Error mitigation for universal gates on encoded qubits

The Eastin-Knill theorem states that no quantum error correcting code can have a universal set of transversal gates. For self-dual 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.