Quantum LDPC memories can encode many logical qubits, but that alone does not make them usable: applications need to know where the logical operators are supported, how they are labeled, and how conjugate $X/Z$ partners pair. For hypergraph-product codes this information follows from row reduction over $\mathbb{F}_2$. For Abelian lifted-product codes, which include prominent high-rate constructions, it does not: the entries of the defining seed matrices live in a group algebra rather than a field, so pivots need not be invertible and row reduction can fail.
To address this problem, we introduce \emph{logical spectroscopy}. For an odd-order Abelian lift group, the Chinese remainder theorem splits the group algebra into finite fields, one for each Frobenius orbit of characters; we call these orbits packets. Packet by packet, we solve ordinary finite-field linear algebra, lift the answers back to the physical code with the associated packet projectors, and pair $X$ and $Z$ logicals between reciprocal packets by trace duality. The result is a complete conjugate logical basis for a finite Abelian lifted product code that is addressable: every logical coordinate carries canonical packet and Künneth-summand labels, deterministic within-block indices, a conjugate partner, and an explicit binary representative. The code's own translation symmetry then acts on these coordinates in closed form. We apply the construction to examples with up to 5000 physical qubits, including high-rate examples whose distance-witness is reported explicitly. We further extend the decomposition to even-order lifts. Logical spectroscopy thus equips Abelian lifted products with an explicit logical coordinate system and an algebraic toolkit for high-rate code search.
Quantum LDPC memories can encode many logical qubits, but that alone does not make them usable: applications need to know where the logical operators are supported, how they are labeled, and how conjugate X/Z partners pair. For hypergraph-product codes this information follows from row reduction over F2. For Abelian lifted-product codes, which include prominent high-rate constructions, it does not: the entries of the defining seed matrices live in a group algebra rather than a field, so pivots need not be invertible and row reduction can fail.
To address this problem, we introduce \emph{logical spectroscopy}. For an odd-order Abelian lift group, the Chinese remainder theorem splits the group algebra into finite fields, one for each Frobenius orbit of characters; we call these orbits packets. Packet by packet, we solve ordinary finite-field linear algebra, lift the answers back to the physical code with the associated packet projectors, and pair X and Z logicals between reciprocal packets by trace duality. The result is a complete conjugate logical basis for a finite Abelian lifted product code that is addressable: every logical coordinate carries canonical packet and Künneth-summand labels, deterministic within-block indices, a conjugate partner, and an explicit binary representative. The code's own translation symmetry then acts on these coordinates in closed form. We apply the construction to examples with up to 5000 physical qubits, including high-rate examples whose distance-witness is reported explicitly. We further extend the decomposition to even-order lifts. Logical spectroscopy thus equips Abelian lifted products with an explicit logical coordinate system and an algebraic toolkit for high-rate code search.
High-rate quantum low-density parity-check (qLDPC) codes encode many logical qubits with low physical-qubit overhead, but realizing efficient fault-tolerant computation on such dense encodings remains a major challenge. Generic, code-agnostic techniques such as code surgery and gate teleportation apply broadly, but are difficult to make modular, low-overhead, and fully certifiable on complex high-rate codes whose structure is left unexploited. Here we overcome these obstacles by co-designing the code together with its logical instruction set for a broad family of canonical lifted-product (LP) codes with cyclic symmetry. We show that these codes admit a canonical logical basis, in which conjugate logical operators are organized into rows and columns of cyclic orbits inherited directly from the underlying classical codes, analogous to the structure that makes hypergraph-product codes so tractable. This canonical basis enables universal logical processing using magic-state input and a constant number of reusable seed surgery gadgets---only two (resp. four) for a [[1122,148,≤20]] (resp. [[4350,1224,≤20]]) LP code. A fixed extractor smaller than half the data block also enables arbitrarily high-weight logical Pauli-product measurements (PPMs) while preserving the code's cyclic symmetry. These codes also enable rigorously fault-tolerant parallel logic, including PPMs across all logical qubits and magic-state injection from many surface codes into a full LP block. For example, a [[468,36,20]] (resp. [[952,112,17]]) code supports simultaneous measurements of arbitrary logically disjoint Pauli products with an ancilla overhead of roughly three (resp. four) times the data block size. These results advance the frontier of fault-tolerant quantum computation on ultra-high-rate quantum architectures.
