We study the computational complexity of the Local Hamiltonian problem under the promise that its ground state is succinctly represented. We show that the Succinct State 2-Local Hamiltonian problem, for qubit Hamiltonians, is (promise) MA-complete. The approach combines a systematic characterisation of succinct quantum states, defined through arithmetic over specific number fields, with a refined reduction that lowers the locality of Feynman-Kitaev circuit-Hamiltonians from 6 to 2, without increasing particle dimension. This reveals a complexity phase transition, parameterised by locality, and extends the scope of previously known MA-complete problem instances. Our results further clarify how succinctness behaves under circuit-based constructions, and progresses toward a better understanding of the boundary between efficiently describable and efficiently verifiable quantum systems.

翻译：我们研究了在基态被简洁表示的承诺下，局域哈密顿量问题的计算复杂性。我们证明，对于量子比特哈密顿量，简洁态2-局域哈密顿量问题是（承诺）MA完全的。该方法结合了通过特定数域上的算术定义的简洁量子态的系统性刻画，以及一个精细化的归约，该归约将费曼-北泽电路-哈密顿量的局域性从6降至2，且不增加粒子维度。这揭示了由局域性参数化的复杂性相变，并扩展了先前已知MA完全问题实例的范围。我们的结果进一步阐明了简洁性在基于电路构造中的行为，并推进了对可有效描述与可有效验证的量子系统之间边界的理解。