Quantum Rank-Metric Codes

• Quantum Rank-Metric Codes are quantum error-correcting codes that use the rank of qubit arrays—rather than Hamming weight—to model and correct multi-layer gate faults in stacked memories.
• They extend classical Gabidulin code concepts by incorporating techniques such as Hermitian orthogonality and CSS stabilizer constructions to achieve maximum rank distance.
• Their design exploits the accumulated-rank bound to guarantee correction of low-rank Pauli errors, supporting efficient syndrome extraction and polynomial-time decoding.

Searching arXiv for papers on quantum rank-metric codes and stacked quantum memory. Quantum rank-metric codes are quantum error-correcting codes designed for stacked quantum memories, in which an $m \times n$ or $\ell \times n$ array of qubits is organized into layers and cells, and the relevant size of a Pauli error is its binary rank on that array rather than its Hamming weight. In the formulation introduced for stacked implementations of Clifford circuits, a small number of faulty multi-layer gates induces a final Pauli error whose rank is bounded linearly by the number of gate faults, making rank distance the natural coding parameter. The initial construction is a quantum generalization of Gabidulin codes for square stacked memories, and a later construction based on Hermitian orthogonality and the Matsumoto–Uyematsu method removes the requirement of odd square layouts while preserving maximum-rank-distance structure and efficient decoding principles (Delfosse et al., 2024, Nizuka et al., 4 May 2026).

1. Stacked-memory model and the rank-based fault picture

A stacked quantum memory of size $\ell \times n$ is a register of $\ell n$ qubits organized into $n$ cells of $\ell$ qubits each. The $i$-th layer is obtained by selecting qubit $i$ in every cell, so layers are length-$n$ rows and cells are length-$\ell$ columns. In a stacked implementation of an $\ell \times n$0-qubit Clifford circuit $\ell \times n$1, each gate is executed in parallel across layers: a 1-qubit gate on qubit $\ell \times n$2 is realized as $\ell \times n$3 on cell $\ell \times n$4, and a 2-qubit gate on qubits $\ell \times n$5 and $\ell \times n$6 is realized as $\ell \times n$7 acting jointly on cells $\ell \times n$8 and $\ell \times n$9 pairwise across layers (Delfosse et al., 2024).

The circuit fault model places a Pauli fault after each multi-layer gate $\ell \times n$0 with probability $\ell \times n$1. The fault is drawn uniformly from the non-identity Paulis supported on the cells touched by the gate: one cell for a 1-qubit gate and two cells for a 2-qubit gate. This models intra-cell crosstalk and pairwise faults spanning two directly operated cells, while assuming low crosstalk between distinct cells except when a two-cell gate acts on them. In the later matrix-based notation, an $\ell \times n$2 stacked memory is represented by $\ell \times n$3, with the left $\ell \times n$4 columns encoding $\ell \times n$5 components and the right $\ell \times n$6 columns encoding $\ell \times n$7 components; the rank of the Pauli is then $\ell \times n$8 (Delfosse et al., 2024, Nizuka et al., 4 May 2026).

The central algebraic fact is the accumulated-rank bound. If at most $\ell \times n$9 gates are faulty, then the total stacked error $\ell n$0 satisfies

$\ell n$1

In the square-memory construction, the proof uses four ingredients: Clifford conjugation maps Paulis to Paulis, Clifford conjugation preserves rank, a single fault after a 1- or 2-qubit gate generates a stacked error of rank at most $\ell n$2, and rank is subadditive under multiplication of stacked Pauli errors. In the later formulation, the same phenomenon is expressed as invariance of $\ell n$3 under $\ell n$4 for Clifford $\ell n$5, together with confinement of a faulty gate’s effect to at most four columns of the $\ell n$6 binary matrix (Delfosse et al., 2024, Nizuka et al., 4 May 2026).

This establishes the physical motivation for the rank metric. The relevant error morphology is not sparsity of qubit support but low-dimensionality of the row space induced by layerwise correlations and shared-gate faults. A plausible implication is that code design should optimize rank distance whenever the hardware and compilation strategy actually generate this fault geometry.

