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arXiv:2608.03825

CNOT-Distance is NP-complete under all-to-all connectivity

Antonio Acuaviva, Arturo Acuaviva, Pablo Acuaviva

quant-phcs.CCSubmitted Aug 4, 2026• Announced Aug 5, 2026• 0 citations
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Abstract

Given $A\in\operatorname{GL}(N,2)$ and an integer $K$, we ask whether $A$ can be implemented by at most $K$ CNOT gates on fixed labelled wires with all-to-all connectivity. We prove that this problem is NP-complete. From a finite simple graph $G=(V,E)$, we construct an upper-unitriangular matrix $A_G\in\operatorname{GL}(2|V|+|E|+1,2)$ satisfying $\ell_{\mathrm{CNOT}}(A_G)=2|V|+2|E|+τ(G)$, where $τ(G)$ is the minimum vertex-cover size. Each target matrix has $O(N)$ nonzero entries and row Hamming weight at most four. The lower bound unfolds an arbitrary CNOT circuit into an XOR directed acyclic graph and applies projection--contraction operations, allowing cancellation and unrestricted reuse of intermediate parities. For this family, the optimum is unchanged by any finite number of clean or borrowed ancillary wires that must be restored. A polynomial-time decoder further yields NP-hardness of approximation within every fixed additive constant and, through an L-reduction from Minimum Vertex Cover on cubic graphs, APX-hardness of the associated CNOT-circuit optimisation problem.

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Given A∈GL⁡(N,2)A\in\operatorname{GL}(N,2)A∈GL(N,2) and an integer KKK, we ask whether AAA can be implemented by at most KKK CNOT gates on fixed labelled wires with all-to-all connectivity. We prove that this problem is NP-complete. From a finite simple graph G=(V,E)G=(V,E)G=(V,E), we construct an upper-unitriangular matrix AG∈GL⁡(2∣V∣+∣E∣+1,2)A_G\in\operatorname{GL}(2|V|+|E|+1,2)AG​∈GL(2∣V∣+∣E∣+1,2) satisfying ℓCNOT(AG)=2∣V∣+2∣E∣+τ(G)\ell_{\mathrm{CNOT}}(A_G)=2|V|+2|E|+τ(G)ℓCNOT​(AG​)=2∣V∣+2∣E∣+τ(G), where τ(G)τ(G)τ(G) is the minimum vertex-cover size. Each target matrix has O(N)O(N)O(N) nonzero entries and row Hamming weight at most four. The lower bound unfolds an arbitrary CNOT circuit into an XOR directed acyclic graph and applies projection--contraction operations, allowing cancellation and unrestricted reuse of intermediate parities. For this family, the optimum is unchanged by any finite number of clean or borrowed ancillary wires that must be restored. A polynomial-time decoder further yields NP-hardness of approximation within every fixed additive constant and, through an L-reduction from Minimum Vertex Cover on cubic graphs, APX-hardness of the associated CNOT-circuit optimisation problem.
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The CNOT-complexity of an invertible matrix over F2\mathbb{F}_2F2​ is the minimum number of CNOT gates needed to synthesize the corresponding linear reversible operator. While the maximum CNOT-complexity over all n×nn \times nn×n matrices is known to be Θ(n2/log⁡n)Θ(n^2 / \log n)Θ(n2/logn), no explicit family of matrices requiring a superlinear number of CNOT gates is known, and until now the hardest explicitly known family has been the cyclic permutations, with CNOT-complexity 3(n−1)3(n-1)3(n−1). We show that lower bounds for the additive complexity of not-necessarily-reversible linear operators can be lifted to the reversible setting with only a small loss. As an application, we use this to describe an explicit family of matrices, constructed from parity-check matrices of error-correcting codes, with CNOT-complexity at least 4n−o(n)4n - o(n)4n−o(n), asymptotically surpassing the cyclic permutations. Moreover, this construction yields an explicit matrix A∈GLn(F2)A \in \mathrm{GL}_{n}(\mathbb{F}_2)A∈GLn​(F2​), n=17167n = 17167n=17167, whose CNOT-complexity exceeds that of the cyclic permutation on nnn symbols.
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