Nested CSS Code Pairs

• The paper demonstrates that nested CSS pairs can be constructed via classical code inclusion, chain maps, and geometric methods, enabling efficient logical operator transfer and fault-tolerant quantum error correction.
• It outlines algebraic, dual-containing, and homological formulations that unify cyclic, toric, and finite-degree LDPC constructions to achieve near-optimal code parameters.
• The discussion highlights open challenges in extending these techniques to non-CSS measurements, sparse decoding strategies, and practical implementations in quantum computing.

Nested Calderbank–Shor–Steane code pairs are families of CSS constructions in which one code, code space, or chain complex is related to another by an inclusion, a dual-containment condition, or a homological map that preserves the logical sector. In the most classical formulation, the relevant data are linear codes $C_1 \subsetneq C_0 \subseteq \mathbb{F}_q^n$, or the dual-containing specialization $C^\perp \subseteq C$, from which CSS stabilizer codes are obtained. In more recent homological formulations, a “nested pair” may instead mean two CSS chain complexes linked by a chain map, a mapping cone, or a pushout, so that logical operators, code parameters, or measurement outcomes are transferred between distinct codes rather than extracted from a single dual-containing classical code. Across these formulations, nested CSS pairs underpin logical measurements, code surgery, code embeddings, communication-efficient quantum secret sharing, and finite-degree quantum LDPC constructions (Guardia, 2017, Huang et al., 2022, Cowtan et al., 2023, Yuan, 7 Jul 2025, Kasai, 25 Mar 2026).

1. Algebraic foundations of nesting in CSS theory

A binary CSS code with parameters $[[n,k,d]]$ can be specified by parity-check matrices

$$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$

satisfying

$H_X H_Z^T = 0.$

The $X$-stabilizers are rows of $H_X$, the $Z$-stabilizers are rows of $H_Z$, logical $X$-operators are representatives in $C^\perp \subseteq C$0, and logical $C^\perp \subseteq C$1-operators are in $C^\perp \subseteq C$2. The corresponding distances are

$C^\perp \subseteq C$3

Equivalently, the code is a length-3 chain complex

$C^\perp \subseteq C$4

with $C^\perp \subseteq C$5, $C^\perp \subseteq C$6, and $C^\perp \subseteq C$7 (Huang et al., 2022).

A second, widely used formulation starts from nested classical codes. If $C^\perp \subseteq C$8, then $C^\perp \subseteq C$9 is an $[[n,k,d]]$0 stabilizer code with

$[[n,k,d]]$1

The dual-containing specialization $[[n,k,d]]$2 yields an $[[n,k,d]]$3 stabilizer code that is pure to $[[n,k,d]]$4 (Senthoor et al., 2022, Guardia, 2017).

These two formulations are compatible. The parity-check condition $[[n,k,d]]$5 is the matrix form of the chain-complex relation $[[n,k,d]]$6, while the classical inclusion $[[n,k,d]]$7 is the conventional CSS orthogonality condition. Nested CSS pairs therefore include both code pairs defined by subspace containment and code pairs defined by morphisms between distinct chain complexes (Huang et al., 2022, Yuan, 7 Jul 2025).

2. Dual-containing and geometric realizations

In algebraic coding theory, the most direct nested CSS pair is the chain

$[[n,k,d]]$8

The paper on cyclic codes restricts precisely to this situation and uses the CSS lemma: if there exists a classical linear $[[n,k,d]]$9 code $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$0 such that $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$1, then there exists an $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$2 stabilizer code that is pure to $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$3 (Guardia, 2017).

