Coset Lifted Product Codes: Constructing Quantum LDPC Codes from Non-Normal Subgroup Actions

AI summary

Overview

• Research area: Quantum error correction, specifically quantum low-density parity-check (LDPC) codes constructed with the CSS (Calderbank–Shor–Steane) framework and algebraic tools from finite group theory (group algebras, coset actions, lifted products).
• Technical level: Advanced. The paper assumes familiarity with CSS stabilizer codes, group algebras over 𝔽₂, permutation representations, and the lifted/hypergraph product. However, the core construction is describable in plain terms and this summary isolates it.
• Scope (one sentence): The paper defines a new family of quantum LDPC codes, "coset lifted product" (CLP) codes, that combine protograph-based lifted products with the action of a group on the cosets of a subgroup that need not be normal, proves the CSS orthogonality condition holds in general, gives a dimension formula for groups of odd order, and reports a computer search over 14,843 (G, H) pairs that finds short-length codes exceeding the best published values of Q = kd²/n at check weights 6, 7 and 9.

What This Paper Is About

Quantum LDPC codes need sparse parity checks plus as much encoded information (k) and as much error tolerance (distance d) as possible for a given number of physical qubits n. Most practical short-length constructions — two-block group algebra codes, bivariate bicycle codes, and lifted products — let a group act on itself, which limits which codes are reachable. Prior work by Aydin, Tamo and Barg showed that letting 𝔽₂[G] act on the coset space G/H of a possibly non-normal subgroup produces a larger design space of "coset" two-block codes. This paper asks whether that coset idea can be pushed from single-block (1×1 protograph) codes up to full protograph-based lifted products, and whether the resulting enlarged search space yields better short codes.

Key Contributions

1. Definition of coset lifted product (CLP) codes. Given a finite group G, a subgroup H ≤ G (not necessarily normal) with normalizer N = N_G(H), a protograph A over 𝔽₂[G] and a protograph B over 𝔽₂[N], the code CLP(A, B; G, H) is formed by letting A act on G/H from the left and B from the right, then taking the lifted product of the resulting block matrices. The code has n = m(n_{A1}n_{B0} + n_{A0}n_{B1}) qubits, where m = |G|/|H|.

2. Proof that the CSS condition holds for every subgroup, normal or not. Lemma III.2 establishes that L(g)R(n) = R(n)L(g) for g ∈ G, n ∈ N, and Theorem III.3 uses this commuting property to prove H_X H_Z^T = 0 block by block over 𝔽₂. The authors note the construction extends to 𝔽_q (qudit codes) if the first block of H_Z is negated, though all codes in the paper are binary.

3. A dimension formula for groups of odd order, expressed as a sum over the irreducible representations ρ of G of hypergraph product dimensions, along with identification of special cases (Proposition III.4): H = {e} recovers the standard lifted product; 1×1 protographs recover the Aydin–Tamo–Barg coset two-block code; normal H collapses to a lifted product over the quotient G/H; and every CLP code equals one with a core-free subgroup via G → G/K where K = Core_G(H).

4. A search over 14,843 pairs (G, H) with H restricted to be non-normal and block lengths 120 ≤ n ≤ 400, producing certified codes that exceed the best published Q = kd²/n at check weights 6, 7 and 9, with all codes printed in full and reproducible from the reported data.

Main Findings

• Best new codes by check weight w: at w = 6, a [[360, 20, 20]] code with Q = 22.2; at w = 7, a [[360, 24, 20]] code with Q = 26.67; at w = 9, a [[288, 16, 20]] code with Q = 22.22, alongside a [[252, 16, 18]] code at the same weight.

• Additional high-distance codes: [[168, 10, 14]] and [[240, 8, 20]] at w = 6, and [[216, 8, 18]] at w = 7.

• Upper-bound candidates: at w ∈ {7, 9, 10}, candidates whose distance upper bounds give Q up to 40.5.

• Which subgroup pairs can yield genuinely new codes: by Proposition III.4, normal H (N = G) only reproduces lifted products over a quotient group, and self-normalizing H (N = H) makes every R(n) the identity so the right protograph contributes only entry-weight parities. The productive regime is therefore H core-free with H < N < G.

• Worked example (Example III.5): with G = A₄ (|G| = 12), H = ⟨(12)(34)⟩ of order 2, m = 6 cosets, Core_G(H) = {e}, and N_G(H) ≅ ℤ₂ × ℤ₂, the 2×2 protographs A and B yield H_X and H_Z of size 24 × 48 on n = 48 qubits; ranks are 21 each, giving k = 6, and the integer linear program of Section IV-B gives d_X = d_Z = 4, so the code is [[48, 6, 4]].

• Degeneracy is characterized: Lemma III.6 extends the equivalence lemmas of Aydin–Tamo–Barg from 1×1 protographs to general CLP codes, showing that row/column multiplication of protograph entries by group elements, and application of a group automorphism to A, B and H, produce equivalent codes with the same parameters and check weights.

• Distance certification: code distances in the search are computed with an integer linear program rather than left as estimates, which is what makes the reported parameters "certified."

Methodology in Plain English

The authors build codes in layers. First they take a finite group G and a subgroup H, and consider the set of left cosets G/H as the "wiring diagram" on which everything acts. Elements of G permute these cosets by left multiplication, and elements of the normalizer N_G(H) permute them by right multiplication; crucially, left and right actions commute, which is exactly the algebraic fact needed for X-type and Z-type quantum checks to commute.

