  We present methods for emulating the matter power spectrum by combining
information from cosmological $N$-body simulations at different resolutions. An
emulator allows estimation of simulation output by interpolating across the
parameter space of a limited number of simulations. We present the first
implementation in cosmology of multi-fidelity emulation, where many
low-resolution simulations are combined with a few high-resolution simulations
to achieve an increased emulation accuracy. The power spectrum's dependence on
cosmology is learned from the low-resolution simulations, which are in turn
calibrated using high-resolution simulations. We show that our multi-fidelity
emulator predicts high-fidelity counterparts to percent-level relative accuracy
when using only $3$ high-fidelity simulations and outperforms a single-fidelity
emulator that uses $11$ simulations, although we do not attempt to produce a
converged emulator with high absolute accuracy. With a fixed number of
high-fidelity training simulations, we show that our multi-fidelity emulator is
$\simeq 100$ times better than a single-fidelity emulator at $k \leq 2 h
\mathrm{Mpc}^{-1}$, and $\simeq 20$ times better at $3 \leq k < 6.4 h
\mathrm{Mpc}^{-1}$. Multi-fidelity emulation is fast to train, using only a
simple modification to standard Gaussian processes. Our proposed emulator shows
a new way to predict non-linear scales by fusing simulations from different
fidelities.
