  Naturally occurring background radiation is a source of correlated
decoherence events in superconducting qubits that will challenge
error-correction schemes. To characterize the radiation environment in an
unshielded laboratory, we performed broadband, spectroscopic measurements of
background events in silicon substrates located inside a millikelvin
refrigerator, an environment representative of superconducting qubit systems.
We measured the background spectra in silicon substrates of two thicknesses,
0.5 mm and 1.5 mm, and obtained the average event rate and the integrated power
deposition. In a 25 mm^2 area and the thinner substrate, these values are 0.023
events per second and 4.9 keV/s, counting events that deposit at least 40 keV.
We find the background spectrum to be nearly featureless. Its intensity
decreases by a factor of 40,000 between 100 keV and 3 MeV for silicon
substrates 0.5 mm thick. We find the cryogenic measurements to be in good
agreement with predictions based on measurements of the terrestrial gamma-ray
flux, published models of cosmic-ray fluxes, a crude model of the cryostat, and
radiation-transport simulations. No free parameters are required to predict the
background spectra in the silicon substrates. The good agreement between
measurements and predictions allow assessment of the relative contributions of
terrestrial and cosmic background sources and their dependence on substrate
thickness. Our spectroscopic measurements are performed with superconducting
microresonators that transduce deposited energy to a readily detectable
electrical signal. We find that gamma-ray emissions from radioisotopes are
responsible for the majority of events depositing E<1.5 MeV, while nucleons
among the cosmic-ray secondary particles cause most events that deposit more
energy. These results suggest several paths to reducing the impact of
background radiation on quantum circuits.