2. Classical rank metric and Gabidulin structure

For vectors $\ell n$7 with $\ell n$8, the rank metric is

$\ell n$9

where each coordinate of $n$0 is expanded as an $n$1-dimensional vector over $n$2. Equivalently, $n$3 is embedded as a matrix over $n$4, and its matrix rank is taken. In the matrix formulation, for $n$5 one has $n$6; the vector and matrix viewpoints are equivalent, and the induced rank metric is basis-independent (Delfosse et al., 2024, Nizuka et al., 4 May 2026).

Gabidulin codes are the canonical maximum-rank-distance family in this setting. If $n$7 are $n$8-linearly independent and

$n$9

then

$\ell$0

is an $\ell$1 code with minimum rank distance

$\ell$2

meeting the Singleton bound for the rank metric and hence being MRD. A standard generator matrix is the $\ell$3-Moore matrix whose rows are successive Frobenius powers of $\ell$4 (Delfosse et al., 2024, Nizuka et al., 4 May 2026).

In the square binary construction, a trace-orthogonal normal basis of $\ell$5 over $\ell$6 plays a further role. With such a basis, the dual of $\ell$7 is

$\ell$8

This duality is one of the mechanisms by which the classical MRD structure is transferred into a stabilizer construction (Delfosse et al., 2024).

Decoding is also inherited from the classical theory. Syndrome-based decoders for Gabidulin codes, formulated through linearized-polynomial algebra, recover errors of rank $\ell$9 in roughly $i$0 operations over the extension field, with subquadratic variants available. The 2024 quantum construction leaves the concrete design and optimization of a full quantum decoder as future work, but it explicitly relies on the fact that the underlying Gabidulin structure admits polynomial-time syndrome decoding (Delfosse et al., 2024, Nizuka et al., 4 May 2026).

3. CSS-type quantum Gabidulin codes on square stacked memories

The first explicit quantum rank-metric family is a stabilizer construction on an $i$1 stacked memory, with $i$2 odd. One chooses a trace-orthogonal normal basis

$i$3

of $i$4 over $i$5 satisfying $i$6. Each $i$7 is embedded into a binary $i$8 array, and from this embedding one defines $i$9 and $i$0 operators on the square array of qubits. The relevant bilinear form is

$i$1

and $i$2 and $i$3 commute if and only if $i$4 (Delfosse et al., 2024).

For integers $i$5 with $i$6, the quantum Gabidulin code $i$7 is defined by taking all $i$8 with $i$9 as $n$0-stabilizers and all $n$1 with $n$2 as $n$3-stabilizers. Commutation follows from the duality relation $n$4. In the symmetric family $n$5 and $n$6, the code has parameters

$n$7

where the distance is a quantum rank distance: the minimum rank of a nontrivial Pauli that commutes with all stabilizers but is not in the stabilizer group (Delfosse et al., 2024).

The correction guarantee is expressed in rank rather than Hamming weight. Since $n$8, the code corrects any Pauli of rank

$n$9

Combining this with the circuit bound $\ell$0 yields the guarantee

$\ell$1

under ideal syndrome extraction. In symplectic matrix language, if $\ell$2 and $\ell$3 are binary parity-check matrices for the embedded Gabidulin spaces, then the commutation condition is

$\ell$4

over $\ell$5 (Delfosse et al., 2024).

This CSS-type construction has a strict algebraic limitation. As later emphasized, the use of a self-dual normal basis over $\ell$6 is possible if and only if $\ell$7 is odd, so the construction applies only to square $\ell$8 memories with odd $\ell$9. The later Hermitian-based work identifies this as the main layout restriction of the original approach (Nizuka et al., 4 May 2026).

4. Hermitian orthogonality and generalized quantum rank-metric constructions

A broader framework starts from a binary symplectic self-orthogonal linear space. If $\ell \times n$00 is linear with $\ell \times n$01, $\ell \times n$02, and

$\ell \times n$03

where $\ell \times n$04 reshapes a length-$\ell \times n$05 vector into an $\ell \times n$06 matrix, then there exists a quantum rank-metric stabilizer code with parameters

$\ell \times n$07

This theorem turns the rank of the reshaped symplectic representative into the quantum distance parameter directly (Nizuka et al., 4 May 2026).

