Daniel Nagaj

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 QMA_1-hard, in that (2,5)-QSAT is QMA_1-complete. (QMA_1 is a quantum analogue of MA with perfect completeness.) In contrast, (2,2)-QSAT (i.e. Quantum 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 = O(1) is also QMA_1-hard. Finally, we initiate the study of (2,d)-QSAT on the 1D line by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state. As implied by our title, our first result uses a direct embedding: We combine a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset and 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 1D (3,d)-QSAT instance, for d = 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 QMA_1-hardness result.