  The Eastin-Knill theorem states that no quantum error correcting code can
have a universal set of transversal gates. For CSS codes that can implement
Clifford gates transversally it suffices to provide one additional non-Clifford
gate, such as the T-gate, to achieve universality. Common methods to implement
fault-tolerant T-gates like magic state distillation generate a significant
hardware overhead that will likely prevent their practical usage in the
near-term future. Recently methods have been developed to mitigate the effect
of noise in shallow quantum circuits that are not protected by error
correction. Error mitigation methods require no additional hardware resources
but suffer from a bad asymptotic scaling and apply only to a restricted class
of quantum algorithms. In this work, we combine both approaches and show how to
implement encoded Clifford+T circuits where Clifford gates are protected from
noise by error correction while errors introduced by noisy encoded T-gates are
mitigated using the quasi-probability method. As a result, Clifford+T circuits
with a number of T-gates inversely proportional to the physical noise rate can
be implemented on small error-corrected devices without magic state
distillation. We argue that such circuits can be out of reach for
state-of-the-art classical simulation algorithms.
