Factory-based Fault-tolerant Preparation of Quantum Polar Codes Encoding One logical Qubit

International audienceA fault-tolerant way to prepare logical code-states of Q1 codes, i.e., quantum polar codes encoding one qubit, has been recently proposed. The fault tolerance therein is guaranteed with the help of an error detection gadget, where if an error is detected during the preparation, one declares a preparation failure and discards entirely the preparation. This makes the preparation probabilistic, whose success rate is referred to as the preparation rate. In this paper, to improve the preparation rate, we consider a factory preparation of Q1 code-states, where one attempts to prepare several copies of Q1 code-states in parallel. With the help of an extra scheduling step, we can avoid discarding the preparation entirely, every time an error is detected, hence, we may achieve an increased preparation rate. We further provide a theoretical method to estimate preparation and logical error rates of the Q1 codes, prepared using factory preparation, which is shown to tightly fit the Monte-Carlo simulation based numerical results. Therefore, our theoretical method is useful for providing estimates for large code-lengths, where Monte-Carlo simulations are practically not feasible. Our numerical results, for a circuit-level depolarizing noise model, indicate that the preparation rate increases significantly, especially for large N. For example, for N = 256, it increases from 0.02% to 27% for a practically interesting physical error rate of p = 10^{-3}. Remarkably, a Q1 code of length N = 256 achieves logical error rates around 10^{-11} and 10^{-15} for the physical error rates of p = 10^{-3} and p = 3 \times 10^{-4}, respectively, hence, showing the promise of the proposed scheme for large-scale fault-tolerant quantum computing