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Even the Plain CNOT Circuit Problem Is NP-Hard
A reduction from Hamiltonian paths closes the complexity question for exact synthesis with labeled qubits, all-to-all links and no ancillas.
Summary
A reduction from Hamiltonian paths closes the complexity question for exact synthesis with labeled qubits, all-to-all links and no ancillas.
Earlier hardness proofs needed restricted connectivity, encoded inputs or extra intermediate variables. The new proof uses recorder qubits to force required intermediate visits into the final parity transformation, reducing a grid-graph Hamiltonian path to the vanilla synthesis problem. The decision form is NP-complete and optimization is NP-hard, with consequences for related shortest-word, Cayley-graph distance and XOR-program problems. Complexity hardness describes worst-case computation; it does not say useful circuits cannot be optimized in practice.
Why it matters
A reduction from Hamiltonian paths closes the complexity question for exact synthesis with labeled qubits, all-to-all links and no ancillas.
Limits and context
- Complexity hardness describes worst-case computation; it does not say useful circuits cannot be optimized in practice.
Key claims
A reduction from Hamiltonian paths closes the complexity question for exact synthesis with labeled qubits, all-to-all links and no ancillas.
Qualification: Complexity hardness describes worst-case computation; it does not say useful circuits cannot be optimized in practice.
Evidence: source-2026-09-06-014
Sources
- arXiv preprint 2609.04160arXiv · primary research
Corrections
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