  In this work, we develop the theory of quasi-exact fault-tolerant quantum
(QEQ) computation, which uses qubits encoded into quasi-exact quantum
error-correction codes ("quasi codes"). By definition, a quasi code is a
parametric approximate code that can become exact by tuning its parameters. The
model of QEQ computation lies in between the two well-known ones: the usual
noisy quantum computation without error correction and the usual fault-tolerant
quantum computation, but closer to the later. Many notions of exact quantum
codes need to be adjusted for the quasi setting. Here we develop quasi
error-correction theory using quantum instrument, the notions of quasi
universality, quasi code distances, and quasi thresholds, etc. We find a wide
class of quasi codes which are called valence-bond-solid codes, and we use them
as concrete examples to demonstrate QEQ computation.
