A robust phase of continuous transversal gates in quantum stabilizer codes

Abstract

A quantum error correcting code protects encoded logical information against errors. Transversal gates are a naturally fault-tolerant way to manipulate logical qubits but cannot be universal themselves. Protocols such as magic state distillation are needed to achieve universality via measurements and postselection. A phase is a region of parameter space with smoothly varying large-scale statistical properties except at its boundaries. Here, we find a phase of continuously tunable logical unitaries for the surface code implemented by transversal operations and decoding that is robust against dephasing errors. The logical unitaries in this phase have an infidelity that is exponentially suppressed in the code distance compared to their rotation angles. We exploit this to design a simple fault-tolerant protocol for continuous-angle logical rotations. This lowers the overhead for applications requiring many small-angle rotations such as quantum simulation.

Figures (4)

• Figure 1: (a) A CSS code state $\ket{\overline{\psi}}$ subjected to uniform coherent $Z$ rotations by $\theta$, followed by projective measurement of $X$-checks $\Pi_s$ to give a syndrome $s$ and application of the correction $C_s$ from the decoder results in a logical channel that is a known coherent $Z$ rotation by angle $\phi_s(\theta)$. (b) Repeating $t$ rounds of coherent rotation by optimally chosen physical rotation angles $\theta_j$ with error correction implements a potentially non-Clifford logical rotation by $\Phi_t=\theta_1+\cdots + \theta_t$. The classical angle optimizer (Opt.) is shaded in pink. (c) The $d=3$ square rotated surface code with qubits on vertices, $X$-checks on dark faces, and $Z$-checks generators on light faces.
• Figure 2: (a) Phase diagram in physical chanel parameter space $(p,\theta)$. In the robust logical coherent phase (shaded pink), the mean relative dephasing $\mathbb{E}[q_s/|\phi_s|]\to 0$ as $d\to\infty$ using the PyMatching decoder higgottSparseBlossomCorrecting2023. It is contained within the correctable phase for conventional QEC storage (shaded green) where $\left\|\overline{\mathcal{E}}\ - \mathcal{I}\right\|_\diamond\to 0$ as $d\to\infty$. (b) For a fixed physical dephasing $p=0.001$, the mean relative dephasing has a minimum value at a $\theta$ in the robust phase that is bounded from above and below by phase transitions. (c) The logical relative dephasing at the target angle that succeeds with 50% probability is exponentially suppressed with the distance of the code.
• Figure 3: Performance of the adaptive protocol with resets for preparing magic states with logical target angle $\Phi_T$ at physical dephasing rate $p=0.001$. (a) The expected number of rounds $\mathbb{E}[T]$ including resets is independent of the tolerance $\epsilon$, but increases with target angle $\Phi_T$. (b) The total logical dephasing $\mathbb{E}[Q_T]$ increases with the target angle $\Phi_T$. (c) There is a target angle where the relative logical dephasing $\mathbb{E}[Q_T/|\Phi_T|]$ is minimal. The dashed lines indicate the target $\Phi_T$ at which the protocol will succeed in one just round with 50% probability with the color corresponding to the code distance.
• Figure 4: Repeated rounds of measurement to tolerate measurement faults.