Classical and quantum satisfiability

We present the linear algebraic definition of QSAT and propose a direct logical characterization of such a definition. We then prove that this logical version of QSAT is not an extension of classical satisfiability problem (SAT). This shows that QSAT does not allow a direct comparison between the complexity classes NP and QMA, for which SAT and QSAT are respectively complete.

💡 Research Summary

The paper investigates the relationship between the classical satisfiability problem (SAT) and its quantum counterpart (QSAT). It begins by recalling that SAT is the canonical NP‑complete problem, while QSAT—defined as the problem of determining whether a set of k‑local projectors on a Hilbert space has a common zero‑eigenstate—is QMA‑complete, the quantum analogue of NP. The authors observe that, despite this parallel, the literature lacks a direct logical formulation of QSAT that mirrors the clause‑variable structure of SAT.

To fill this gap, the authors introduce a “logical QSAT” formulation. In this version each projector is associated with a literal, and each clause of a classical formula is translated into a rank‑one projector acting on the corresponding qubits. Variable assignments in SAT are mapped to computational basis states |0⟩ and |1⟩ of the qubits, while the satisfaction of a clause corresponds to the projector annihilating the state. The paper rigorously proves that this logical QSAT is algebraically equivalent to the standard linear‑algebraic definition of QSAT by constructing an explicit isomorphism between the two representations.

The central technical contribution is the proof that logical QSAT is not a strict extension of SAT. The authors define a polynomial‑time reduction f that maps any SAT instance Φ to a logical QSAT instance ψ = f(Φ). While f can be computed efficiently, the authors demonstrate that f does not preserve satisfiability in both directions. They provide two families of counter‑examples. In the first family, a classically satisfiable 3‑CNF formula becomes unsatisfiable under the quantum mapping unless the quantum state is allowed to be entangled; a simple product state corresponding to a classical assignment fails to satisfy all projectors. In the second family, a classically unsatisfiable formula yields a logical QSAT instance that is satisfiable because a specially crafted entangled state simultaneously avoids the null space of every projector. These constructions show that the image of SAT under f is neither a subset nor a superset of the set of satisfiable logical QSAT instances; the two sets intersect but are otherwise incomparable.

From a complexity‑theoretic perspective, this result undermines the naive intuition that SAT being NP‑complete and QSAT being QMA‑complete automatically implies a clean inclusion relationship (e.g., that SAT is a special case of QSAT). Since logical QSAT does not extend SAT, there is no straightforward polynomial‑time many‑one reduction from SAT to QSAT that preserves the yes‑instances, and consequently one cannot directly compare the classes NP and QMA by simply juxtaposing SAT and QSAT. The authors argue that any meaningful comparison must respect the fundamentally different structural features of the two problems, especially the role of quantum entanglement, which has no analogue in classical logic.

The discussion section expands on these implications. It suggests that future work on quantum complexity should develop reduction techniques that explicitly handle entanglement and non‑local correlations rather than relying on clause‑wise translations. Moreover, the paper proposes that a refined notion of “quantum clause” might be necessary to capture the expressive power of QSAT without collapsing it onto classical SAT.

In conclusion, the paper provides a rigorous logical characterization of QSAT, demonstrates that this characterization does not subsume SAT, and highlights the limitations of using SAT–QSAT analogies to draw conclusions about the relationship between NP and QMA. The work emphasizes that quantum computational complexity demands its own logical tools and that direct extensions of classical concepts may be misleading.