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 Calderbank-Shor-Steane 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, e.g., 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 Letter, we combine both approaches and show how to implement encoded $\text{Clifford}+T$ circuits where Clifford gates are protected from noise by error correction while errors introduced by noisy encoded $T$ gates are mitigated using the quasiprobability method. As a result, $\text{Clifford}+T$ 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.

Figure 1

Implementation of a $T$ gate via a magic state $|\pi /4⟩$ and Clifford operations. The $S$ and $X$ gates are only performed if the measurement outcome is 1.

Figure 2

Logical $T$ gate by code switching. (a) Surface code ${\mathcal{S}}_1$ with the distance $d=3$. Black and white circles indicate data and syndrome qubits. Red and blue faces indicate $X$ and $Z$ stabilizers. (b) Asymmetric surface code ${\mathcal{S}}_2$ exhibiting a single-qubit logical $T$ gate at the northwest corner of the lattice. Code switching ${\mathcal{S}}_1\to {\mathcal{S}}_2$ is performed by turning off $X$ stabilizer $F_1$ (dark red triangle) and turning on $Z$ stabilizer $G_1$ (dark blue triangle). (c) The syndrome extraction cycle consists of four rounds of CNOTs R1, R2, R3, R4. To avoid clutter, we only show a local schedule of CNOTs for each stabilizer. Schedules for weight-3 stabilizers are properly truncated. The schedule extends to the full lattice in a translation invariant fashion.

Figure 3

Maximal $T$ count that can be performed via the magic state method with a total sampling overhead ${\mathrm{\Gamma }}^2\in \{{10}^2,{10}^3,{10}^4\}$. We assume a depolarizing noise model with only two-qubit errors, i.e., ${\gamma }_{ϵ}$ is given in Eq. (7) for $\kappa =2/5$, where we neglect second order error terms.