ZSZ Codes: Non-Abelian Quantum LDPC CSS

• ZSZ codes are a non-abelian quantum LDPC CSS family that generalizes bivariate bicycle codes by replacing the abelian translation group with a semidirect product.
• They leverage Cayley graph geometries to enable exponential small-set growth while maintaining constant check weights for efficient passive local decoding.
• Designed for neutral-atom arrays, they incorporate optimized routing protocols and controlled qubit interactions to achieve competitive error thresholds.

ZSZ codes are a family of quantum LDPC CSS codes obtained by a minimal, yet structurally consequential, non-abelian generalization of bivariate bicycle (BB) codes. They replace the abelian translation group $G=\mathbb{Z}_{\ell}\times\mathbb{Z}_{m}$ underlying BB codes with a semidirect product $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$, introducing a controlled non-commutativity between horizontal and vertical translations. In the formulation introduced in “Towards self-correcting quantum codes for neutral atom arrays,” this twist changes local propagation rules on the code’s Cayley graphs, enhances small-set growth, preserves constant-weight LDPC structure, and empirically improves performance under passive local decoding while retaining competitive thresholds under conventional circuit-level decoding (Guo et al., 29 Jul 2025).

1. Group-theoretic definition and CSS construction

The defining algebraic object is the semidirect product

$$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$

where $\ell,m,q$ are positive integers satisfying $q^m\equiv 1 \pmod{\ell}$. A BB code is recovered as the special case $q=1$, so that $x$ and $y$ commute and $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$. The ZSZ family takes $q>1$, yielding a non-abelian generalization of two-block group algebra CSS codes (Guo et al., 29 Jul 2025).

Writing group elements in canonical form $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$0, the group law is

$G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$1

with indices modulo $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$2 and $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$3. The corresponding push-through relations are

$G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$4

These relations encode the twist: moving $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$5 through $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$6 multiplies its exponent by a power of $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$7.

ZSZ codes are constructed as 2BGA CSS codes over $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$8. One chooses sparse group-ring elements $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$9, typically with three terms each, and forms their left- and right-regular binary representations

$$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$0

The parity-check matrices are

$$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$1

with orthogonality constraint

$$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$2

In group-ring language this is $$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$3, where $$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$4 denotes the anti-involution induced by group inversion. Operationally, the choice $$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$5, $$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$6 ensures $$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$7 by associativity of group multiplication, which implies CSS commutativity.

The paper also gives an explicit illustrative instance with $$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$8, $$\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m} := \langle x,y \mid x^{\ell}=y^{m}=1,\ yxy^{-1}=x^q\rangle,$$9, and $\ell,m,q$0. In that example, $\ell,m,q$1, $\ell,m,q$2, and the check neighborhoods become twisted block-circulants rather than simple translates because $\ell,m,q$3 (Guo et al., 29 Jul 2025).

2. Geometry, locality, and LDPC parameters

The canonical labeling $\ell,m,q$4 suggests an $\ell,m,q$5 rectangular embedding. Because the construction is two-block, the code places

$\ell,m,q$6

data qubits on the horizontal and vertical links of a toroidal grid: horizontal links correspond to $\ell,m,q$7, and vertical links to $\ell,m,q$8. There are $\ell,m,q$9 $q^m\equiv 1 \pmod{\ell}$0-type checks and $q^m\equiv 1 \pmod{\ell}$1 $q^m\equiv 1 \pmod{\ell}$2-type checks. For 3-term choices of $q^m\equiv 1 \pmod{\ell}$3 and $q^m\equiv 1 \pmod{\ell}$4, every check has weight $q^m\equiv 1 \pmod{\ell}$5, and every data qubit participates in $q^m\equiv 1 \pmod{\ell}$6 checks, namely $q^m\equiv 1 \pmod{\ell}$7 $q^m\equiv 1 \pmod{\ell}$8-type and $q^m\equiv 1 \pmod{\ell}$9 $q=1$0-type checks. Hook errors in single-ancilla extraction have weight at most $q=1$1 due to circuit depth $q=1$2 (Guo et al., 29 Jul 2025).

The number of logical qubits is

$q=1$3

Small examples found by computer search include the following instances.

Code	Parameters	$q=1$4
ZSZ80	$q=1$5	$q=1$6
ZSZ180	$q=1$7	$q=1$8
ZSZ360-1	$q=1$9	$x$0
ZSZ756	$x$1	$x$2

The listed minimum distances were estimated with QDistRnd (GAP), using probabilistic searches for low-weight logical operators (Guo et al., 29 Jul 2025).

