  Multi-objective optimization is a ubiquitous problem that arises naturally in
many scientific and industrial areas. Network routing optimization with
multi-objective performance demands falls into this problem class, and finding
good quality solutions at large scales is generally challenging. In this work,
we develop a scheme with which near-term quantum computers can be applied to
solve multi-objective combinatorial optimization problems. We study the
application of this scheme to the network routing problem in detail, by first
mapping it to the multi-objective shortest path problem. Focusing on an
implementation based on the quantum approximate optimization algorithm (QAOA)
-- the go-to approach for tackling optimization problems on near-term quantum
computers -- we examine the Pareto plot that results from the scheme, and
qualitatively analyze its ability to produce Pareto-optimal solutions. We
further provide theoretical and numerical scaling analyses of the resource
requirements and performance of QAOA, and identify key challenges associated
with this approach. Finally, through Amazon Braket we execute small-scale
implementations of our scheme on the IonQ Harmony 11-qubit quantum computer.
