Hypergraph Product Codes and Quantum LDPC

• Hypergraph product codes are CSS quantum LDPC codes derived by taking the Kronecker product of two classical parity-check matrices, achieving constant rate and distance scaling as √n.
• They synthesize concepts from Tanner graphs, stabilizer formalisms, and chain-complex constructions, offering an explicit alternative to geometric topological codes like the surface code.
• These codes demonstrate robust decoding performance with specialized linear-time and hybrid decoders, while variations reduce physical overhead and enhance fault tolerance.

Hypergraph product codes are CSS quantum low-density parity-check codes obtained from two classical linear codes by a Kronecker-product construction of parity-check matrices. They are a central family of quantum LDPC codes because, for suitable classical inputs, they combine finite asymptotic rate with distance scaling as $\Theta(\sqrt{n})$, thereby providing an explicit alternative to zero-rate geometric topological codes such as the surface code (Yi et al., 2024). They also sit at a conceptual junction between classical Tanner graphs, CSS stabilizer formalisms, and chain-complex or homological constructions; in particular, the $m=2$ case of higher-dimensional product complexes recovers the original Tillich–Zémor hypergraph product, while special choices of input matrices reproduce toric-code families (Zeng et al., 2018).

1. Construction and algebraic form

Let

$$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$

be parity-check matrices of two classical binary linear codes. A standard CSS presentation of the hypergraph product defines physical qubits on the disjoint union of “check of $H_1$ $\times$ variable of $H_2$” and “variable of $H_1$ $\times$ check of $H_2$,” so that

$n = m_1 n_2 + n_1 m_2,$

with

$m=2$0

and $m=2$1 by the mixed Kronecker identities (Yi et al., 2024). An equivalent presentation used elsewhere associates physical qubits to “variable-type” qubits in $m=2$2 and “check-type” qubits in $m=2$3, with the same essential Kronecker-product structure for $m=2$4- and $m=2$5-stabilizers (Manes et al., 2023).

The Tanner-graph interpretation is equally standard. Each $m=2$6 is the biadjacency matrix of a bipartite graph with variable and check nodes, and the hypergraph product is a Cartesian-like product of those Tanner graphs. This viewpoint makes clear why the LDPC property transfers: if $m=2$7 and $m=2$8 have bounded row and column weight, then every stabilizer has bounded support and every qubit participates in $m=2$9 stabilizers (Yi et al., 2024).

A symmetric specialization, frequently used in the literature, starts from one classical code $$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$0 with full-rank parity-check matrix $$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$1. In that case the resulting HGP code has parameters

$$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$2

and the qubits may be viewed as living on $$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$3 and $$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$4, where $$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$5 and $$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$6 are the variable and check node sets of the classical Tanner graph (Berthusen et al., 2023).

2. Parameters, asymptotics, and comparison with topological codes

If the constituent classical codes have parameters $$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$7, then a typical hypergraph-product code has asymptotic behavior

$$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$8

Choosing classical LDPC codes with $$H_1 \in \mathbb{F}_2^{m_1 \times n_1}, \qquad H_2 \in \mathbb{F}_2^{m_2 \times n_2}$$9 and $H_1$0 yields a quantum LDPC code with constant rate and

$H_1$1

which is the canonical asymptotic regime of HGP codes (Yi et al., 2024). Multiple expositions in the supplied literature emphasize the same point in slightly different language: by choosing the right classical codes, HGP codes can have

$H_1$2

(Manes et al., 2023).

This tradeoff sharply contrasts with the two-dimensional surface code. The latter has parameters

$H_1$3

so that $H_1$4, whereas hypergraph product codes preserve the $H_1$5-distance scaling while increasing the number of logical qubits to a constant fraction of $H_1$6 (Manes et al., 2023). A central price is nonlocality: unlike the surface code, finite-rate HGP stabilizers are not geometrically local in 2D and are instead defined combinatorially by sparse product graphs (Manes et al., 2023).

The symmetric hypergraph-product formulas can also be written in terms of the transpose codes. If $H_1$7 defines a code with parameters $H_1$8, then the hypergraph product has parameters

$H_1$9

making explicit that both the original and transpose classical codes enter the quantum dimension and distance (Zhao et al., 2024).

3. Homological formulation and higher-dimensional extensions

Hypergraph product codes admit a natural chain-complex formulation. A classical parity-check matrix $\times$0 defines a $\times$1-complex

$\times$2

and tensor products of such complexes generate product complexes whose adjacent boundary maps furnish CSS codes (Zeng et al., 2018). In this language, a general finite chain complex

$\times$3

satisfies $\times$4, and the $\times$5-th homology group has rank

$\times$6

Choosing $\times$7 and $\times$8 yields a CSS code whose logical operators are the homology and cohomology classes of the complex (Zeng et al., 2018).

