Even more efficient magic state distillation by zero level distillation on the square lattice

We propose zero-level distillation as a cost-effective method to reduce the logical error rate to O(p²) in the first stage of magic state distillation (MSD). Instead of using multiple logical magic states with O(p) error at the first stage of MSD, we generate a magic state encoded in surface code with the logical error rate of O(p²) at physical level, which we call zero-level distillation.More precisely, we first encode a magic state using the Steane code and perform Hadamard-test-type distillation using a cat state. Furthermore, by utilizing lattice surgery, we can teleport the magic state from the Steane code to the surface code, thus creating a magic state on the surface code. We conducted a full simulation that originally required 47 qubits by reducing the number of qubits have to be allocated at the same time to 22 qubits. As a result, we confirmed that the logical error rate decreases with the square of the physical error rate, and when the physical error rate reaches around 10^-4, we can reduce the logical error rate several order of magnitudes.By further combining our method with existing MSD approaches, such as 15-to-1 distillation reducing logical error rate by O(p³), the proposed zero-level MDS can lower the logical error rate to O(p⁶), allowing us to quickly reach the desired error rate. Therefore, our method is expected to reduce the overhead of MSD, which is one of the major challenges in FTQC.