  We present a new quantum adiabatic theorem that allows one to rigorously
bound the adiabatic timescale for a variety of systems, including those
described by unbounded Hamiltonians. Our bound is geared towards the qubit
approximation of superconducting circuits, and presents a sufficient condition
for remaining within the $2^n$-dimensional qubit subspace of a circuit model of
$n$ qubits. The novelty of this adiabatic theorem is that unlike previous
rigorous results, it does not contain $2^n$ as a factor in the adiabatic
timescale, and it allows one to obtain an expression for the adiabatic
timescale independent of the cutoff of the infinite-dimensional Hilbert space
of the circuit Hamiltonian. As an application, we present an explicit
dependence of this timescale on circuit parameters for a superconducting flux
qubit, and demonstrate that leakage out of the qubit subspace is inevitable as
the tunnelling barrier is raised towards the end of a quantum anneal. We also
discuss a method of obtaining a $2^n\times 2^n$ effective Hamiltonian that best
approximates the true dynamics induced by slowly changing circuit control
parameters.
