which asserts that no Turing machine can solve the problem for all instances, and axiomatic independence, which shows that for some Hamiltonians, the gap status is not determined by the standard axioms of mathematics. These results impose fundamental limits on numerical simulation and phase classification, but they do not render all systems intractable. They demarcate a boundary between decidable and undecidable behavior, revealing deep connections between physics, computation, and logic.