  Single-photon detectors based on the superconducting transition-edge sensor
(TES) are used in a number of visible to near-infrared (VNIR) applications,
particularly for photon-number-resolving measurements in quantum information
science. To be practical for large-scale photonic quantum computing or for
future spectroscopic imaging applications in astronomy, the size of VNIR TES
arrays must be increased from a few pixels to many thousands. Historically, TES
arrays have been read out with multiplexed superconducting quantum interference
devices (SQUIDs), but the microsecond-duration pulse signals of VNIR TESs are
notoriously difficult to multiplex. In this manuscript, we introduce the
kinetic inductance current sensor (KICS), a more readily scalable readout
technology that exploits the nonlinear kinetic inductance in a superconducting
resonator to make sensitive current measurements. KICS devices can replace
SQUIDs for many applications because of their ability to measure fast, high
slew-rate signals, their compatibility with standard microwave
frequency-division multiplexing techniques, and their relatively simple
fabrication. Here, we demonstrate the readout of a VNIR TES using a KICS with
$3.7$ $\text{MHz}$ of bandwidth. We measure a readout noise of $1.4$
$\text{pA}/\sqrt{\text{Hz}}$, considerably below the TES noise at frequencies
of interest, and a TES energy resolution of $(0.137 \pm 0.001)$ $\text{eV}$ at
$0.8$ $\text{eV}$, comparable to resolutions observed with non-multiplexed
SQUID readouts.