That construction is realized through $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$4-cyclotomic cosets. If a cyclic code has a defining set consisting of a single $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$5-coset $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$6 containing $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$7 consecutive integers, then the BCH bound gives $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$8. Under the conditions of Theorem 3.1, there exists an $$H_X : F^n \to F^{r_X},\qquad H_Z : F^n \to F^{r_Z},$$9 cyclic code, where $H_X H_Z^T = 0.$0 is the cardinality of the $H_X H_Z^T = 0.$1-coset containing those consecutive integers. If additionally $H_X H_Z^T = 0.$2, the code is dual-containing and produces an $H_X H_Z^T = 0.$3 quantum code (Guardia, 2017). The explicit examples include $H_X H_Z^T = 0.$4, $H_X H_Z^T = 0.$5, $H_X H_Z^T = 0.$6, and $H_X H_Z^T = 0.$7; some of these satisfy $H_X H_Z^T = 0.$8, so they lie close to the Quantum Singleton Bound (Guardia, 2017).

A geometric variant appears in toric-surface constructions. There, the classical code $H_X H_Z^T = 0.$9 is an evaluation code obtained from sections of $X$0 on a smooth complete toric surface, evaluated on a finite support set $X$1. The key duality mechanism is a dualizing differential form

$X$2

whose residues define a weighted dual. The resulting restricted code satisfies

$X$3

where $X$4. If the weights are squares, a coordinatewise rescaling produces $X$5, hence a dual-containing CSS pair in the usual Euclidean sense (Hansen, 2012). On toric Hirzebruch surfaces, the construction is made explicit through toric divisors, intersection theory, and residue calculations (Hansen, 2012).

These two lines of work illustrate that “nested CSS pair” need not refer to a single combinatorial paradigm. In one case the nesting is encoded by cyclotomic-coset combinatorics; in the other it is encoded by divisor theory and residues. The invariant feature is the same: a classical code contains an appropriate dual, so that the CSS orthogonality constraint is satisfied (Guardia, 2017, Hansen, 2012).

3. Homological and categorical formulations

A broader notion of nested CSS pair arises when CSS codes are treated as chain complexes rather than merely as dual-containing classical codes. In the homomorphic-measurement framework, one has a data code $X$6, an ancilla code $X$7, and a binary matrix

$X$8

describing the CNOT pattern. The defining compatibility conditions are

$X$9

These conditions imply the existence of maps

$H_X$0

with

$H_X$1

so the ancilla and data codes form a commutative diagram of chain complexes. The ancilla–data pair is therefore “nested” by a chain homomorphism rather than by literal subspace inclusion (Huang et al., 2022).

The embedding framework of 2025 makes this point systematic. A CSS code with parity-check matrices $H_X$2 is represented as

$H_X$3

with $H_X$4 encoding $H_X$5-type logicals and $H_X$6 encoding $H_X$7-type logicals. The central construction is a multi-level cone whose differential is block lower triangular. In the 3-level case,

$H_X$8

and the “embedded” code is the column complex

$H_X$9

Under the regularity assumption $Z$0 and $Z$1, Theorem 1.1 gives a natural isomorphism

$Z$2

In this sense, nested CSS pairs consist of an input code and a larger output code with canonically isomorphic logical sectors, even though the output may have more qubits, more checks, different geometry, or lower check weight (Yuan, 7 Jul 2025).

Code surgery provides a categorical analogue. There, a code map is a chain map $Z$3, and surgery is expressed by a pushout in the category of chain complexes. Merging two codes along a shared logical operator subcomplex $Z$4 produces a new code $Z$5 that identifies the logicals represented on $Z$6 and $Z$7. This formulation treats nested CSS pairs as monic chain maps, subcomplex inclusions, and universal colimits rather than only as dual-containing classical code pairs (Cowtan et al., 2023).

4. Nested pairs in logical measurement, surgery, and code modification

Homomorphic logical measurement makes the operational role of nested CSS pairs explicit. The ancilla is prepared in

$Z$8

whose $Z$9-stabilizer space is $H_Z$0. After the CNOT pattern $H_Z$1, any $H_Z$2 transforms to $H_Z$3. If $H_Z$4, the outcome is classical syndrome information. If $H_Z$5, then $H_Z$6 is a nontrivial ancilla logical $H_Z$7, and $H_Z$8 is a nontrivial $H_Z$9-type logical operator of the data code. Thus the logical $X$0-operators implementable by a gadget are

$X$1

modulo stabilizers (Huang et al., 2022).