The tensor product of a general complex $\times$9 with a $H_2$0-complex $H_2$1 produces the higher-dimensional quantum hypergraph-product family studied by Zeng and Pryadko. If $H_2$2 with $H_2$3, then

$H_2$4

and the exact distance formula becomes

$H_2$5

which is the principal technical statement of that work for products with $H_2$6-complexes (Zeng et al., 2018). The case $H_2$7 recovers the original Tillich–Zémor hypergraph product, while suitable choices of $H_2$8 reproduce all families of toric codes on $H_2$9-dimensional hypercubic lattices (Zeng et al., 2018).

The same homological viewpoint underlies more recent fault-tolerance constructions. In particular, the initialization protocol based on “dimension jump” thickens an HGP chain

$H_1$0

by taking a homological product with a repetition-code chain, producing a higher-dimensional code with metachecks and soundness and then collapsing back to the original HGP block (Hong, 2024).

4. Decoding theory and empirical decoding performance

A distinctive feature of HGP codes is the coexistence of rigorous decoding theory on special subclasses and strong empirical performance under several heuristic or hybrid decoders. For quantum expander codes, a subclass of HGP codes built from classical expander codes, the small-set-flip decoder runs in linear time and corrects adversarial errors up to

$H_1$1

under the expansion hypotheses stated in the literature (Grospellier et al., 2020). In the phenomenological setting with noisy syndrome bits, a threshold theorem due to Fawzi–Grospellier–Leverrier underlies later work on HGP fault tolerance (Berthusen et al., 2023).

At the numerical level, Grospellier and Krishna studied two prominent HGP families under independent bit- and phase-flip noise. For a $H_1$2-biregular family with rate $H_1$3, qubits of weight $H_1$4 or $H_1$5, and stabilizers of weight $H_1$6, they reported a threshold near $H_1$7 for the small-set-flip decoder; for a $H_1$8-biregular family with rate $H_1$9, qubits of weight $\times$0 and $\times$1, and stabilizers of weight $\times$2, they reported a threshold near $\times$3 (Grospellier et al., 2018). The same work also found that, for similar rate, the hypergraph product code outperformed the toric code once the number of logical qubits exceeded roughly $\times$4, and that for $\times$5 logical qubits the logical error rate was several orders of magnitude smaller (Grospellier et al., 2018).

Hybrid decoding materially improved this picture. Combining belief propagation with small-set-flip, the $\times$6-regular HGP family studied in “Combining hard and soft decoders for hypergraph product codes” achieved an empirical threshold of roughly $\times$7 under ideal syndrome extraction and remained close to $\times$8 in the presence of syndrome noise (Grospellier et al., 2020). This work emphasized that low-complexity heuristic decoders can substantially narrow the practical gap between finite-rate quantum LDPC codes and surface-code-style decoders (Grospellier et al., 2020).

For the quantum erasure channel, specialized HGP structure can be exploited more directly. The “Fast erasure decoder for hypergraph product codes” proposed a decoder whose numerical performance is close to maximum-likelihood while requiring $\times$9 bit operations, with a probabilistic version in $H_2$0 bit operations (Connolly et al., 2022). More recently, optimization of the underlying classical Tanner graphs against erasure noise using random walks, simulated annealing, and reinforcement learning produced symmetric HGP codes that outperform PEG-based baselines under erasure ML decoding, with improvement also transferring to bit-flip noise under BP+OSD decoding (Freire et al., 16 Jan 2025).

A separate development concerns noisy-syndrome decoding via classical reductions. For HGP codes built from classical codes whose code and dual are “simultaneously nice,” noisy-syndrome quantum decoding can be reduced to noisy or exact syndrome decoding of the underlying classical codes; the paper establishing this gives near-linear-time reductions for families including expander codes, Reed–Solomon codes, and folded Reed–Solomon codes (Gandikota et al., 8 Oct 2025).

5. Syndrome extraction, effective distance, and initialization

Because finite-rate HGP stabilizers are nonlocal in two-dimensional hardware, syndrome extraction is a central fault-tolerance issue. A naive ancilla-based measurement of a weight-$H_2$1 stabilizer can spread a single ancilla fault to as many as $H_2$2 data errors, which in general can reduce the effective distance to

$H_2$3

For HGP codes, however, “Distance-preserving stabilizer measurements in hypergraph product codes” proves a much stronger statement: HGP codes are distance-robust, meaning that any valid stabilizer measurement circuit preserves the effective distance of the code (Manes et al., 2023). In particular, the depth-optimal constant-depth circuits of Tremblay et al. are also effective-distance-optimal, so there is no hidden tradeoff between measurement depth and circuit-level distance for this family (Manes et al., 2023).

The same nonlocality motivates partial-syndrome strategies in hardware-constrained settings. “Partial Syndrome Measurement for Hypergraph Product Codes” considers schemes in which only a subset of generators are measured in a given round. Under a simplified random-masking model and a modified small-set-flip decoder, the work proves that a threshold still exists when a sufficiently high percentage of generators is measured, and numerically finds exponential suppression of the logical error rate even when a large constant fraction of generators is not measured (Berthusen et al., 2023). For one $H_2$4-LDPC-derived HGP family at $H_2$5, the extracted suppression parameter satisfies $H_2$6 with full syndrome, $H_2$7 at $H_2$8 masking, and crosses below $H_2$9 at $n = m_1 n_2 + n_1 m_2,$0 masking under simple scheduling, while iterative scheduling restores $n = m_1 n_2 + n_1 m_2,$1 at $n = m_1 n_2 + n_1 m_2,$2 masking (Berthusen et al., 2023).

