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arXiv · 2604.19717

Qubit Routing for (Almost) Free

Arianne Meijer-van de Griend
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Abstract

In this paper, we give a mathematical proof that bounds the number of CNOT gates required to synthesize an $n$ qubit phase polynomial with $g$ terms to be at least $O(\frac{gn}{\max (\log g, 1)})$ and at most $O(gn)$. However, when targeting restricted hardware, not all CNOTs are allowed. If we were to use SWAP-based methods to route the qubits on the architecture such that the earlier synthesized gates are natively allowed, we increase the number of CNOTs by a routing overhead factor of $O(\log n) \leq \alpha \leq O(n \log^2 n)$. However, if we only synthesize allowed gates, we do not need to route any qubits. Moreover, in that case the routing overhead factor is $1 \leq \alpha \leq 4 \simeq O(1)$. Additionally, since phase polynomials and Hadamard gates together form a universal gate set, we get qubit routing for almost free.

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Arianne Meijer-van de Griend. 2026-04-21. Qubit Routing for (Almost) Free. https://arxiv.org/abs/2604.19717

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