  We investigate a two-electron double quantum dot with both spin and valley
degrees of freedom as they occur in graphene, carbon nanotubes, or silicon, and
regard the 16-dimensional space with one electron per dot as a four-qubit logic
space. In the spin-only case, it is well known that the exchange coupling
between the dots combined with arbitrary single-qubit operations is sufficient
for universal quantum computation. The presence of the valley degeneracy in the
electronic band structure alters the form of the exchange coupling and in
general leads to spin-valley entanglement. Here, we show that universal quantum
computation can still be performed by exchange interaction and single-qubit
gates in the presence of the additional (valley) degree of freedom. We present
an explicit pulse sequence for a spin-only controlled-NOT consisting of the
generalized exchange coupling and single-electron spin and valley rotations. We
also propose state preparations and projective measurements with the use of
adiabatic transitions between states with (1,1) and (0,2) charge distributions
similar to the spin-only case, but with the additional requirement of
controlling the spin and the valley Zeeman energies by an external magnetic
field. Finally, we demonstrate a universal two-qubit gate between a spin and a
valley qubit, allowing universal gate operations on the combined spin and
valley quantum register.
