  Detection mechanisms for low mass bosonic dark matter candidates, such the
axion or hidden photon, leverage potential interactions with electromagnetic
fields, whereby the dark matter (of unknown mass) on rare occasion converts
into a single photon. Current dark matter searches operating at microwave
frequencies use a resonant cavity to coherently accumulate the field sourced by
the dark matter and a near standard quantum limited (SQL) linear amplifier to
read out the cavity signal. To further increase sensitivity to the dark matter
signal, sub-SQL detection techniques are required. Here we report the
development of a novel microwave photon counting technique and a new exclusion
limit on hidden photon dark matter. We operate a superconducting qubit to make
repeated quantum non-demolition measurements of cavity photons and apply a
hidden Markov model analysis to reduce the noise to 15.7 dB below the quantum
limit, with overall detector performance limited by a residual background of
real photons. With the present device, we perform a hidden photon search and
constrain the kinetic mixing angle to $\epsilon \leq 1.68 \times 10^{-15}$ in a
band around 6.011 GHz (24.86 $\mu$eV) with an integration time of 8.33 s. This
demonstrated noise reduction technique enables future dark matter searches to
be sped up by a factor of 1300. By coupling a qubit to an arbitrary quantum
sensor, more general sub-SQL metrology is possible with the techniques
presented in this work.