The key new ingredient is Hermitian orthogonality. Over $\ell \times n$08 with even $\ell \times n$09, the Hermitian product is

$\ell \times n$10

A basis $\ell \times n$11 of $\ell \times n$12 over $\ell \times n$13 is self-dual if $\ell \times n$14; such a basis exists when $\ell \times n$15 is even, and in particular over $\ell \times n$16. With $\ell \times n$17 even and $\ell \times n$18 a self-dual basis, the Islam–Horlemann lemma states that if $\ell \times n$19 then

$\ell \times n$20

The Matsumoto–Uyematsu method then maps Hermitian self-orthogonality to binary symplectic self-orthogonality through an explicit coordinate expansion $\ell \times n$21 (Nizuka et al., 4 May 2026).

Applied to Gabidulin codes, this yields a new family. One works over $\ell \times n$22, selects a self-dual $\ell \times n$23-basis $\ell \times n$24, takes the classical MRD code $\ell \times n$25 with $\ell \times n$26, and maps it to a binary symplectic code $\ell \times n$27. The resulting quantum rank-metric code has parameters

$\ell \times n$28

The distance claim follows from preservation of duality under $\ell \times n$29, preservation of rank under $\ell \times n$30 and reshaping, and the fact that $\ell \times n$31 is again MRD with distance $\ell \times n$32 (Nizuka et al., 4 May 2026).

The main structural consequences are summarized below.

Construction	Layout condition	Parameters
CSS quantum Gabidulin	odd square $\ell \times n$33	$\ell \times n$34
Hermitian/MU quantum Gabidulin	removes odd-square restriction; explicit $\ell \times n$35 family	$\ell \times n$36

For comparable rates, the later paper states that the relative minimum rank distance per physical qubit is approximately doubled. In its notation, if $\ell \times n$37, then for the proposed family

$\ell \times n$38

while for the CSS family on the nearest odd square

$\ell \times n$39

so $\ell \times n$40 tends to $\ell \times n$41 as $\ell \times n$42 grows. The same paper also states that this construction eliminates the conventional requirement that the number of cells and layers be odd, thereby increasing design freedom for stacked memories (Nizuka et al., 4 May 2026).

5. Decoding, syndrome extraction, and representative parameters

In the circuit-level protocol, the stacked implementation of a Clifford circuit maps the initial stabilizer code to an image code by conjugation. If the initial code is $\ell \times n$43 and the circuit is applied layerwise, the conjugated image code $\ell \times n$44 has stabilizers obtained from those of the original code by Clifford conjugation. Because Clifford conjugation permutes or phase-rotates Pauli operators, it preserves both rank and the minimum quantum rank distance. Syndrome measurement is then performed on $\ell \times n$45 rather than on the original code (Delfosse et al., 2024).

The syndrome model separates $\ell \times n$46 and $\ell \times n$47 components. In the binary embedding, an unknown Pauli error decomposes as $\ell \times n$48, each component being an $\ell \times n$49 binary matrix in the square construction. Measuring $\ell \times n$50-type stabilizers yields syndrome information for $\ell \times n$51 through $\ell \times n$52, and measuring $\ell \times n$53-type stabilizers yields syndrome information for $\ell \times n$54 through $\ell \times n$55, with duality ensuring consistency. The end-to-end protocol is: encode a code state, apply the stacked Clifford circuit, measure the stabilizers of the conjugated code, decode to a minimum-rank Pauli $\ell \times n$56 consistent with the syndrome, and apply $\ell \times n$57 as recovery. Under ideal syndrome extraction, any pattern with

$\ell \times n$58

faulty gates is guaranteed correctable in the CSS family (Delfosse et al., 2024).

