Quantum 2-SAT on low dimensional systems is \(\mathsf{QMA}_1\)-complete: Direct embeddings and black-box simulation

Despite the fundamental role the Quantum Satisfiability (QSAT) problem has played in quantum complexity theory, a central question remains open: At which local dimension does the complexity of QSAT transition from "easy" to "hard"? Here, we study QSAT with each constraint acting on a \(k\)-dimensional and \(l\)-dimensional qudit pair, denoted \((k,l)\)-QSAT. Our first main result shows that, surprisingly, QSAT on qubits can remain \(\mathsf{QMA}_1\)-hard, in that \((2,5)\)-QSAT is \(\mathsf{QMA}_1\)-complete. In contrast, \(2\)-SAT on qubits is well-known to be poly-time solvable [Bravyi, 2006]. Our second main result proves that \((3,d)\)-QSAT on the 1D line with \(d\in O(1)\) is also \(\mathsf{QMA}_1\)-hard. Finally, we initiate the study of 1D \((2,d)\)-QSAT by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state. Our first result uses a direct embedding, combining a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset, Nagaj, 2013]. Of note is a new simplified and analytic proof for the latter (as opposed to a partially numeric proof in [GN13]). This exploits Unitary Labelled Graphs [Bausch, Cubitt, Ozols, 2017] together with a new "Nullspace Connection Lemma", allowing us to break low energy analyses into small patches of projectors, and to improve the soundness analysis of [GN13] from \(\Omega(1/T^6)\) to \(\Omega(1/T^2)\), for \(T\) the number of gates. Our second result goes via black-box reduction: Given an arbitrary 1D Hamiltonian \(H\) on \(d'\)-dimensional qudits, we show how to embed it into an effective null-space of a 1D \((3,d)\)-QSAT instance, for \(d\in O(1)\). Our approach may be viewed as a weaker notion of "simulation" (à la [Bravyi, Hastings 2017], [Cubitt, Montanaro, Piddock 2018]). As far as we are aware, this gives the first "black-box simulation"-based \(\mathsf{QMA}_1\)-hardness result, i.e. for frustration-free Hamiltonians.