  The Swap gate is a ubiquitous tool for moving information on quantum
hardware, yet it can be considered a classical operation because it does not
entangle product states. Genuinely quantum operations could outperform Swap for
the task of permuting qubits within an architecture, which we call routing. We
consider quantum routing in two models: (1) allowing arbitrary two-qubit
unitaries, or (2) allowing Hamiltonians with norm-bounded interactions. We
lower bound the circuit depth or time of quantum routing in terms of spectral
properties of graphs representing the architecture interaction constraints, and
give a generalized upper bound for all simple connected $n$-vertex graphs. In
particular, we give conditions for a superpolynomial classical-quantum routing
separation, which exclude graphs with a small spectral gap and graphs of
bounded degree. Finally, we provide examples of a quadratic separation between
gate-based and Hamiltonian routing models with a constant number of local
ancillas per qubit and of an $\Omega(n)$ speedup if we also allow fast local
interactions.