Second, instead of a single group-algebra element per block (the two-block case), they use small matrices — protographs — whose entries are sums of group elements, one matrix over 𝔽₂[G] for the left side and one over 𝔽₂[N] for the right side. Turning each group element into a permutation matrix of the cosets expands these protographs into large sparse binary block matrices.

Third, they assemble the quantum check matrices H_X and H_Z in the standard lifted-product pattern, with the left-acting blocks in one set of columns and the right-acting blocks in another. The proof that the two sets of checks commute reduces to the block-level identity L(α)R(β) = R(β)L(α), and over 𝔽₂ the two equal terms sum to zero.

Fourth, to make the search tractable and the results interpretable, they show that codes from normal subgroups or self-normalizing subgroups are not new, so the search can be restricted to core-free H with H < N < G. Distances are certified by solving an integer linear program rather than by random sampling.

Why This Matters

• Expands the design space of short quantum LDPC codes. Only a small number of code families are competitive at the block lengths current hardware can contemplate, and CLP codes strictly generalize the 2BGA, bivariate bicycle, lifted product and coset two-block families, so previously unreachable codes become accessible.

• Gives a rigorous guarantee rather than a numerical check. The CSS condition is proved for arbitrary G and H via Theorem III.3, in contrast to search-based approaches that retain a candidate only after verifying H_X H_Z^T = 0 numerically.

• Improves the key figure of merit at fixed check weight. The reported codes exceed the best published Q = kd²/n at w = 6, 7 and 9, which is directly relevant to the overhead-versus-distance tradeoff that dominates fault-tolerance resource estimates.

Real-world applications (of quantum LDPC codes generally):

• Quantum memory and logical storage in fault-tolerant quantum processors, where constant-rate codes with efficient decoders permit constant space overhead rather than polylogarithmic overhead.
• Reducing qubit overhead in near-term and early fault-tolerant hardware, since codes with circuit-level thresholds approaching the surface code's can do so at an order of magnitude fewer physical qubits (as reported for bivariate bicycle codes).
• Distributed quantum computing and modular architectures, where sparse checks that couple limited sets of qubits map naturally onto limited connectivity.
• Quantum communication and entanglement distribution, where block codes with certified distance bound the number of physical qubits needed per protected logical channel.

Industry relevance: The work targets the practical regime of block lengths 120 ≤ n ≤ 400, which is where hardware-motivated code selection happens, and provides fully printed, reproducible code data — a prerequisite for adoption by groups building compilers and decoders around specific code instances.

Future Directions

• Decoder design and thresholds. The paper reports code parameters and Q values but does not report circuit-level noise thresholds or decoder performance for the new codes; establishing these, as has been done for bivariate bicycle codes, is the natural next step.

• Larger and differently structured searches. The search is confined to 14,843 pairs with 120 ≤ n ≤ 400; extending it to longer blocks, larger protographs, or structured families of core-free subgroups (with H < N < G) could reveal whether the improvements scale.

• Asymptotic analysis. The paper's theory is a dimension formula for odd-order groups plus a finite search; whether CLP families can achieve constant rate with linear distance, in the spirit of the non-abelian lifted product result that settled the qLDPC conjecture, is not addressed.

• Non-binary and qudit generalizations. The authors note that Definition III.1 and Theorem III.3 hold over any finite field 𝔽_q once the first block of H_Z is negated, yielding qudit codes; all codes reported here are binary, leaving the qudit regime unexplored.

Target Audience

Researchers and graduate students in quantum information and coding theory who work on quantum LDPC constructions, particularly those already familiar with CSS codes, group algebras and lifted products. It is also useful for hardware-oriented architects selecting specific short codes with certified distance, since the paper prints complete code data, and for mathematicians interested in applications of non-normal subgroup actions and core-free subgroups to combinatorial code design. Readers without a group-theory background will find the construction conceptually accessible through the A₄ worked example but will need the algebraic machinery to follow the proofs.

Authors’ abstract

We introduce coset lifted product (CLP) codes, a family of quantum LDPC codes that combine the protograph structure of lifted product codes with the coset action of Aydin, Tamo, and Barg \cite{AydinTamoBarg2026}. Given a finite group $G$, a subgroup $H \le G$ that need not be normal, a protograph $A$ over $\mathbb F_2[G]$, and a protograph $B$ over $\mathbb F_2[N_G(H)]$, the two protographs act on the cosets $G/H$ from the left and from the right to produce quantum CSS codes. Further, for groups of odd order, we give an explicit formula for the dimension as a sum over the irreducible representations $ρ$ of $G$ of hypergraph product dimensions. A search over $14,843$ pairs $(G,H)$ with $H$ restricted to be non-normal and $120 \le n \le 400$ yields certified codes that exceed the best published value of $Q=kd^2/n$ at check weights $6,7$ and $9$, i.e., we obtain $[[360,20,20]]$ at $w=6$, a $[[360,24,20]]$ code at $w=7$, and codes $[[252,16,18]]$ and $[[288,16,20]]$ at $w=9$. Our search also produce high-distance codes with parameters $[[168,10,14]], [[240,8,20]]$ at $w=6$ and $[[216,8,18]]$ at $w=7$.