Efficient algorithm for a quantum analogue of 2-SAT
Sergey Bravyi
TL;DR
The paper defines quantum 2-SAT via forbidden subspaces represented by 2-qubit projectors and proves a polynomial-time classical algorithm to decide satisfiability, using a rank-based reduction and a local constraint-generation lemma. It then extends the complexity landscape by proving that quantum k-SAT is $\mathrm{QMA}_1$-complete for $k\ge4$, via a 4-local Hamiltonian construction that encodes quantum circuits as history states. The results illuminate when quantum satisfiability problems can be solved efficiently and establish a sharp boundary in hardness for larger k, tying quantum SAT to the quantum analogue of NP via QMA$_1$. The approach leverages both reductions that shrink the system size and a complete constraint framework that guarantees product-state solutions in the homogeneous case, alongside a robust circuit-to-Hamiltonian mapping using $4$-qubit projectors.
Abstract
Complexity of a quantum analogue of the satisfiability problem is studied. Quantum k-SAT is a problem of verifying whether there exists n-qubit pure state such that its k-qubit reduced density matrices have support on prescribed subspaces. We present a classical algorithm solving quantum 2-SAT in a polynomial time. It generalizes the well-known algorithm for the classical 2-SAT. Besides, we show that for any k>=4 quantum k-SAT is complete in the complexity class QMA with one-sided error.