FaultTolerant Preparation of Quantum Polar Codes Encoding One Logical Qubit
Abstract
This paper explores a new approach to faulttolerant quantum computing, relying on quantum polar codes. We consider quantum polar codes of CalderbankShorSteane type, encoding one logical qubit, which we refer to as $\mathcal{Q}_1$ codes. First, we show that a subfamily of $\mathcal{Q}_1$ codes is equivalent to the wellknown family of Shor codes. Moreover, we show that $\mathcal{Q}_1$ codes significantly outperform Shor codes, of the same length and minimum distance. Second, we consider the faulttolerant preparation of $\mathcal{Q}_1$ code states. We give a recursive procedure to prepare a $\mathcal{Q}_1$ code state, based on twoqubit Pauli measurements only. The procedure is not by itself faulttolerant, 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 faulttolerant error correction of $\mathcal{Q}_1$ codes. We use Steane's error correction technique, which incorporates the proposed faulttolerant code state preparation procedure. We provide numerical estimates of the logical error rates for $\mathcal{Q}_1$ and Shor codes of length $16$ and $64$ qubits, assuming a circuitlevel depolarizing noise model. Remarkably, the $\mathcal{Q}_1$ code of length $64$ qubits achieves a pseudothreshold value slightly below $1\%$, demonstrating the potential of polar codes for faulttolerant quantum computing.
 Publication:

arXiv eprints
 Pub Date:
 September 2022
 arXiv:
 arXiv:2209.06673
 Bibcode:
 2022arXiv220906673G
 Keywords:

 Quantum Physics;
 Computer Science  Information Theory