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

TL;DR

This paper investigates the unavoidable depth overhead caused by limited qubit connectivity in near-term quantum computers, proving a logarithmic lower bound and presenting a graph and algorithm achieving this bound.

Contribution

It establishes a logarithmic lower bound on depth overhead for limited connectivity and provides a construction and algorithm that attain this bound.

Findings

Depth overhead is at least logarithmic in the number of qubits for finite degree graphs.

A 4-regular interaction graph can support any circuit with logarithmic depth overhead.

An algorithm is proposed that achieves the logarithmic overhead bound.

Abstract

This paper addresses the problem of finding the depth overhead that will be incurred when running quantum circuits on near-term quantum computers. Specifically, it is envisaged that near-term quantum computers will have low qubit connectivity: each qubit will only be able to interact with a subset of the other qubits, a reality typically represented by a qubit interaction graph in which a vertex represents a qubit and an edge represents a possible direct 2-qubit interaction (gate). Thus the depth overhead is unavoidably incurred by introducing swap gates into the quantum circuit to enable general qubit interactions. This paper proves that there exist quantum circuits where a depth overhead in $\Omega(\log n)$ must necessarily be incurred when running quantum circuits with $n$ qubits on quantum computers whose qubit interaction graph has finite degree, but that such a logarithmic depth overhead is achievable. The latter is shown by the construction of a 4-regular qubit interaction graph and associated compilation algorithm that can execute any quantum circuit with only a logarithmic depth overhead.