Cyclic Hypergraph Product Code

• The paper introduces cyclic hypergraph product codes that leverage cyclic matrices to impose global symmetries, enabling an exhaustive search over structured instances.
• The paper demonstrates improved [N,K,D] tradeoffs with logical error rates up to three orders of magnitude lower than those of ML-optimized hypergraph product codes.
• The paper outlines a constant-depth syndrome extraction protocol using a two-row QCCD layout that supports scalable implementations in trapped-ion and flying-qubit hardware.

Cyclic hypergraph product codes are a symmetry-constrained family of quantum low-density parity-check CSS codes obtained by taking the hypergraph product of two cyclic codes. In the formulation introduced in "Cyclic Hypergraph Product Code" (Aydin et al., 12 Nov 2025), the cyclic restriction defines a class denoted CxC, together with two subfamilies, C2 and CxR. The construction is motivated by a contrast with prior optimization of hypergraph product codes through progressive edge growth, random search, simulated annealing, and reinforcement learning: instead of local transformations, it imposes additional global symmetries and permits exhaustive search over structured instances. The reported consequence is a family of codes with improved $[N,K,D]$ tradeoffs, logical error rates per logical qubit up to three orders of magnitude lower than previously optimized HGP codes, and a planar QCCD layout supporting constant-depth syndrome extraction.

1. Hypergraph-product setting

The cyclic construction is a specialization of the general hypergraph-product CSS construction. Let $A\in\mathbb{F}_2^{m_a\times n_a}$ and $B\in\mathbb{F}_2^{m_b\times n_b}$ be classical parity-check matrices. The associated CSS code has block length

$N=n_a n_b + m_a m_b,$

with sparse check matrices

$$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$

These have dimensions $(m_a n_b)\times N$ and $(n_a m_b)\times N$, respectively, and satisfy $H_XH_Z^T=0$ by construction (Aydin et al., 12 Nov 2025).

Within the Tillich–Zémor framework quoted in the exposition, if $r_a=\mathrm{rank}\,A$ and $r_b=\mathrm{rank}\,B$, then

$A\in\mathbb{F}_2^{m_a\times n_a}$0

and

$A\in\mathbb{F}_2^{m_a\times n_a}$1

This places cyclic hypergraph product codes squarely inside the standard HGP paradigm rather than outside it: the novelty lies in the symmetry of the seed matrices, not in a different commutativity mechanism or a different CSS assembly.

As context, HGP codes are presented as one of the most popular families of quantum LDPC codes, and circuit-level simulations are cited as showing that they can achieve the same logical error rate as surface codes with reduced qubit overhead. The cyclic construction is therefore best viewed as a structured refinement of an already competitive LDPC family.

2. Cyclic specialization and the C2/CxR subfamilies

The cyclic ingredient is introduced through binary cyclic matrices. A binary cyclic matrix of size $A\in\mathbb{F}_2^{m_a\times n_a}$2 has the form

$A\in\mathbb{F}_2^{m_a\times n_a}$3

with coefficients $A\in\mathbb{F}_2^{m_a\times n_a}$4. By construction,

$A\in\mathbb{F}_2^{m_a\times n_a}$5

In the cyclic hypergraph-product construction, one takes

$A\in\mathbb{F}_2^{m_a\times n_a}$6

so that

$A\in\mathbb{F}_2^{m_a\times n_a}$7

Because $A\in\mathbb{F}_2^{m_a\times n_a}$8 and similarly for $A\in\mathbb{F}_2^{m_a\times n_a}$9, the full parity-check structure is doubly block-circulant: shifting the $B\in\mathbb{F}_2^{m_b\times n_b}$0 row-blocks by one and the $B\in\mathbb{F}_2^{m_b\times n_b}$1 column-blocks by one leaves $B\in\mathbb{F}_2^{m_b\times n_b}$2 and $B\in\mathbb{F}_2^{m_b\times n_b}$3 invariant (Aydin et al., 12 Nov 2025).

Two distinguished subfamilies are isolated.

1. C2 codes: these are the symmetric cyclic instances obtained by choosing

$B\in\mathbb{F}_2^{m_b\times n_b}$4

so the HGP is taken between a cyclic code and itself.

2. CxR codes: these are repeated cyclic instances obtained by choosing

$B\in\mathbb{F}_2^{m_b\times n_b}$5

where $B\in\mathbb{F}_2^{m_b\times n_b}$6 is identified as the $B\in\mathbb{F}_2^{m_b\times n_b}$7 repetition code.

