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

TL;DR

This work investigates fault-tolerant quantum computation via CSS quantum polar codes encoding a single logical qubit ($\\mathcal{Q}_1$). It shows a subfamily of $\\mathcal{Q}_1$ codes is equivalent to Shor codes, yet $\\mathcal{Q}_1$ generally yields lower logical error rates at the same length and distance due to channel polarization and SC decoding; it also develops a recursive, measurement-based state-preparation scheme with error-detection gadgets to achieve fault tolerance, and applies Steane error correction to quantify logical error rates under circuit-level depolarizing noise. Numerical results for code lengths $N=16$ and $N=64$ indicate that $\\mathcal{Q}_1$ can achieve very low LERs (near $10^{-6}$ at $p=10^{-3}$ for $N=64$), highlighting the potential of polar-code–based FTQC. The work integrates classical and quantum polar code theories, explicit code-state preparation methods, and practical error-correction performance analyses to advance fault-tolerant quantum computation using structured polar codes.

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.

Figures (17)

• Figure 1: The quantum polar code, with $N=2^3$, frozen sets $\mathcal{Z}=\{1,2,3\}$, $\mathcal{X}=\{6,7,8\}$ (chosen only for the simplicity of illustration), and frozen states ${\lvert\bm{u}\rangle}_\mathcal{Z} ={\lvert0,0,0\rangle}$, and ${\lvert\bm{\overline{v}}\rangle}_\mathcal{X}= {\lvert+,+,+\rangle}$.
• Figure 1: $(a)$ Polar transform recursion: $P_N$ in terms of $P_{N/2}$. $(b)$ Example of a classical polar code encoding $|\mathcal{I}| = 5$ bits into $N = 2^3$ bits, with frozen set $\mathcal{F}= \{1,2,3\}$, and frozen vector $u_\mathcal{F}=(0,0,0)$. Here, the set $\mathcal{F}$ (thus, $\mathcal{I}$) is chosen only for the purpose and the simplicity of the illustration. In general, it needs not consist of consecutive positions.
• Figure 2: LER of $\mathcal{Q}_1$ and Shor codes, for the depolarizing channel.
• Figure 2: Classical polar codes in (a) $Z$ basis, and (b) $X$ basis, induced by the CSS quantum polar code $\mathcal{Q}(N,\mathcal{Z},\mathcal{X},{\lvert\bm{u}\rangle}_\mathcal{Z} ,{\lvert\bm{\overline{v}}\rangle}_\mathcal{X})$. The permutation $\pi$ in (b) is the reverse order permutation. (Note that in this example the two codes are actually the same.)
• Figure 3: Measurement-based preparation of ${\lvert q_N\rangle}_\mathcal{S}$ in (\ref{['eq:q-prep-state']}), with $N = 8, i(n) = 3$, where slightly flattened circles connected by a vertical wire denote either a $X \otimes X$ or a $Z \otimes Z$ measurement on the corresponding qubits, and ${\lvert q_{2^k}\rangle}$ are equivalent $\mathcal{Q}_1$ states of length $2^k$ (${\lvert q_{2^0}\rangle}$ is a Pauli $Z$ basis state).

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Lemma 1
• Theorem 3
• Lemma 2
• proof
• Lemma 3
• Lemma 4
• proof
• Lemma 5