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arXiv:2606.05044

Generalized Bicycle Codes as Cyclic Submodules and their Automorphism Structure

AJ Davenport, John Blue, Isaac Chuang

quant-phSubmitted Jun 3, 2026• Announced Jun 4, 2026• 0 citations
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Abstract

Automorphisms of quantum codes, when they exist, offer a pathway toward fault-tolerant gate implementation via qubit relabeling. Although useful, the conditions under which automorphisms appear in a given code remain poorly understood. In this paper, we develop an algebraic framework for systematically analyzing and engineering automorphisms in Generalized Bicycle (GB) codes. Central to our approach is the derivation of a three-space dependency between the polynomial ring space, the parity check matrix space, and the $\mathbb{F}_2^{2\ell}$ qubit space, similar to the structure found in the study of classical cyclic codes. By expressing GB codes as a pair of cyclic submodules of $R_\ell^2$, where $R_\ell \cong \mathbb{F}_2[x]/\langle x^\ell-1\rangle$, we reduce the search for code automorphisms to a deterministic algebraic problem, deriving necessary and sufficient conditions for the existence of block-separable automorphisms built from cyclic shifts, ring automorphisms and block-swaps. We connect these conditions to the fold-transversal gate framework, providing explicit criteria for the existence of $H$-, $S$-, and $CX$-type fold-transversal gates. We further discuss structured bases for logical operators in order to determine the logical action of a given automorphism. Finally, we introduce the Maximal Cube Root (MCR) code family, a family of GB codes constructed around the principle of maximizing automorphism flexibility and fold-CX gates. We demonstrate a collection of $k=2$ MCR codes up to $d=13$ generating the 2-qubit Clifford group via automorphism and fold-transversal gates, with stabilizer weight ranging from 8 to 16, and $k>2$ MCR codes with a minimum of 20 distinct logical gates achievable from automorphisms. This serves as a first demonstration of inverse design: using these methods to build codes around a rich automorphism structure from the ground up.