This framework unifies Shor and Steane measurements. Shor measurement is a degenerate homomorphic gadget with ancilla distance $X$2, while Steane measurement is the identity case $X$3 and $X$4. The intermediate regime uses ancilla codes with $X$5, $X$6, and LDPC structure, thereby “filling the space between Shor and Steane” (Huang et al., 2022). The related syndrome-extraction work on toric codes states that blocks of size $X$7 can be used to decode errors in $X$8 rounds of measurements, which makes the interpolation between Shor-style and Steane-style ancillas explicit at the circuit level (Huang et al., 2020).

For surface codes, the ancilla can be constructed from a covering space. If the data lattice is $X$9 on a closed surface $C^\perp \subseteq C$00, a chosen logical loop $C^\perp \subseteq C$01 determines a subgroup $C^\perp \subseteq C$02, an intermediate quotient $C^\perp \subseteq C$03, and a finite subcomplex $C^\perp \subseteq C$04 encoding exactly one logical qubit with distance $C^\perp \subseteq C$05. The induced map

$C^\perp \subseteq C$06

yields chain maps $C^\perp \subseteq C$07, with $C^\perp \subseteq C$08, satisfying the homomorphic conditions. Conventional surface-code decoders, such as minimum-weight perfect matching, can be directly applied to these constructions (Huang et al., 2022).

The same operational theme appears in code surgery. If two CSS codes share a separated logical operator subcomplex, their $C^\perp \subseteq C$09-merge identifies the corresponding logical $C^\perp \subseteq C$10’s. Under separation, the merged code has

$C^\perp \subseteq C$11

and, if no extra logical qubits are introduced,

$C^\perp \subseteq C$12

For LDPC families, the merged code remains LDPC, with

$C^\perp \subseteq C$13

$C^\perp \subseteq C$14

The sandwiched merge protocol then gives a distance-$C^\perp \subseteq C$15 measurement of $C^\perp \subseteq C$16 or, dually, $C^\perp \subseteq C$17, provided the gauge-fixing and distance-bounded-below conditions hold (Cowtan et al., 2023).

This suggests that nested CSS pairs serve not only as static code constructions but also as transition objects between codes. In homomorphic measurement they transfer a logical observable to an ancilla. In surgery they transfer a shared logical degree of freedom to a pushout code. In both cases, the nesting is the mechanism that controls fault propagation and logical equivalence (Huang et al., 2022, Cowtan et al., 2023).

5. Extended, concatenated, and communication-efficient nested pairs

In communication-efficient quantum secret sharing, nested CSS pairs are extended rather than merely embedded. The basic object is the extended CSS code

$C^\perp \subseteq C$18

built from a nested pair $C^\perp \subseteq C$19 and a matrix $C^\perp \subseteq C$20 with row space $C^\perp \subseteq C$21, subject to

$C^\perp \subseteq C$22

The resulting classical codes on $C^\perp \subseteq C$23 qudits are

$C^\perp \subseteq C$24

so the logical dimension remains $C^\perp \subseteq C$25 (Senthoor et al., 2022).

The access structure is determined by further nested pairs derived from $C^\perp \subseteq C$26 and subcodes $C^\perp \subseteq C$27. With prior access to $C^\perp \subseteq C$28 extension qudits, the recovery threshold is

$C^\perp \subseteq C$29

The full CE-QSS construction then uses a nested chain

$C^\perp \subseteq C$30

with

$C^\perp \subseteq C$31

An outer $C^\perp \subseteq C$32 is concatenated with an inner $C^\perp \subseteq C$33. In the GRS-based instantiation, the resulting CE-QSS scheme meets both the storage and communication cost bounds with equality (Senthoor et al., 2022).

