  Solid state quantum bits are promising candidates for the realization of a
{\em scalable} quantum computer. However, they are usually strongly limited by
decoherence due to the many extra degrees of freedom of a solid state system.
We investigate a system of two solid state qubits that are coupled via
$\sigma_z^{(i)} \otimes \sigma_z^{(j)}$ type of coupling. This kind of setup is
typical for {\em pseudospin} solid-state quantum bits such as charge or flux
systems. We evaluate decoherence properties and gate quality factors in the
presence of a common and two uncorrelated baths coupling to $\sigma_z$,
respectively. We show that at low temperatures, uncorrelated baths do degrade
the gate quality more severely. In particular, we show that in the case of a
common bath, optimum gate performance of a CPHASE gate can be reached at very
low temperatures, because our type of coupling commutes with the coupling to
the decoherence, which makes this type of coupling attractive as compared to
previously studied proposals with $\sigma_y^{(i)} \otimes \sigma_y^{(j)}$
-coupling. Although less pronounced, this advantage also applies to the CNOT
gate.
