Symmetry-Based Quantum Codes Beyond the Pauli Group

TL;DR

The paper develops a unifying symmetry-based framework for quantum error correction by encoding logical information in the $G$-symmetric subspace $\mathrm{Sym}_G$ defined by a finite group representation $W:G\to U(\mathcal{H})$. It generalizes stabilizer syndrome extraction to isotypic syndrome extraction using projections $\Pi_\lambda$ onto irreducible components and a group Fourier transform $\mathrm{QFT}_G$, enabling symmetry-resolved error diagnosis. Stabilizer codes appear as the abelian $G=\mathbb{Z}_2^{\oplus n}$ case, while nonabelian constructions (e.g., a single-qubit dihedral $D_n$ code) illustrate broader applicability, with explicit circuit and complexity analyses. Logical operations are characterized via the normalizer of $W(G)$, and a generalized Knill-Laflamme condition with $G$-weight $w_G$ and $d_G$ governs correctability, paving the way for symmetry-aware fault-tolerance designs. The framework unifies existing codes and offers pathways to tailor codes to specific hardware, with potential impact on fault-tolerance thresholds and symmetry-driven quantum simulations.

Abstract

Typical stabilizer codes aim to solve the general problem of fault-tolerance without regard for the structure of a specific system. By incorporating a broader representation-theoretic perspective, we provide a generalized framework that allows the code designer to take this structure into account. For any representation of a finite group, we produce a quantum code with a code space invariant under the group action, providing passive error mitigation against errors belonging to the image of the representation. Furthermore, errors outside this scope are detected and diagnosed by performing a projective measurement onto the isotypic components corresponding to irreducible representations of the chosen group, effectively generalizing syndrome extraction to symmetry-resolved quantum measurements. We show that all stabilizer codes are a special case of this construction, including qudit stabilizer codes, and show that there is a natural one logical qubit code associated to the dihedral group. Thus we provide a unifying framework for existing codes while simultaneously facilitating symmetry-aware codes tailored to specific systems.

Figures (5)

• Figure 1: $G$-Bose Symmetry Test. The gate $\mathcal{E}_G$ prepares the uniform superposition over group algebra basis for $G$ and the controlled $W(g)$ operation triggers when the ancillary register is in the state $\ket{g}$.
• Figure 2: Isotypic Syndrome Extraction Procedure. The state $\ket{+_G}$ is the uniform superposition over $G$, $M_\lambda$ denotes the measurement with respect to the isotypic label basis, and $\mathcal{R}$ denotes the recovery operation.
• Figure 3: State creation circuit used to generate $\ket{+_{D_3}}$ from single-qubit unitaries of the form \ref{['eq:gen-unitary']} and a single CNOT gate. In the circuit, $\gamma = 2 \cos^{-1} \left ( \frac{{\csc{\frac{\pi}{8}}}\sec{\frac{\pi}{8}}}{2 \sqrt{3}} \right) = 1.23096$.
• Figure 4: Full $D_3$ (equivalently, $S_3$) circuit according to the error detection method. $\ket{\psi}_L$ is the input logical state. The value of $\gamma$ is given by $\gamma = 2 \cos^{-1} \left ( \frac{{\csc{\frac{\pi}{8}}}\sec{\frac{\pi}{8}}}{2 \sqrt{3}} \right) = 1.23096$.
• Figure 5: The $\operatorname{QFT}_{D_n}$ circuit where $\ket{k}$ designates the rotational register and $\ket{\alpha}$ denotes the flip register.

Theorems & Definitions (15)

• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Theorem 5
• Theorem 6: Generalized Knill-Laflamme bound