Quantum

Clifford synthesis is NP-complete

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Based on a feed excerpt; full article content has not been acquired. Source evidence retrieved 02/11/2026, 05:13:25 UTC.

Briefings

A 2026 arXiv preprint establishes that synthesizing quantum circuits for Clifford operators is NP-complete under several practical resource models. The study proves that deciding if a target circuit can be implemented using a limited number of CNOT gates is NP-complete, even when all two-qubit Clifford gates are allowed and SWAP gates are free. This result relies on polynomial-time reductions from Hamiltonian path problems in grid graphs, where the number of qubits required scales quadratically with the grid vertices.

Key points

• The problem of synthesizing a quantum circuit for a given Clifford operator is NP-complete for single-qubit gates and two-qubit CNOT gates with full connectivity.
• The problem remains NP-complete when any two-qubit Clifford gate is allowed and the budget counts all such gates, even with free SWAP insertions.

Context

This research builds on recent work by Li and Guan, demonstrating that Hamiltonian path reductions in grid graphs can be used to prove the NP-completeness of these synthesis problems. The findings are significant for quantum algorithm design as they establish computational hardness limits for resource-constrained circuit synthesis.

Limitations

The evidence is an unverified arXiv preprint (arXiv:2610.40886v1) from November 2026, so the results are preliminary. The study focuses on specific resource models (single-qubit, CNOT, SWAP) and does not address other gate types or resource constraints. The reduction proof relies on grid graph properties which may not generalize to all quantum circuit topologies.

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