Initialization exhibits an analogous tension between shallow depth and noisy measurements. “Single-shot preparation of hypergraph product codes via dimension jump” shows that one can prepare the codespace of constant-rate HGP codes in constant circuit depth with $n = m_1 n_2 + n_1 m_2,$3 spatial overhead, and that the protocol remains robust in the presence of measurement errors (Hong, 2024). The thickened homological product used there has parameters

$n = m_1 n_2 + n_1 m_2,$4

and the protocol replaces the usual $n = m_1 n_2 + n_1 m_2,$5-depth repeated-measurement bottleneck with a constant-depth preparation procedure followed by a controlled dimensional collapse (Hong, 2024).

6. Variants, optimization, and overhead reduction

A longstanding theme in the HGP literature is that the core product idea admits substantial variation without abandoning LDPC structure. An early systematic example is “Improved quantum hypergraph-product LDPC codes,” which introduced rotated toric and checkerboard lattices and several algebraic modifications of the hypergraph product. Among its geometric outcomes is the family of minimal single-qubit-encoding toric codes

$n = m_1 n_2 + n_1 m_2,$6

and among its algebraic outcomes are square, symmetric, and two-tile product constructions that can improve the rate of the original hypergraph-product family by up to about a factor of four while retaining LDPC structure (Kovalev et al., 2012).

At the opposite end of the tradeoff space, the TGRE-hypergraph-product construction targets unusually high coding rate. Using Z-TGRE and X-TGRE component codes of rate $n = m_1 n_2 + n_1 m_2,$7, the resulting code has

$n = m_1 n_2 + n_1 m_2,$8

and a code-capacity depolarizing threshold around $n = m_1 n_2 + n_1 m_2,$9 under FDBP decoding, but its component-code distances scale only as $m=2$00, so the quantum distance is $m=2$01, and the stabilizer weights scale as $m=2$02 rather than $m=2$03 (Yi et al., 2024). The paper explicitly notes that this makes the family non-LDPC in the conventional constant-check-weight sense, despite the high rate and strong code-capacity behavior (Yi et al., 2024).

Global symmetry constraints have recently produced another direction of improvement. “Cyclic Hypergraph Product Code” restricts the classical inputs to cyclic codes, enabling exhaustive search over a symmetry-reduced space. In that framework, C2 codes and C$m=2$04R codes were found that outperform previously optimized HGP codes; concrete examples include $m=2$05 and $m=2$06 C2 codes, with logical error rate per logical qubit approximately $m=2$07 and below $m=2$08, respectively, at physical $m=2$09, and a constant-depth syndrome extraction layout on a $m=2$10 QCCD architecture with per-round depth

$m=2$11

(Aydin et al., 12 Nov 2025).

The most direct attack on physical-qubit overhead appears in “Spatial overhead reduction for 2D hypergraph product codes.” That work gives reductions that preserve the code dimension, canonical logical basis, and minimum distances of the HGP code, and also provides distance-preserving syndrome measurement schedules. Reported examples include

$m=2$12

and

$m=2$13

with memory simulations under circuit-level depolarizing noise showing similar subthreshold performance to unreduced codes while using fewer physical qubits (Pabla et al., 11 May 2026).

7. Energy barriers and generalized product-code phases

Distance is a static parameter; the energy barrier probes the dynamical difficulty of implementing a logical error through local error paths. For an HGP code built from classical parity-check matrices $m=2$14, “On the energy barrier of hypergraph product codes” proves that, under a mild condition automatically satisfied for LDPC families with growing classical barriers,

$m=2$15

Thus the hypergraph product does not create an energy barrier larger than the minimum barrier already present in the classical codes and their transpose codes (Zhao et al., 2024). In particular, hypergraph products of classical LDPC codes with non-growing energy barrier necessarily have constant quantum energy barrier, whereas expander-based choices can inherit macroscopic barriers (Zhao et al., 2024). This sharply constrains proposals to use standard HGP codes as self-correcting quantum memories.

Generalized product constructions also connect HGP ideas to fracton and higher-order topological phenomena. The orthoplex models introduced as a class of generalized HGP codes place qubits on all odd-dimensional cells of a product complex and stabilize them with orthoplex-shaped interactions. In three dimensions, the resulting model has

$m=2$16

exhibits lineons and planons, and supports non-Abelian lattice defects; in four dimensions it supports “fragmented topological excitations,” namely point-like excitations whose projections onto lower-dimensional subsystems form connected loop-like objects (Li et al., 14 Jan 2026). These generalized HGP codes lie beyond standard CSS hypergraph products, but they show that product-code techniques form an analytically tractable platform not only for quantum LDPC constructions but also for exotic many-body order (Li et al., 14 Jan 2026).