Fault tolerance of stabilizer channels

TL;DR

This work develops a rigorous, general formalism to analyze fault tolerance of stabilizer channels under broad noise models by viewing any stabilizer channel as a sequence of code-deformation rounds and introducing the fault-distance $d(\mathcal F)$ and generalized hook faults. It then provides a comprehensive analytic framework (code-deformation perspective, channel checks, and logical-action representations) and practical algorithms to compute fault distance and identify problematic hook faults, including graph-based methods for graph-like channel checks and extensions beyond graphs. The authors demonstrate the approach with surface-code examples (e.g., XX measurement and Hadamard channel), revealing both spatial/temporal variations and the need for carefully designed, stage-wise implementations to preserve full distance. The framework is broadly applicable to stabilizer codes, including surface codes, Floquet codes, and LDPC codes, and lays groundwork for future extensions to non-Clifford operations, adaptive protocols, and scalable decoders with potential impact on overhead and reliability in fault-tolerant quantum computing.

Abstract

Stabilizer channels are stabilizer circuits that implement logical operations while mapping from an input stabilizer code to an output stabilizer code. They are widely used to implement fault tolerant error correction and logical operations in stabilizer codes such as surface codes and LDPC codes, and more broadly in subsystem, Floquet and space-time codes. We introduce a rigorous and general formalism to analyze the fault tolerance properties of any stabilizer channel under a broad class of noise models. This includes rigorous but easy-to-work-with definitions and algorithms for the fault distance and hook faults for stabilizer channels. The generalized notion of hook faults which we introduce, defined with respect to an arbitrary subset of a circuit's faults rather than a fixed phenomenological noise model, can be leveraged for fault-tolerant circuit design. Additionally, we establish necessary conditions such that channel composition preserves the fault distance. We apply our framework to design and analyze fault tolerant stabilizer channels for surface codes, revealing novel aspects of fault tolerant circuits.

Figures (24)

• Figure 1: Any stabilizer circuit $\mathcal{C}$ is equivalent to a general form circuit $\mathcal{C}_\text{gen}$ as shown here, which consists of five steps: i) initial Clifford $L$, ii) destructive measurements, iii) ancilla preparation, iv) final Clifford $R$, v) conditional Pauli. The conditional Pauli is controlled by the general form's circuit outcome vector$V$, consisting of random bits (represented by a black disk) and measurement outcomes. Changing the conditional Paulis in $\mathcal{C}$ only changes the conditional Pauli in $\mathcal{C}_\text{gen}$ (not the parts in the dashed box).
• Figure 2: (a) The logical action of a stabilizer channel, which is a stabilizer circuit that takes input states encoded in the code $\mathcal{S}_\text{in}$ and produces output states encoded in $\mathcal{S}_\text{out}$. We take the convention that all conditional Paulis are at the end of the stabilizer channel. The Clifford unitaries $C_\text{in}$ and $C_\text{out}$ fix the logical basis of $\mathcal{S}_\text{in}$ and $\mathcal{S}_\text{out}$. (b) A general form circuit equivalent to the logical action. Note there are no random bits or classically controlled Paulis in this general form, which can be ensured by an appropriate choice of classically controlled Paulis for the stabilizer channel. The outcome vector $\mathfrak{f}(O)$ of the general form is a function of the outcome vector $O$ of the stabilizer channel.
• Figure 3: A code basis conversion is the logical action of a trivial circuit with respect to different code bases. This requires that the stabilizer group of the output code is contained in that of the input code $\mathcal{S}_\text{out} \subseteq \mathcal{S}_\text{in}$. The logical action is a simple general form circuit specified by a $k_\text{out}$-qubit Clifford $R$, where $k_\text{out}$ is the number of logical qubits in $\mathcal{S}_\text{out}$.
• Figure 4: The effect on the logical action of a logical operator $\overline{E}$ for the output code. Applying $\overline{E}$ at the end of a stabilizer channel is equivalent to inserting Pauli operators $X^a$, $Z^b$, $X^c$ in the the general form circuit as shown. We use the bitstrings $a(\overline{E})$, $b(\overline{E})$ and $c(\overline{E})$ to uniquel capture the action of $\overline{E}$ on the channel.
• Figure 5: Different classes of hook faults illustrated using a single round of surface code stabilizer measurements, with $X$ and $Z$ stabilizers corresponding to blue and yellow plaquettes respectively. (a) $\mathcal{F}_\text{sub}$ is single-qubit $Z$ faults (red squares). $F$ is a minimum-weight logical fault configuration for $\mathcal{F}_\text{sub}$ (green), such that $d(\mathcal{F}_\text{sub})=5$. Now consider adding an in-equivalent fault $f$ to $\mathcal{F}_\text{sub}$ to form $\mathcal{F}$. (b) If $f$ (red) is equivalent to the combination of $w>1$ faults in $F$, then the fault distance is reduced to $d(\mathcal{F}) = d(\mathcal{F}_\text{sub}) - w +1$ and we say $f$ is brazen for $\mathcal{F}$ with respect to $\mathcal{F}_\text{sub}$. (c) Non-brazen faults can also reduce the fault distance however. If $f$ (red) is in a logical fault configuration $F'$ for fault set $\mathcal{F}$ as shown, we say $f$ is hazardous for $\mathcal{F}$ with respect to $\mathcal{F}_\text{sub}$. In this example $F'$ does not correspond to a minimum-weight logical fault for $\mathcal{F}_\text{sub}$, and the hazardous $f$ reduces the distance. (d) Hazardous faults may not reduce the fault distance however. In this example we add two faults $f_1$ (green) and $f_2$ (red) to $\mathcal{F}_\text{sub}$ to form $\mathcal{F}$, which results in $d(\mathcal{F}) = d(\mathcal{F}_\text{sub})-1$. Removing $f_1$ from $\mathcal{F}$ (but leaving $f_2$) recovers the distance.

Theorems & Definitions (5)

• Definition 3.1: Fault distance
• Definition 4.1: Hazardous and brazen hook faults
• Definition 6.1: Time-local stabilizer channels
• Theorem 6.2
• proof