Quantum Tanner Codes

• Quantum Tanner Codes are quantum LDPC codes constructed using robust local classical codes and highly expanding Cayley complexes, achieving constant encoding rate and linear minimum distance.
• They leverage left–right Cayley complex structures and Ramanujan generators to ensure strong expansion properties that prevent low-weight logical errors.
• Efficient decoding strategies, including small-set-flip and parallel algorithms, enable single-shot error correction and fault tolerance in quantum architectures.

Quantum Tanner codes are a family of quantum low-density parity-check (qLDPC) codes constructed via the interplay of robust local classical codes and the high-expansion combinatorics of algebraically-defined square complexes, most notably the left–right Cayley complex. These codes are the first explicit, constructive examples of quantum LDPC codes achieving both constant encoding rate and linear minimum distance, thereby closing a long-standing gap between classical and quantum LDPC theory. Quantum Tanner codes are intrinsically linked to advances in the theory of locally testable codes (LTCs), expander graphs, and group-theoretic/graph-theoretic combinatorics. Their constructions, parameter analyses, decoding algorithms, and generalizations mark a central development in modern quantum error correction.

1. Theoretical Framework: Left–Right Cayley Complex and Classical Components

Quantum Tanner codes leverage the structure of two symmetric generating sets $A, B \subseteq G$ in a finite group $G$, satisfying the “total no-conjugacy condition” ($\forall g, a, b: a\,g \neq g\,b$), to construct a double-covered left–right Cayley 2-complex. The complex consists of:

• Vertex sets $V_0 = V_1 = G$
• Edges: type- $A$ edges ($(g,0)$ to $(a g,1)$) and type-$B$ edges ($(g,0)$ to $(g b,1)$)
• Squares $Q = \{(g,0),(a g,1),(g b,1),(a g b,0)\}$

Two diagonal graphs, $G' = (V_0, Q)$ and $G'' = (V_1,Q)$, each $\Delta^2$-regular, are obtained by viewing faces as edges on each partition class. The expansion properties are optimized by choosing Ramanujan generators for $A, B$, ensuring a spectral gap ($\lambda \leq 2 \sqrt{\Delta}$) that is crucial for linear-distance proofs.

Classical codes $C_A \subseteq \mathbb{F}_2^\Delta$, $C_B \subseteq \mathbb{F}_2^\Delta$ (with rates $p = k_A / \Delta$, $1-p$) are imposed locally:

• Local code at each vertex: tensor and dual-tensor codes, $C_0 = C_A \otimes C_B$, $$C_1 = (C_A \otimes C_B)^\perp + (C_A^\perp \otimes C_B^\perp)$$
• Parity checks are constructed from Kronecker products of local code check matrices.

This yields a bipartite, LDPC cell complex where qubits are associated to the squares and stabilizers to the vertices, with each check having constant weight $O(1)$ and each qubit participating in exactly four stabilizers.

Key Parameter Scaling (using appropriate robust and expanding choices): $$n = |Q| = \Delta^2 |G|, \quad k \geq (1-2p)n, \quad d = \Omega(\Delta^{3/2+\epsilon} n)$$ for appropriate choices of $p, \epsilon > 0$ and robust local codes (Gu et al., 2022, Leverrier et al., 23 Dec 2025, Leverrier et al., 2022).

2. Quantum Tanner Code Construction: Generalizations and Variants

The foundational construction applies equally to more general frameworks:

• Schreier complex generalization replaces group multiplication with general group actions, allowing construction over arbitrary bipartite square complexes with two commuting Schreier graphs. This greatly broadens the landscape of structures in which quantum Tanner-like codes can be instantiated. The only requirement for “quantum Tanner machinery” is the presence of two commuting, regular expanders on a common bipartition (Mostad et al., 13 May 2024).
• Lifted/tiled families utilize covers of smaller seed complexes, with local codes constructed from cyclic or double-circulant codes. Lifted quantum Tanner codes can realize moderate-length examples with nontrivial degeneracy and distances exceeding $\sqrt{n}$ (Guemard et al., 27 Feb 2025).
• Recursive expansion variants (e.g., XZ-Type and hypergraph-product “TGRE” codes) exhibit superior trade-offs in rate/distance for moderate blocklengths, achieving rates up to 0.2 and code distances scaling as $O(\log \sqrt{N})$ (Yi et al., 12 Feb 2024).

Quantum Tanner Color Codes extend the paradigm to higher-dimensional simplicial complexes, allowing constructions with transversal logical gates and self-dual structure, with parameter regimes (rates, stabilizer weight, conjectural distances) suitable for fault-tolerant architecture (Gulshen et al., 9 Oct 2025).

3. Distance, Rate, and Expander Analysis

Quantum Tanner code parameter analysis is predicated on:

• Robustness of local codes: The dual-tensor product code $$(C_A \otimes \mathbb{F}_2) + (\mathbb{F}_2 \otimes C_B)$$ must be robust (product-expanding). For random code choices of fixed rate, robustness holds with high probability, enabling linear-distance proofs (Leverrier et al., 2022, Leverrier et al., 23 Dec 2025, Radebold et al., 7 Aug 2025).
• Expansion of the square complex: The expander-mixing lemma is applied on the diagonal ($\Delta^2$-regular) graphs, leveraging the Ramanujan property ($\lambda \ll \Delta^2$), which, together with robust local codes, forbids low-weight logical operators and ensures $d = \Omega(n)$.
• Parameter formulas: For generic local codes with parameters $(\Delta,k_A,d_A), (\Delta,k_B,d_B)$, and robust product expansion, codes achieve: $k/n \geq (1-2p)^2$

$d \geq c n ~~~ \text{for some } c > 0$

The precise constants depend on the chosen local codes and expansion parameters. Code families constructed with small groups yield explicit instances such as $[[144,12,11]]$, $[[432,20,\leq 22]]$, etc., with check weights as low as 9 (Leverrier et al., 23 Dec 2025, Guemard et al., 27 Feb 2025).

