  Models for quantum computation with circuit connections subject to the
quantum superposition principle have been recently proposed. There, a control
quantum system can coherently determine the order in which a target quantum
system undergoes $N$ gate operations. This process, known as the quantum
$N$-switch, is a resource for several information-processing tasks. In
particular, it provides a computational advantage -- over fixed-gate-order
quantum circuits -- for phase-estimation problems involving $N$ unknown unitary
gates. However, the corresponding algorithm requires an experimentally
unfeasible target-system dimension (super)exponential in $N$. Here, we
introduce a promise problem for which the quantum $N$-switch gives an
equivalent computational speed-up with target-system dimension as small as 2
regardless of $N$. We use state-of-the-art multi-core optical-fiber technology
to experimentally demonstrate the quantum $N$-switch with $N=4$ gates acting on
a photonic-polarization qubit. This is the first observation of a quantum
superposition of more than $N=2$ temporal orders, demonstrating its usefulness
for efficient phase-estimation.
