Fault-Tolerant Preparation of Quantum Polar Codes Encoding One Logical Qubit
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This paper explores a new approach to fault-tolerant quantum computing, relying on quantum polar codes. We consider quantum polar codes of Calderbank-Shor-Steane type, encoding one logical qubit, which we refer to as 𝒬_1 codes. First, we show that a subfamily of 𝒬_1 codes is equivalent to the well-known family of Shor codes. Moreover, we show that 𝒬_1 codes significantly outperform Shor codes, of the same length and minimum distance. Second, we consider the fault-tolerant preparation of 𝒬_1 code states. We give a recursive procedure to prepare a 𝒬_1 code state, based on two-qubit Pauli measurements only. The procedure is not by itself fault-tolerant, however, the measurement operations therein provide redundant classical bits, which can be advantageously used for error detection. Fault tolerance is then achieved by combining the proposed recursive procedure with an error detection method. Finally, we consider the fault-tolerant error correction of 𝒬_1 codes. We use Steane's error correction technique, which incorporates the proposed fault-tolerant code state preparation procedure. We provide numerical estimates of the logical error rates for 𝒬_1 and Shor codes of length 16 and 64 qubits, assuming a circuit-level depolarizing noise model. Remarkably, the 𝒬_1 code of length 64 qubits achieves a pseudothreshold value slightly below 1%, demonstrating the potential of polar codes for fault-tolerant quantum computing.
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