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This paper addresses the problem of finding the depth overhead that will beincurred when running quantum circuits on near-term quantum computers.Specifically, it is envisaged that near-term quantum computers will have lowqubit connectivity: each qubit will only be able to interact with a subset ofthe other qubits, a reality typically represented by a qubit interaction graphin which a vertex represents a qubit and an edge represents a possible direct2-qubit interaction (gate). Thus the depth overhead is unavoidably incurred byintroducing swap gates into the quantum circuit to enable general qubitinteractions. This paper proves that there exist quantum circuits where a depthoverhead in $\Omega(\log n)$ must necessarily be incurred when running quantumcircuits with $n$ qubits on quantum computers whose qubit interaction graph hasfinite degree, but that such a logarithmic depth overhead is achievable. Thelatter is shown by the construction of a 4-regular qubit interaction graph andassociated compilation algorithm that can execute any quantum circuit with onlya logarithmic depth overhead.

On the depth overhead incurred when running quantum algorithms on near-term quantum computers with limited qubit connectivity