Classical spin Hamiltonians are a powerful tool to model complex systems, characterized by a local structure given by the local Hamiltonians. One of the best understood local structures is the grammar of formal languages, which are central in computer science and linguistics, and have a natural complexity measure given by the Chomsky hierarchy. If we see classical spin Hamiltonians as languages, what grammar do the local Hamiltonians correspond to? Here, we cast classical spin Hamiltonians as formal languages, and classify them in the Chomsky hierarchy. We prove that the language of (effectively) zero-dimensional spin Hamiltonians is regular, one-dimensional spin Hamiltonians is deterministic context-free, and higher-dimensional and all-to-all spin Hamiltonians is context-sensitive. This provides a new complexity measure for classical spin Hamiltonians, which captures the hardness of recognizing spin configurations and their energies. We compare it with the computational complexity of the ground state energy problem, and find a different easy-to-hard threshold for the Ising model. We also investigate the dependence on the language of the spin Hamiltonian. Finally, we define the language of the time evolution of a spin Hamiltonian and classify it in the Chomsky hierarchy. Our work suggests that universal spin models are weaker than universal Turing machines.

No 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.

We introduce a new complexity measure of classical spin Hamiltonians in which they are described as automata. Specifically, we associate a classical spin Hamiltonian to the formal language consisting of pairs of spin configurations and the corresponding energy, and classify this language in the Chomsky hierarchy. We prove that the language associated to local one-dimensional (1D) classical spin Hamiltonians is deterministic context-free, and the one associated to the two-dimensional (2D) case is context-sensitive. It follows that the Ising model without fields is easy / hard if defined on a 1D / 2D lattice, in contrast to the computational complexity of its ground state energy problem, which is easy / hard (namely in P / NP-complete) if defined on a planar / non-planar graph. We also prove that only highly non-physical spin Hamiltonians, namely totally unbounded ones, correspond to Turing machines. Our work puts classical spin Hamiltonians at the same level as automata, and paves the road toward a rigorous comparison of universal spin models and universal Turing machines.