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Quantum homological complexity measure

Develop a quantum homological complexity measure h_q(L) such that BQP = { L : h_q(L) = 0 } and QMA = { L : 0 < h_q(L) < ∞ }, and prove the domination bound h_q(L) ≤ ½ · h(L) for all problems L.

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Background

Extending the classical homological framework, the authors hypothesize a quantum variant h_q capturing the algorithmic power of quantum classes. The proposed correspondences mirror classical results (P via trivial homology), assigning BQP to quantum‑trivial homology and QMA to finite nonzero quantum homology.

They also conjecture a quantitative relation h_q ≤ ½ h that aligns with known quadratic quantum speedups.

References

Conjecture [Quantum Homological Complexity] There exists a quantum homological complexity measure h_q(L) such that: BQP = {L : h_q(L) = 0} \quad QMA = {L : 0 < h_q(L) < \infty} Moreover, h_q(L) \leq \frac{1}{2}h(L) for all L. This conjecture is grounded in several deep principles:

— A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework (2510.17829 - Tang, 2 Oct 2025) in Conjecture (Quantum Homological Complexity), Subsection "Homological Theory of Quantum Computation" of Section 10 (Conclusions and Future Work)