The central geometric distinction from BB codes lies in local growth on the underlying left/right Cayley graphs. In BB codes with $x$3, the number of distinct vertices reachable by short words grows polynomially, $x$4. In suitable ZSZ instances, the corresponding growth is exponential,

$x$5

For example, with $x$6, constructing $x$7 via binary decomposition requires $x$8 steps, yielding $x$9. For suitable parameters such as $y$0 and generators that do not commute pairwise, the qubit adjacency graph has diameter $y$1, whereas abelian 2BGA codes with check weight $y$2 have diameter $y$3. At the same time, ZSZ Cayley graphs have constant girth, at most $y$4 by constructive cycles, and are not global expanders (Guo et al., 29 Jul 2025).

3. Decoding behavior and threshold phenomena

Under circuit-level depolarizing noise, the paper uses a single-parameter local depolarizing model simulated with Stim: a single-qubit gate, including idle, fails with depolarizing probability $y$5; a two-qubit gate fails with probability $y$6; ancilla measurement and reset fail with probability $y$7. Syndrome extraction alternates $y$8-type then $y$9-type rounds, while the final data-qubit $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$0 measurement is noiseless for readout and a final $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$1-syndrome is inferred (Guo et al., 29 Jul 2025).

For global decoding over $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$2 rounds, the decoder stacks $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$3 copies of the $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$4-syndrome Tanner graph, inserts repetition-code detectors across time, and applies min-sum belief propagation for up to $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$5 iterations followed by combination-sweep OSD of order $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$6. Across several codes, the block logical error rate curves cross at $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$7. Below threshold, BLER decays exponentially in system size. Under the same noise model, the quoted threshold for unrotated surface codes decoded by MWPM is approximately $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$8, so the ZSZ family is presented as competitive given its nonlocal connectivity. The study also notes that ancilla hook errors of weight at most $G\cong \mathbb{Z}_{\ell}\times \mathbb{Z}_{m}$9 explain some non-monotonic BLER ordering with distance (Guo et al., 29 Jul 2025).

For single-logical error rates, high-rate ZSZ codes are reported as comparable to BB codes and surface codes of similar effective distance. Two explicit comparisons are highlighted: the LER slope of ZSZ360-1 matches that of a distance-19 surface code, and ZSZ288-1 matches BB288 and a distance-15 surface code.

The most distinctive numerical result concerns local “self-correcting” decoding. The decoder is a greedy single-shot local sweep: for $q>1$0-errors, an $q>1$1 is applied to a data qubit if doing so reduces the local $q>1$2-syndrome weight, with ties broken randomly. Qubits are partitioned into $q>1$3 non-overlapping color classes of the qubit adjacency graph, with $q>1$4 by Brooks’s theorem and $q>1$5–$q>1$6 in practice. This sweep is repeated every cycle, and at the end the final $q>1$7-syndrome is decoded using BP+LSD with $q>1$8 min-sum iterations and order-5 combination-sweep LSD (Guo et al., 29 Jul 2025).

Under this passive scheme, ZSZ codes show a sustainable threshold

$q>1$9

defined as the critical value below which the BLER per cycle stabilizes and decreases with increasing code size over long times. Finite-size scaling shows stabilization for at least $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$00 cycles. Under the same local decoder and noise model, the corresponding estimate for the four-dimensional toric code is approximately $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$01. BB codes show no evidence of a sustainable threshold under the same passive decoder (Guo et al., 29 Jul 2025).

The greedy update also admits a Glauber-dynamics interpretation. At inverse temperature $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$02,

$G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$03

where $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$04 is the change in syndrome weight. A uniform failure probability $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$05 can be upper-bounded through

$G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$06

with $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$07 for one Pauli sector in these codes. This links the observed sustainable threshold to effective temperatures and to linear confinement on small sets (Guo et al., 29 Jul 2025).

4. Neutral-atom realization in movable tweezer arrays

ZSZ codes are designed with neutral-atom implementation in mind. The $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$08 rectangular embedding places horizontal and vertical data qubits on links and uses $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$09 ancillas, one per check. A single-ancilla syndrome extraction round initializes ancillas, applies Hadamard gates on ancilla and data for $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$10-syndrome extraction, schedules CZ gates by routing, and then applies Hadamards again; the $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$11-syndrome round is analogous but omits Hadamards on data. Entangling gates are Rydberg-mediated, and single-qubit gates are implemented by optical pulses (Guo et al., 29 Jul 2025).

Routing exploits AOD grid transfers. For left-action routing associated with $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$12, a factor $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$13 is implemented by global vertical cyclic shifts together with a horizontal grid-type permutation of columns, described as riffle-shuffle style, in $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$14 AOD moves using auxiliary SLM traps; a factor $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$15 is a horizontal cyclic shift in $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$16 grid transfers. For right-action routing associated with $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$17, $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$18 requires distinct horizontal cyclic shifts per row, realized by picking up the whole grid and dropping each row with its shift to scratch columns in $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$19 moves, while $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$20 is a global vertical cyclic shift in $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$21 (Guo et al., 29 Jul 2025).

