Let L be a computational problem, let h(L) be its ordinary homological complexity, and let h_q(L) be a proposed quantum homological complexity measure. Quantum homological complexity conjecture. There exists a measure h_q such that BQP=L:h_q(L)=0,qquad QMA=L:0<h_q(L)<∞, and h_q(L)≤frac12h(L) for every L. The proposed measure is intended to encode quantum complexity classes and a quadratic quantum speedup, but neither its construction nor the asserted characterizations is established in the source.

The quantum homological complexity conjecture

Let $L$ be a computational problem, let $h(L)$ be its ordinary homological complexity, and let $h_q(L)$ be a proposed quantum homological complexity measure. Quantum homological complexity conjecture. There exists a measure $h_q$ such that

\mathcal{BQP}=\{L:h_q(L)=0\},\qquad \mathcal{QMA}=\{L:0<h_q(L)<\infty\},

and

h_q(L)\leq\frac{1}{2}h(L)

for every $L$ . The proposed measure is intended to encode quantum complexity classes and a quadratic quantum speedup, but neither its construction nor the asserted characterizations is established in the source.

Sources & referencesView supporting material

Primary source

Jian-Gang Tang, “A Homological Separation of P from NP via Computational Topology and Category Theory”, arXiv:2510.17829 (2025).

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Progress summary

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Open

The conjecture remains an unproved proposal: related quantum-homology results have appeared, but nobody has constructed the required measure or verified its complexity-class claims.

The conjecture proposes a quantum homological measure $h_q$ that characterizes $\mathcal{BQP}$ and $\mathcal{QMA}$ and is at most half the ordinary measure $h$ . The available statement presents this as conjectural and gives neither a construction nor a proof.

2024–2025 related results

A 2024 paper proves $\mathsf{QMA}$ -hardness for a local Cohomology problem and $\mathsf{DQC1}$ -hardness for estimating normalized Betti numbers; some instance classes lie in $\mathsf{QMA}$ or $\mathsf{BQP}$ , but this does not establish the conjecture.
A January 2025 paper studies quantum algorithms and hardness for approximating Khovanov-homology ranks, including $\mathsf{BQP}$ -hardness and conditional quantum algorithms, without defining or proving the proposed $h_q$ .

Current status (as of August 2026): The conjecture remains open; its proposed measure, the $\mathcal{BQP}$ / $\mathcal{QMA}$ characterizations, and the bound $h_q(L)\leq \frac{1}{2}h(L)$ are all unsettled.

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