Fault-Tolerant Preparation of Quantum Polar Codes Encoding One Logical Qubit

TL;DR

This paper introduces a fault-tolerant quantum computing approach using quantum polar codes, demonstrating their advantages over Shor codes and providing a recursive, measurement-based preparation method with promising error rate performance.

Contribution

It presents a new fault-tolerant quantum code construction based on quantum polar codes, with a recursive preparation method and error correction strategy, outperforming traditional Shor codes.

Findings

Quantum polar codes outperform Shor codes of same length.

Recursive preparation method uses only two-qubit Pauli measurements.

Achieved logical error rate close to 10^{-6} at physical error rate 10^{-3}.

Abstract

This paper explores a new approach to fault-tolerant quantum computing (FTQC), relying on quantum polar codes. We consider quantum polar codes of Calderbank-Shor-Steane 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 well-known 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 fault-tolerant preparation of $\mathcal{Q}_1$ code states. We give a recursive procedure to prepare a $\mathcal{Q}_1$ 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 $\mathcal{Q}_1$ codes. We use Steane error correction, which incorporates the proposed fault-tolerant 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 circuit-level depolarizing noise model. Remarkably, the $\mathcal{Q}_1$ code of length $64$ qubits achieves a logical error rate very close to $10^{-6}$ for the physical error rate $p = 10^{-3}$, therefore, demonstrating the potential of the proposed polar codes based approach to FTQC.