4. Decoding and Algorithmic Performance

Quantum Tanner codes admit highly efficient decoders, both sequential and parallel:

• Small-set-flip iterative algorithm: At each round, the decoder examines constant-sized neighborhoods (usually the set of faces incident to a vertex) and greedily applies corrections that reduce a local cost (potential) function. Utilizing the locality and combinatorial structure, the decoder can provably correct adversarial errors of weight up to a constant fraction $\alpha n$ with total runtime $O(n)$ (Gu et al., 2022, Leverrier et al., 2022, Leverrier et al., 2022).
• Parallel decoding: Through synchronous rounds partitioned by vertex type (color class), the decoder achieves $O(\log n)$ convergence. The parallel decoder offers constant-depth implementations suitable for hardware-realistic, low-latency quantum error correction (Leverrier et al., 2022, Gu et al., 2023).
• Single-shot error correction: Quantum Tanner codes, with these decoders, enable single-shot QEC even in the presence of syndrome measurement errors, with only one round of (constant-weight) stabilizer measurements required. The decoders can be adapted to correct time-correlated errors across repeated QEC rounds in constant time per round, making this family especially robust for fault-tolerant protocols (Gu et al., 2023).

Alternative decoders such as belief propagation with ordered statistics decoding (BP+OSD) demonstrate good logical error rates and pseudo-thresholds in circuit-level and phenomenological noise models, showing competitive performance relative to surface codes at similar distance and overhead (Radebold et al., 7 Aug 2025).

5. Explicit Instances, Small-Length and Moderate-Length Codes

Computer-aided searches over small groups and carefully engineered local codes have produced quantum Tanner code instances with high encoding rate and relative distance for small-to-moderate blocklengths. Notable examples include:

• $[[144,12,11]]$, $[[432,20,\leq 22]]$, $[[576,28,\leq 24]]$, realized for groups such as $C_2 \times C_2$, $C_6 \times C_2$, and $C_4 \rtimes C_4$, with generator sets and code structures detailed explicitly (Leverrier et al., 23 Dec 2025).
• The celebrated $[[96,2,12]]$ CSS LDPC code from a lifted double-circulant construction, exhibiting distance surpassing $\sqrt{n}$, check weights of 4 and 8, and nontrivial degeneracy (Guemard et al., 27 Feb 2025).
• Table 1 summarizes small dihedral-group-based constructions, with parameters, stabilizer weights, k/n, and d/n (Radebold et al., 7 Aug 2025):

[[n, k, d]]	Group	Δ	Check Weight(s)	k/n	d/n
[36,8,3]	D₄	3	6	.222	.083
[54,11,4]	D₆	3	6	.204	.074
[72,14,4]	D₈	3	6	.194	.056
[200,10,10]	D₈	5	6,8,9,12	.05	.05
[250,10,15]	D₁₀	5	6,8,9,12	.04	.06

These codes achieve logical error rates and space-time overheads competitive with surface codes up to distances $d = 4$, and explicit check matrices/generating sets are provided for reproducibility.

6. Generalizations and Recent Developments

Recent work generalizes the construction to settings where the underlying combinatorial objects are arbitrary bipartite square complexes built from two commuting Schreier graphs, vastly increasing the available code constructions. This encompasses all quantum Tanner codes defined on Cayley complexes and extends to non-group examples (e.g., combinatorial designs, finite geometries). Critical properties, including LDPC structure, positive rate, and linear distance, are maintained under these generalizations, contingent on expansion and robustness hypotheses (Mostad et al., 13 May 2024).

Quantum Tanner Color Codes, defined via “sheaf-theoretic” LDPC structures on high-dimensional expanding complexes, realize qubit codes with transversal Clifford and non-Clifford gates, constant rate, and conjecturally constant distance. These developments signal the integration of Tanner-like architectures with topological and transversal gate paradigms (Gulshen et al., 9 Oct 2025).

7. Open Questions and Future Directions

Several avenues remain open:

• Determining the full logical operator structure in finite-size examples: While logical operators of quantum Tanner codes often inhabit single “vertical” or “horizontal” slices, in general, not all logicals can be localized; basis classification is incomplete (Leverrier et al., 23 Dec 2025).
• Designing practical, robust decoders for moderate lengths (100–1000 qubits), especially when local robustness fails or short classical codes are used.
• Identifying fully explicit, deterministic local code choices (as opposed to randomly sampled codes) with maximal robustness and expansion properties.
• Further exploration of lifted product, recursive, and Schreier-graph-based variants to optimize rates, distances, and decoding efficiency.
• Systematic benchmarking of new constructions against surface codes and other qLDPC families in realistic hardware models.

Quantum Tanner codes thus establish a unifying framework for quantum LDPC code design, combining asymptotic optimality, efficient decoding, and applicability to a broad class of combinatorial, group-theoretic, and geometric contexts (Leverrier et al., 2022, Leverrier et al., 2022, Gu et al., 2022, Leverrier et al., 23 Dec 2025, Radebold et al., 7 Aug 2025, Mostad et al., 13 May 2024, Gu et al., 2023, Gulshen et al., 9 Oct 2025, Guemard et al., 27 Feb 2025, Yi et al., 12 Feb 2024, Leverrier et al., 2022).