aiqbench analysis
This preprint proves that finding the minimum-cost circuit for a Clifford operator is NP-complete under several practical gate-count models, including CNOT-only budgets, any two-qubit Clifford budgets, and cases with free SWAPs or counted single-qubit gates. The reductions from Hamiltonian path in grid graphs, which use a qubit count quadratic in the grid size, extend earlier work by Li and Guan. For compiler and hardware readers, the result is a formal reason to expect heuristics rather than exact optimal synthesis at scale, though it is an unreviewed v1 preprint.
Clifford synthesis is NP-complete
arXiv:2610.40886v1 Announce Type: new Abstract: We show that the problem of synthesizing a quantum circuit for a given Clifford operator using at most a specified amount of resources is NP-complete for several resource models of practical interest. Specifically, we study synthesis of a given Clifford operator using single-qubit Clifford gates and two-qubit Clifford gates, with full qubit connectivity and no additional ancilla qubits. With single-qubit gates free and CNOT as the only two-qubit gate, we show that deciding whether a given target admits an implementation using at most a specified number of CNOT gates is NP-complete, and we show that the problem remains NP-complete when any two-qubit Clifford gate is allowed and the budget counts all such gates. Moreover, we show that both results continue to hold when SWAP gates may be inserted for free, and when single-qubit gates are also considered part of the budget. In each case, we build on recent work by Li and Guan and show that there are polynomial-time reductions from Hamiltonian path in grid graphs to these decision problems, using a number of qubits quadratic in the number of grid vertices.
Abstract:We show that the problem of synthesizing a quantum circuit for a given Clifford operator using at most a specified amount of resources is NP-complete for several resource models of practical interest. Specifically, we study synthesis of a given Clifford operator using single-qubit Clifford gates and two-qubit Clifford gates, with full qubit connectivity and no additional ancilla qubits. With single-qubit gates free and CNOT as the only two-qubit gate, we show that deciding whether a given target admits an implementation using at most a specified number of CNOT gates is NP-complete, and we show that the problem remains NP-complete when any two-qubit Clifford gate is allowed and the budget counts all such gates. Moreover, we show that both results continue to hold when SWAP gates may be inserted for free, and when single-qubit gates are also considered part of the budget. In each case, we build on recent work by Li and Guan and show that there are polynomial-time reductions from Hamiltonian path in grid graphs to these decision problems, using a number of qubits quadratic in the number of grid vertices.
Quantum Physics (quant-ph); Computational Complexity (cs.CC)
arXiv:2610.40886 [quant-ph]
(or arXiv:2610.40886v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.40886 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 279 words