Comments
59 pages, 11 figures

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Automorphisms of quantum codes, when they exist, offer a pathway toward fault-tolerant gate implementation via qubit relabeling. Although useful, the conditions under which automorphisms appear in a given code remain poorly understood. In this paper, we develop an algebraic framework for systematically analyzing and engineering automorphisms in Generalized Bicycle (GB) codes. Central to our approach is the derivation of a three-space dependency between the polynomial ring space, the parity check matrix space, and the F22ℓ\mathbb{F}_2^{2\ell}F22ℓ​ qubit space, similar to the structure found in the study of classical cyclic codes. By expressing GB codes as a pair of cyclic submodules of Rℓ2R_\ell^2Rℓ2​, where Rℓ≅F2[x]/⟨xℓ−1⟩R_\ell \cong \mathbb{F}_2[x]/\langle x^\ell-1\rangleRℓ​≅F2​[x]/⟨xℓ−1⟩, we reduce the search for code automorphisms to a deterministic algebraic problem, deriving necessary and sufficient conditions for the existence of block-separable automorphisms built from cyclic shifts, ring automorphisms and block-swaps. We connect these conditions to the fold-transversal gate framework, providing explicit criteria for the existence of HHH-, SSS-, and CXCXCX-type fold-transversal gates. We further discuss structured bases for logical operators in order to determine the logical action of a given automorphism. Finally, we introduce the Maximal Cube Root (MCR) code family, a family of GB codes constructed around the principle of maximizing automorphism flexibility and fold-CX gates. We demonstrate a collection of k=2k=2k=2 MCR codes up to d=13d=13d=13 generating the 2-qubit Clifford group via automorphism and fold-transversal gates, with stabilizer weight ranging from 8 to 16, and k>2k>2k>2 MCR codes with a minimum of 20 distinct logical gates achievable from automorphisms. This serves as a first demonstration of inverse design: using these methods to build codes around a rich automorphism structure from the ground up.
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Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉\lceil d/2\rceil⌈d/2⌉ for distance-ddd codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed TTT gate on the [[16,4,4]][[16,4,4]][[16,4,4]] BB code, TTT-type rotations on the [[64,8,8]][[64,8,8]][[64,8,8]] BB code, and CCZ-, CS-, and TTT-type operations on the [[36,6,6]][[36,6,6]][[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields Cr{}^rrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.
Identifying stabilizer codes that admit fault-tolerant implementations of the full logical Clifford group would significantly advance fault-tolerant quantum computation. Motivated by this goal, we study several classes of fault-tolerant gadget constructions consisting of Clifford gates acting on the physical qubits, including transversal gadgets, code automorphisms, and fold-transversal gadgets. While stabilizer codes encoding a single logical qubit, most notably the [[7,1,3]] Steane code, are known to admit transversal implementations of the full logical Clifford group, no analogous examples are known for codes encoding multiple logical qubits. In this work, we prove a no-go theorem establishing that no stabilizer code admits a fully transversal implementation of the Clifford group on more than one logical qubit. We further strengthen this result by showing that fold-transversal implementations of the full logical Clifford group are impossible for stabilizer codes encoding more than two logical qubits. More generally, we introduce the notion of k-fold transversal gadgets and prove that implementing the full Clifford group on k logical qubits requires at least k-fold transversal gadgets at the physical level. In addition, we analyze code-automorphism based constructions and demonstrate that they also fail to realize the full Clifford group on multiple logical qubits for any stabilizer code. Together, these results place fundamental constraints on fault-tolerant Clifford gadget design and show that stabilizer codes supporting the full logical Clifford group on multiple logical qubits via these architectures do not exist. Since the Clifford group is a core component of universal gate sets, our findings imply that quantum computing with codes encoding multiple logical qubits within a single code block necessarily entails more complex constructions for fault tolerance.
We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a two-polynomial search in GB codes to a single-polynomial search, while preserving sparsity. We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices. Finally, simulation results for short- to medium-length UB codes (block lengths ranging from a few hundred to approximately 10310^3103) demonstrate competitive performance relative to existing GB and bivariate bicycle (BB) codes despite the additional algebraic restriction.
Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.
Universal fault-tolerant quantum computing combines versatile but expensive operations with specialized but cheap ones. Its efficiency depends on how much computation can be pushed onto the cheap operations and on the size of the code needed to do so. Automorphism gates provide such cheap operations using only physical single-qubit Clifford gates and qubit permutations. Yet no general theory characterizes their maximum logical power or the minimum code size needed to attain it. We develop such a theory. For stabilizer codes encoding k≥3k\geq3k≥3 logical qubits, we show that the largest logical group attainable by automorphisms is generated by all addressable SSS and CX\mathrm{CX}CX gates, and we construct codes attaining it. While this group contains exponentially fewer gates than the full Clifford group, adding one suitable non-Clifford gate yields universality. We further classify the largest logical groups attainable using qubit permutations, physical single-qubit Cliffords, or both across general stabilizer and CSS codes, and derive refined bounds for self-dual CSS subclasses. Achieving the maximum-size logical group through automorphisms requires n=Θ(2k)n=Θ(2^k)n=Θ(2k) physical qubits. By contrast, all addressable diagonal Clifford gates, generated by SSS and CZ\mathrm{CZ}CZ, require only n=Θ(k2)n=Θ(k^2)n=Θ(k2) physical qubits when implemented using physical single-qubit Cliffords alone. Both bounds are tight. This polynomial qubit cost extends beyond Cliffords to all addressable diagonal gates at any fixed level of the Clifford hierarchy, using physical single-qubit diagonal gates. Thus, for full addressability, the sharpest physical-qubit cost divide lies between diagonal and CX\mathrm{CX}CX-type gates, not between Clifford and non-Clifford gates.