A full $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$22-syndrome round therefore takes $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$23 moves for horizontal coupling and $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$24 for vertical coupling; the $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$25-syndrome round is analogous with transposes and interchanged roles. For logarithmic-diameter instances with $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$26, the overall routing per round is $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$27. With selective transfers, defined by deepening chosen SLM traps to leave atoms behind, arbitrary two-dimensional permutations become $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$28, and ZSZ extraction becomes $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$29 irrespective of $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$30 (Guo et al., 29 Jul 2025).

The resource profile includes $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$31 scratch columns and $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$32 scratch rows at different stages. Scheduling is designed to avoid intersecting parallel AOD rows and columns, thereby minimizing disturbance. The simulations explicitly model Rydberg CZ errors, measurement/reset errors, and idling. In this setting, the work states that a complete round of syndrome extraction can be achieved using simple global motions of the atomic arrays (Guo et al., 29 Jul 2025).

5. Overhead, comparative performance, and unresolved questions

The family is positioned as a high-rate quantum-memory architecture. One explicit example is ZSZ360-1, $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$33, which matches the single-logical error slope of a distance-19 surface code while encoding $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$34 more logical qubits. Under passive decoding, ZSZ756 outperforms four copies of the four-dimensional toric code with $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$35, while using only approximately $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$36 of the physical qubits for the same $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$37. Relative to BB codes, ZSZ codes retain low overhead and long-range-connectivity advantages but exhibit a sustainable passive threshold not observed in BB codes. Their routing cost is $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$38–$G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$39 per round rather than $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$40, though the single-shot property may offset this by requiring fewer rounds per logical cycle (Guo et al., 29 Jul 2025).

The name “ZSZ” reflects the group-theoretic backbone $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$41: a semidirect product of two cyclic groups. The action is

$G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$42

and best-performing examples often use nontrivial $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$43 coprime to $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$44 and satisfying $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$45, such as $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$46 for $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$47 and $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$48 for $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$49 (Guo et al., 29 Jul 2025).

Several limitations remain explicit. ZSZ Tanner and Cayley graphs have constant girth and are not global expanders, so classical expander-code arguments do not directly apply. Formal proofs of linear confinement and of a macroscopic free-energy barrier remain open. The numerical sustainable threshold does not constitute a proof of true self-correction, which would require a lifetime growing with $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$50 under local thermal noise. Decoder robustness under accurate circuit-level local decoding is not yet established; the paper states that incorporating such noise will require state-vector simulations and that performance may improve through reinforcement learning or sweep rules. Performance is also sensitive to scheduling, to the specific choice of $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$51, and to hardware constraints including AOD speed, cross-talk, and erasure detection. Exploiting erasure information mid-circuit or with delayed erasures is expected to increase thresholds significantly. Finally, single-shot state preparation and lattice surgery for ZSZ codes are not yet established, and logical Pauli measurements and adapters likely incur $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$52 temporal overhead absent single-shot surgery (Guo et al., 29 Jul 2025).

6. Terminology and disambiguation across research areas

The term “ZSZ codes” is not unique across disciplines. In the 2025 quantum-error-correction literature, it denotes the non-abelian $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$53-based LDPC CSS family described above (Guo et al., 29 Jul 2025). In communications and sequence design, closely related terminology often refers instead to zero-sidelobe-zone or zero-correlation-zone constructions, including symmetrical Z-complementary code sets (SZCCSs) and cross Z-complementary sequence sets (CZCSSs), where the defining property is summed aperiodic-correlation cancellation in front-end and tail-end lag intervals rather than a quantum stabilizer construction (Zhou et al., 2020, Kumar et al., 2023, Kumar et al., 2022).

Other coding-theoretic usages are also unrelated to the quantum family. Mixed-alphabet algebraic code papers study $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$54-additive cyclic codes, $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$55-cyclic codes, and $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$56-skew cyclic codes, all with distinct module structures, Gray maps, and duality theories (Borges et al., 2017, Aydogdu et al., 2017, 1711.01816, Benbelkacem et al., 2019). The rendering paper “SZ Sequences: Binary-Based $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$57-Sequences” uses “SZ” for two block-matrix alphabets denoted $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$58 and $G=\mathbb{Z}_{\ell}\rtimes_q \mathbb{Z}_{m}$59, and notes that “ZSZ codes” appears informally in discussion but refers there to a low-discrepancy-sequence construction rather than a code in the quantum- or communications-theoretic sense (Ahmed et al., 26 May 2025).

Accordingly, in current arXiv usage, “ZSZ codes” most prominently names the neutral-atom-oriented quantum LDPC family introduced in 2025, but the same letter sequence also appears in several unrelated literatures with different formal meanings, different performance metrics, and different ambient algebraic structures.