A related, but asymptotic, use of nesting appears in partially concatenated CSS codes. There the key matrices are

$C^\perp \subseteq C$34

so that

$C^\perp \subseteq C$35

This nested pair yields a PC-CSS code $C^\perp \subseteq C$36 with

$C^\perp \subseteq C$37

thus asymptotically achieving the quantum Gilbert–Varshamov bound. The same paper combines this with Steane’s enlargement by constructing a nested triple

$C^\perp \subseteq C$38

which yields enlarged stabilizer codes satisfying

$C^\perp \subseteq C$39

It also gives two explicit families,

$C^\perp \subseteq C$40

and

$C^\perp \subseteq C$41

together with the stated $C^\perp \subseteq C$42, $C^\perp \subseteq C$43, and $C^\perp \subseteq C$44 encoding and decoding complexities (Fan et al., 2021).

Across these constructions, the nested pair is the device that separates roles. One pair carries the secret or the data, another pair carries extension or concatenation structure, and the inclusion relations determine which errors are logical, which are degenerate, and which recovery sets are authorized (Senthoor et al., 2022, Fan et al., 2021).

6. Finite-degree quantum LDPC pairs and current directions

The most recent finite-degree LDPC construction makes the nesting explicit at the level of sparse matrices. One starts from

$C^\perp \subseteq C$45

so that

$C^\perp \subseteq C$46

With a square sparse matrix $C^\perp \subseteq C$47, the visible classical codes are

$C^\perp \subseteq C$48

Because $C^\perp \subseteq C$49, one has $C^\perp \subseteq C$50, hence $C^\perp \subseteq C$51 is a CSS pair (Kasai, 25 Mar 2026).

In the homogeneous regular regime, with a balanced triple satisfying $C^\perp \subseteq C$52, the design quantum rate is

$C^\perp \subseteq C$53

so positivity requires $C^\perp \subseteq C$54. For fixed balanced triples with even degrees and $C^\perp \subseteq C$55, the actual rates converge in probability to the design rates, and both constituent classical codes have minimum distance $C^\perp \subseteq C$56 with high probability. Consequently the CSS relative distances are also linear (Kasai, 25 Mar 2026).

For the seven explicit balanced triples

$C^\perp \subseteq C$57

the same paper proves, by a rigorous computer-assisted exponent analysis, that both classical constituents attain the classical GV distance and the resulting CSS code attains the CSS GV bound at finite degree (Kasai, 25 Mar 2026). This moves the notion of nested CSS pair into a regime where bounded check degree, positive rate, and GV-optimal asymptotic distance coexist.

Several open directions are stated explicitly in the literature. Homomorphic logical measurements are left open beyond the CSS setting: “Generalizations of our framework to non-CSS Pauli measurements is a natural direction, which is left for future work” (Huang et al., 2022). For product-based LDPC codes, the same paper states that “It is unlikely that for codes with product constructions, useful homomorphic gadgets have product structure as well” (Huang et al., 2022). The embedding framework identifies a general homological criterion for preserving logical qubits, but it also notes that distance arguments for some subdivisions remain open (Yuan, 7 Jul 2025). The finite-degree LDPC construction leaves sparse syndrome representatives and practical BP decoding as unresolved issues and formulates a conjecture that any balanced triple with $C^\perp \subseteq C$58 should attain finite-degree GV behavior (Kasai, 25 Mar 2026).

A common misconception is that nested CSS code pairs are exhausted by the dual-containing condition $C^\perp \subseteq C$59. The literature here shows a broader picture. Nested pairs include dual-containing cyclic and toric-surface codes, ancilla–data pairs linked by a chain homomorphism, input–output code pairs related by mapping cones, and code families glued by surgery pushouts (Guardia, 2017, Hansen, 2012, Huang et al., 2022, Cowtan et al., 2023, Yuan, 7 Jul 2025). This suggests that “nested CSS pair” is best understood as a structural relation that preserves CSS compatibility while transporting logical information across algebraic, geometric, and operational transformations.