The decoding method is algebraic. Classically, one solves linearized-polynomial key equations for an error-locator and an error-evaluator; quantumly, the same Gabidulin structure is used to decode $\ell \times n$59 from $\ell \times n$60-syndromes and $\ell \times n$61 from $\ell \times n$62-syndromes. The 2024 work leaves an efficient full decoder for the conjugated image code as future work, but states that known Gabidulin decoding methods run in time polynomial in $\ell \times n$63 over $\ell \times n$64 arithmetic. The 2026 work makes the same point more concretely: after converting the symplectic syndrome back to an extension-field syndrome, one may run a Gabidulin or Loidreau-type rank decoder in roughly $\ell \times n$65 operations over $\ell \times n$66, followed by linear basis transforms and multiplication by the fixed matrices associated with the Matsumoto–Uyematsu map (Delfosse et al., 2024, Nizuka et al., 4 May 2026).

The published examples illustrate both the promise and the current scale. In the square CSS family, choosing $\ell \times n$67 and $\ell \times n$68 gives

$\ell \times n$69

This corrects all rank-$\ell \times n$70 Pauli errors on the $\ell \times n$71 array, but it does not guarantee correction of a single faulty gate because $\ell \times n$72. The same paper notes that practical single-fault correction requires $\ell \times n$73, hence odd $\ell \times n$74, giving

$\ell \times n$75

and guaranteeing correction of one faulty gate via $\ell \times n$76. In the Hermitian/MU family, the explicit example $\ell \times n$77, $\ell \times n$78 over $\ell \times n$79 yields an $\ell \times n$80 code (Delfosse et al., 2024, Nizuka et al., 4 May 2026).

A common misunderstanding is that these constructions already supply a complete practical fault-tolerant stack. Both papers are explicit that the formal guarantees assume ideal or noiseless syndrome extraction, and that robust fault-tolerant extraction and optimized decoding remain open engineering and algorithmic tasks.

6. Applications, comparisons, and open problems

The proposed application domain is high-throughput Clifford-dominated computation on stacked memories. The 2024 work specifically envisions stabilizer-state factories producing $\ell \times n$81, $\ell \times n$82, and Bell pairs in parallel across layers, as well as magic-state factories in which the Clifford part is parallelized and protected. It also suggests batched evaluation of Clifford-heavy subcircuits in variational quantum algorithms. In each case, the intended advantage is that shared-gate or crosstalk-induced faults are aligned with the low-rank patterns targeted by the code rather than with sparse Hamming-weight patterns (Delfosse et al., 2024).

This comparison to other code families is important. CSS, LDPC, and topological codes are described as excelling under local and approximately independent, Hamming-like noise. Subsystem codes may localize checks, but still typically optimize Hamming distance. Quantum rank-metric Gabidulin codes, by contrast, are dense and optimized for rank distance, so they are matched to a fault morphology in which errors are correlated across layers and cells but remain low rank as binary matrices. This does not imply superiority under all noise models; it implies specialization to stacked architectures whose circuit faults obey the accumulated-rank picture (Delfosse et al., 2024).

The later Hermitian-based construction strengthens this architectural case by removing the odd-square restriction and supporting code families built from self-dual bases over $\ell \times n$83. The paper states that even numbers of layers and cells are now supported, and also emphasizes explicit rectangular $\ell \times n$84 footprints. A plausible implication is that hardware-software co-design becomes less constrained, since rectangular layouts may better fit routing and packaging constraints, but the paper presents this as design freedom rather than as an experimentally established advantage (Nizuka et al., 4 May 2026).

The main limitations are stated plainly in both works. Practical deployment requires a hardware platform capable of hosting multi-qubit cells with low inter-cell crosstalk, support for simultaneous multi-layer gates with high fidelity, and connectivity for high-weight stabilizer measurement or alternatively LDPC-like rank-metric variants. Fault-tolerant syndrome extraction is still missing in the published protocols, and the required decoders for the conjugated image code need further design and optimization. Additional open directions include broader quantum rank-metric code families with flexible $\ell \times n$85, rigorous fault-tolerance analysis with thresholds and overheads, integration with non-Clifford resource injection, concatenated or subsystem variants, and refined circuit-level noise models that could improve the constant $\ell \times n$86 in the accumulated-rank bound (Delfosse et al., 2024, Nizuka et al., 4 May 2026).