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Abstract

We study the complexity of a class of problems involving satisfying constraints which remain the same under translations in one or more spatial directions. In this paper, we show hardness of a classical tiling problem on an N × N 2-dimensional grid and a quantum problem involving finding the ground state energy of a 1-dimensional quantum system of N particles. In both cases, the only input is N, provided in binary. We show that the classical problem is NEXP-complete and the quantum problem is QMA EXP-complete. Thus, an algorithm for these problems which runs in time polynomial in N (exponential in the input size) would imply that EXP = NEXP or BQEXP = QMA EXP , respectively. Although tiling in general is already known to be NEXP-complete, the usual approach is to require that either the set of tiles and their constraints or some varying boundary conditions be given as part of the input. In the problem considered here, these are fixed, constant-sized parameters of the problem. Instead, the problem instance is encoded solely in the size of the system.

On the NP-completeness of the Hartree-Fock method for translationally invariant systems

“…However, there are some hardness results in this direction for local and translationally invariant systems. 26 FIG. 3.…”

Translationally Invariant Universal Classical Hamiltonians

“…As in Theorem 3.5, the two body interaction h makes up a standard form Hamiltonian which encodes a QTM M 1 dovetailed with the phase-estimation computation from Lemma 3.2. It is based on the construction from [GI09].…”

“…Gottesman and Irani proved in 2009 that TI-L H (poly, 1) is QMA EXPcomplete [GI09], which has since been generalized to systems with lower local dimension [BCO17; BP17], variants of which again introduce a polynomially-scaling local coupling strength. We emphasize that while Gottesman and Irani's definition restricts the bit precision Σ to be constant, the input size to the problem-namely the chain length N-is already of size log N. A poly-time reduction thus does not change the complexity class, and allowing matrix entries of size poly log N is arguably natural.…”

The complexity of translationally invariant low-dimensional spin lattices in 3D

“…that the interactions between equivalent lattice sides are identical. This allows us to define the following variant of the local Hamiltonian problem, where we follow the naming convention in [13], i.e. we abbreviate translationally invariant local Hamiltonian as TILH.…”