  Hyperfine interaction (HF) is of key importance for the functionality of
solid-state quantum information processing, as it affects qubit coherence and
enables nuclear-spin quantum memories. In this work, we complete the theory of
the basic hyperfine interaction mechanisms (Fermi contact, dipolar, orbital) in
carbon nanotube quantum dots by providing a theoretical description of the
orbital HF. We find that orbital HF induces an interaction between the nuclear
spins of the nanotube lattice and the valley degree of freedom of the electrons
confined in the quantum dot. We show that the resulting
nuclear-spin--electron-valley interaction (i) is approximately of Ising type,
(ii) is essentially local, in the sense that a radius- and
dot-length-independent atomic interaction strength can be defined, and (iii)
has an atomic interaction strength that is comparable to the combined strength
of Fermi contact and dipolar interactions. We argue that orbital HF provides a
new decoherence mechanism for single-electron valley qubits and spin-valley
qubits in a range of multi-valley materials. We explicitly evaluate the
corresponding inhomogeneous dephasing time $T_2^*$ for a nanotube-based valley
qubit.
