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Quantum homological obstruction

Prove that for any language L decidable in bounded‑error quantum polynomial time (BQP), the homological complexity satisfies h(L) ≤ 2, i.e., quantum computers cannot efficiently solve problems whose homological complexity exceeds 2.

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Background

The paper proposes an intrinsic topological limitation on quantum computation within the homological framework: that BQP algorithms can only resolve problems whose computational complexes have nontrivial homology in dimensions at most two.

The conjecture is motivated by representations of quantum computation via 2D topological quantum field theories and by evidence that higher‑dimensional topological features are not efficiently accessible to standard quantum models.

References

Conjecture [Quantum Homological Obstruction] If L ∈ BQP, then h(L) ≤ 2. That is, quantum computers cannot efficiently solve problems with homological complexity greater than 2. This conjecture is based on fundamental limitations of quantum mechanics:

— A Homological Proof of $\mathbf{P} \neq \mathbf{NP}$: Computational Topology via Categorical Framework (2510.17829 - Tang, 2 Oct 2025) in Conjecture (Quantum Homological Obstruction), Subsection "Relations with Quantum Complexity Theory" of Section 9 (Connections with Existing Theories)