The role of symmetry is central. The work explicitly contrasts this with machine-learning-based searches that improve code performance through local transformations. A plausible implication is that the cyclic ansatz restricts the search space in a way that exposes globally organized high-performance instances that may be difficult to reach through unconstrained local optimization.

3. Parameter formulas

Let the classical cyclic code $B\in\mathbb{F}_2^{m_b\times n_b}$8 have parameters $B\in\mathbb{F}_2^{m_b\times n_b}$9, so that

$N=n_a n_b + m_a m_b,$0

If $N=n_a n_b + m_a m_b,$1 has parameters $N=n_a n_b + m_a m_b,$2 and rank $N=n_a n_b + m_a m_b,$3, then the exposition gives the CxC parameters as

$N=n_a n_b + m_a m_b,$4

using the stated facts that for a cyclic parity check $N=n_a n_b + m_a m_b,$5 and $N=n_a n_b + m_a m_b,$6, and that $N=n_a n_b + m_a m_b,$7. The logical-qubit count and minimum distance are

$N=n_a n_b + m_a m_b,$8

For the two subfamilies, these formulas simplify substantially (Aydin et al., 12 Nov 2025).

For C2 codes,

$N=n_a n_b + m_a m_b,$9

where $$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$0 is the number of nonzero terms in the seed polynomial of $$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$1.

For CxR codes,

$$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$2

These closed forms are important for two reasons. First, they make the effect of the cyclic seed transparent at the level of $$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$3 and check weight $$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$4. Second, they explain why exhaustive search is feasible in the reported study: the search is over a small-parameter, symmetry-constrained family rather than over arbitrary sparse parity-check matrices.

4. Exemplary codes and comparative tradeoffs

The exposition highlights three codes from an exhaustive search, all simulated with check weight $$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$5. The comparison baseline "ML-HGP $$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$6" is identified as the state-of-the-art HGP code optimized by simulated annealing and reinforcement learning in Freire et al. (Aydin et al., 12 Nov 2025).

Code	Parameters	Comparison
C2 $$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$7	$$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$8, $$H_X=\bigl[A\otimes I_{n_b}\;\big|\;I_{m_a}\otimes B^T\bigr], \qquad H_Z=\bigl[I_{n_a}\otimes B\;\big|\;A^T\otimes I_{m_b}\bigr].$$9, rate $(m_a n_b)\times N$0, $(m_a n_b)\times N$1	Same $(m_a n_b)\times N$2 as ML-HGP $(m_a n_b)\times N$3, shorter $(m_a n_b)\times N$4, more $(m_a n_b)\times N$5; at $(m_a n_b)\times N$6, $(m_a n_b)\times N$7
C2 $(m_a n_b)\times N$8	$(m_a n_b)\times N$9, $(n_a m_b)\times N$0, rate $(n_a m_b)\times N$1, $(n_a m_b)\times N$2	Up to 3 orders better $(n_a m_b)\times N$3 than ML-HGP at $(n_a m_b)\times N$4
CxR $(n_a m_b)\times N$5	$(n_a m_b)\times N$6, $(n_a m_b)\times N$7, rate $(n_a m_b)\times N$8, $(n_a m_b)\times N$9	Outperforms ML-HGP $H_XH_Z^T=0$0 despite lower rate

The comparison is not limited to prior HGP optimization. The abstract further states that some C2 codes achieve simultaneously a lower logical error rate and a smaller qubit overhead than state-of-the-art LDPC codes such as bivariate bicycle codes, with the qualification that this comes at the price of a larger block length. That qualification is material: the reported advantage is not a blanket dominance over all competing LDPC families on every axis.

A common misconception would be to treat the cyclic restriction as merely aesthetic or as a source of implementation convenience only. The examples instead show that the same symmetry constraint can improve both code parameters and circuit-level behavior relative to earlier optimized HGP instances.

5. Decoding, syndrome extraction, and QCCD layout

Decoding is performed with belief propagation plus ordered statistics decoding, BP+OSD, following Panteleev–Kovalev (2021). The implementation emphasis, however, is not only on decoding but on a syndrome-extraction schedule tailored to the cyclic structure.

The proposed QCCD layout uses two rows. All data qubits are placed in an upper $H_XH_Z^T=0$1 row, and all ancilla qubits in a lower $H_XH_Z^T=0$2 row. A cyclic-shift primitive $H_XH_Z^T=0$3 permutes ancilla indices so as to align each ancilla with the appropriate block of data qubits. The exposition states that trapped-ion or other "flying-qubit" hardware can realize these long-range cyclic permutations with shuttling (Aydin et al., 12 Nov 2025).

