  We consider superconducting circuits for the purpose of simulating the
spin-boson model. The spin-boson model consists of a single two-level system
coupled to bosonic modes. In most cases, the model is considered in a limit
where the bosonic modes are sufficiently dense to form a continuous spectral
bath. A very well known case is the ohmic bath, where the density of states
grows linearly with the frequency. In the limit of weak coupling or large
temperature, this problem can be solved numerically. If the coupling is strong,
the bosonic modes can become sufficiently excited to make a classical
simulation impossible. Here, we discuss how a quantum simulation of this
problem can be performed by coupling a superconducting qubit to a set of
microwave resonators. We demonstrate a possible implementation of a continuous
spectral bath with individual bath resonators coupling strongly to the qubit.
Applying a microwave drive scheme potentially allows us to access the
strong-coupling regime of the spin-boson model. We discuss how the resulting
spin relaxation dynamics with different initialization conditions can be probed
by standard qubit-readout techniques from circuit quantum electrodynamics.
