Qubit Routing for (Almost) Free

TL;DR

This paper proves bounds on CNOT gate counts for synthesizing n-qubit phase polynomials and shows that, with certain restrictions, qubit routing can be achieved with minimal overhead due to the universality of phase polynomials and Hadamard gates.

Contribution

It provides mathematical bounds on CNOT gate complexity and demonstrates near-free qubit routing under hardware restrictions using phase polynomial synthesis.

Findings

CNOT gate count bounds are between O(gn / log g) and O(gn).

Routing overhead varies from constant to O(n log^2 n) depending on hardware restrictions.

Allowed gate synthesis can eliminate routing overhead, enabling almost free qubit routing.

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.