Data qubits are indexed as

$H_XH_Z^T=0$4

and ancillas as

$H_XH_Z^T=0$5

A syndrome round decomposes into four phases:

1. Prepare all ancilla in $H_XH_Z^T=0$6.
2. For each monomial $H_XH_Z^T=0$7 in $H_XH_Z^T=0$8, perform a $H_XH_Z^T=0$9 cyclic shift, then simultaneous CX gates

$r_a=\mathrm{rank}\,A$0

and CZ gates

$r_a=\mathrm{rank}\,A$1

followed by measurement/reset of the $r_a=\mathrm{rank}\,A$2-ancilla in the $r_a=\mathrm{rank}\,A$3-basis.

1. For each monomial $r_a=\mathrm{rank}\,A$4 in $r_a=\mathrm{rank}\,A$5, perform a $r_a=\mathrm{rank}\,A$6 cyclic shift, then simultaneous CX gates

$r_a=\mathrm{rank}\,A$7

and CZ gates

$r_a=\mathrm{rank}\,A$8

followed by measurement/reset of the $r_a=\mathrm{rank}\,A$9-ancilla in $r_b=\mathrm{rank}\,B$0.

2. Repeat $r_b=\mathrm{rank}\,B$1 rounds for fault tolerance.

Because the CX and CZ gates associated with each monomial are non-overlapping, each loop executes in a single layer. The resulting two-qubit-gate depth per syndrome round is

$r_b=\mathrm{rank}\,B$2

independent of $r_b=\mathrm{rank}\,B$3 and $r_b=\mathrm{rank}\,B$4. Including ancilla preparation and measurement, the total depth for $r_b=\mathrm{rank}\,B$5 rounds is

$r_b=\mathrm{rank}\,B$6

The paper therefore characterizes the syndrome extraction as constant-depth at LDPC scale. The significance is not merely asymptotic: the depth bound is directly tied to seed-polynomial weights rather than to the block dimensions.

6. Circuit-level performance and comparative interpretation

The reported simulations use standard depolarizing circuit noise in which gate, measure, reset, and idle all occur at physical rate $r_b=\mathrm{rank}\,B$7. The logical error rate $r_b=\mathrm{rank}\,B$8 is normalized per round per logical qubit, and a heuristic fit of the form

$r_b=\mathrm{rank}\,B$9

is given, with coefficients $A\in\mathbb{F}_2^{m_a\times n_a}$00 tabulated in Appendix Table A1 (Aydin et al., 12 Nov 2025).

At $A\in\mathbb{F}_2^{m_a\times n_a}$01, the key values singled out in the exposition are: $A\in\mathbb{F}_2^{m_a\times n_a}$02

$A\in\mathbb{F}_2^{m_a\times n_a}$03

$A\in\mathbb{F}_2^{m_a\times n_a}$04

These figures support the stated claim of up to three orders of magnitude suppression of logical error at comparable or lower qubit overhead.

The comparison against the surface-code heuristic

$A\in\mathbb{F}_2^{m_a\times n_a}$05

is reported to show that an LDPC advantage emerges already for modest $A\in\mathbb{F}_2^{m_a\times n_a}$06. In addition, plots of $A\in\mathbb{F}_2^{m_a\times n_a}$07 versus inverse rate $A\in\mathbb{F}_2^{m_a\times n_a}$08 and versus $A\in\mathbb{F}_2^{m_a\times n_a}$09 are said to confirm that the C2 family, and even the simpler CxR family, outperform both ML-optimized HGP codes and leading bivariate bicycle codes in many regimes.

The comparative picture is therefore specific rather than absolute. The cyclic construction is reported to improve on earlier HGP codes under full circuit-level noise, and in some cases to exceed leading bivariate bicycle codes in logical error rate and qubit overhead, but the abstract explicitly notes the larger block length of those C2 instances. This suggests that cyclic hypergraph product codes are best understood as a high-performing structured corner of the quantum LDPC design space, distinguished by the joint appearance of exhaustive searchability, favorable $A\in\mathbb{F}_2^{m_a\times n_a}$10 tradeoffs with $A\in\mathbb{F}_2^{m_a\times n_a}$11, and constant-depth syndrome extraction in a two-row QCCD architecture.