Efficient fault-tolerant quantum computing architectures benefit from low-weight logical operators. However, reported canonical bases for high-rate, high-distance codes can be several times heavier than the code distance, and fundamental obstructions can prevent minimum-weight logical operators from forming a complete canonical basis. To address this issue, we develop design principles for group-valued quantum error-correcting codes with low-weight full regular canonical logical bases, generated by translating a set of seed operators. We introduce frame width f, the smallest weight cutoff that permits a complete canonical logical basis, as an additional code parameter alongside distance d and check weight w. Cataloguing these parameters together with explicit logical bases can inform resource estimates and the compilation of fault-tolerant logical operations.
First, we identify a structural obstruction: for fully populated binary monomial CSS checks over a group of odd order, full regularity forces distance two, while for groups of power-of-two order, full row rank of both check matrices after replacing every group element by 1 guarantees full regularity. Second, we demonstrate our principles through explicit high-rate lifted-product (LP) and pair-partition (PP) codes: an [[1088,128,22]] LP code with f=22 and an [[1024,256,24]] PP code with f≤27, both at w=10. At w=11, our PP constructions attain either d=25 with f≤27 or d=24 with the proved optimum f=25, demonstrating both a tradeoff among these parameters and a strict separation between distance and frame width. Finally, we propose a method combining symmetry reduction and stabilizer-based pruning for exhaustive distance certification, and use it to efficiently establish exact distances for LP examples with more than a thousand qubits.
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≤G that need not be normal, a protograph A over F2[G], and a protograph B over F2[NG(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≤n≤400 yields certified codes that exceed the best published value of Q=kd2/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.
Code surgery measures a logical operator of a quantum LDPC code with O(d) rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to 13 where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of A4, S4 and SL(2,3) with gain three for a best logical, up to [[336,26,12]]. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.
Lifted product (LP) codes form an important class of quantum error correcting codes with favorable code parameters. We introduce a systematic construction of LP code families based on group extensions and graph lifts. The construction increases the code size while preserving the local structure of the Tanner graph, and relates code parameters, logical operators, and fault-tolerant logical-operation gadgets within the families through chain and cochain maps.
As a first application, we obtain LP codes with better code parameters than previously reported ones. We then demonstrate that code-surgery gadgets can be transferred across the selected finite lifts through chain maps and, in several cases, implemented with lower space overhead. We also develop parallel product surgery for lifted clustered cyclic codes. Finally, we propose lifting as a systematic first step toward defining thermodynamic families for algebraically defined qLDPC codes without an underlying Euclidean lattice. For several base codes and selected lifts, coherent information exhibits finite-size crossings, while our results also indicate that additional conditions are needed to determine a unique family.
Bivariate bicycle (BB) quantum codes are a prominent finite-length family of quantum low-density parity-check codes, but their minimum distance is usually established numerically rather than read from the defining polynomials. We study the Z-logical quotient K/S over F2[x,y]/(xℓ−1,ym−1) and show that it fits into a short exact sequence with an annihilator quotient as kernel and a colon quotient as cokernel. The sequence gives an explicit logical basis, a dimension formula, and a componentwise distance identity dZ=min(dA,dC). Using the Frobenius structure of the finite group algebra, we prove rA=rC=k/2 for every BB code, including repeated-root cases. The algebraic component of a logical class is distinct from the support shape of its lightest representatives: an annihilator class can have a lighter two-block representative, and a colon class can have a one-sided minimum. For lower bounds we show that every proper subset of a minimum-weight logical operator has nonzero syndrome, and that this property persists inside the colon component but not inside the annihilator component. A translation-anchored cluster search built on it proves the distances 4,6,10,10,12,18 of the six standard BB codes of lengths 18 to 288 and enumerates every minimum-weight logical operator. The resulting census shows that [[108,8,10]] is the only one of the six whose distance is attained in a single component, with dC=10 and dA=12.