CSS-Preserving Equivalence Relation

• CSS-preserving equivalence relation is a framework where transformations must preserve designated CSS structures such as the X/Z split, Tanner graph, or Borel complexity.
• It employs mechanisms including restriction to large sets, local relabeling with marginalization, and simplicial isomorphism to maintain invariant structural presentations.
• This concept unifies approaches across descriptive set theory, quantum information, and coding theory by converting complex invariants into consistent, preservable forms.

“CSS-preserving equivalence relation” is not a single standardized term in the supplied literature. Instead, it names a recurring pattern: an equivalence or preservation notion that keeps a distinguished CSS-type structure invariant under transformation. In descriptive set theory, the relevant structure is a “complexity simplification somewhere” phenomenon, where analytic or coanalytic equivalence relations with Borel classes become Borel on a large $I^+$ Borel set under proper idealized forcing (Chan, 2015). In quantum information, the term is used more literally for Calderbank–Shor–Steane structure: equivalent factor-graph formulations of CSS decoding related by relabeling and marginalization (Kasai, 6 May 2026), coverings and simplicial isomorphisms of Tanner cone-complexes preserving the partition into $X$-checks, qubits, and $Z$-checks (Guemard, 2024), finite-field extensions preserving support and orthogonality (Kasai, 29 Oct 2025), and CSS-preserving stabilizer channels characterized by purely linear quadratic-form data and exact classical rewritings (Yashin et al., 7 Nov 2025). This suggests a general schema in which the preserved object is not merely an equivalence class of states, but a constrained structural presentation.

1. General schema and scope

Across the supplied sources, the preserved structure varies, but the formal pattern is stable. One starts with a class of objects carrying an explicitly designated decomposition, support pattern, or definability profile, and then defines transformations that preserve that extra structure. In the quantum-coding papers, the preserved data are the $X/Z$ split, orthogonality, Tanner incidence pattern, or the affine-linear form of a CSS-preserving channel. In the descriptive-set-theoretic paper, the preserved datum is the fact that an equivalence relation with Borel classes can be canonized to a Borel restriction on a large positive set. In each case, equivalence is stronger than mere behavioral or set-theoretic coincidence because it is required to respect additional internal organization (Chan, 2015).

The literature therefore uses several concrete mechanisms for “preservation.” One mechanism is restriction to a large set, as in canonization under proper forcing. Another is local relabeling plus marginalization, as in the equivalence of joint BP and four-state BP for CSS decoding. A third is simplicial isomorphism or covering, as in lifts of Tanner cone-complexes. A fourth is support-preserving relabeling of matrix entries, as in non-binary extensions of binary CSS pairs. A fifth is equality of linear form data $(H,s,V)$, or equivalently equality of a hidden-variable transition kernel, for CSS-preserving stabilizer channels (Kasai, 6 May 2026).

A plausible unifying interpretation is that a CSS-preserving equivalence relation identifies two presentations exactly when the transformation between them preserves the designated CSS structure and the invariants computed from it. The sources differ on what those invariants are—Borel complexity, posterior distribution, homology, orthogonality, or classical simulation kernel—but each source insists that preservation is stricter than extensional coincidence.

2. Descriptive-set-theoretic canonization as “complexity simplification somewhere”

For a Polish space $X$, a $\sigma$-ideal $I$, and the idealized forcing

$$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$

Chan studies when an analytic or coanalytic equivalence relation with all classes Borel becomes Borel on a large Borel subset. The arrow notation

$\Lambda \rightarrow_I \Gamma$

means that for every $X$0 Borel $X$1 and every equivalence relation $X$2 on $X$3 such that $X$4, there is a Borel $X$5 set $X$6 with $X$7 (Chan, 2015).

The central analytic theorem states: if $X$8 is a $X$9-ideal on a Polish space $Z$0 such that $Z$1 is proper, and $Z$2 is an analytic equivalence relation on $Z$3 with all classes Borel, then—assuming $Z$4 exists for all reals $Z$5 and $Z$6 exists—there is a Borel $Z$7 set $Z$8 such that $Z$9 is Borel. A parallel theorem holds for coanalytic equivalence relations under the same hypotheses. Chan also shows that if sharps exist for all sets in $X/Z$0, then

$X/Z$1

for every $X/Z$2-ideal $X/Z$3 with $X/Z$4 proper (Chan, 2015).

The proof uses several specific ingredients. Properness is characterized via the Borel $X/Z$5 sets of generics

$X/Z$6

For analytic $X/Z$7, Burgess approximations $X/Z$8 satisfy

$X/Z$9

with club many $(H,s,V)$0 equivalence relations; for coanalytic $(H,s,V)$1, there is an increasing sequence with union $(H,s,V)$2. Chan also proves that “all classes of $(H,s,V)$3 are Borel” can be reduced to a $(H,s,V)$4 statement, which is then handled באמצעות $(H,s,V)$5-absoluteness under sharps. The restriction $(H,s,V)$6 is then identified with some Borel $(H,s,V)$7 (Chan, 2015).

Interpreting this as a CSS-preserving equivalence is explicitly a reinterpretation in the source. There, “CSS” is read as a “complexity simplification somewhere” phenomenon: on a large Borel positive set, the definability complexity of the equivalence relation collapses to Borel. The paper also delineates the boundary of that phenomenon. In $(H,s,V)$8, there is a $(H,s,V)$9 equivalence relation $X$0 with all classes countable such that for every $X$1-ideal $X$2 and every $X$3 Borel $X$4, the restriction $X$5 is not Borel. Thus canonization is robust at the $X$6 level under sharps, but can fail already at projective level $X$7 (Chan, 2015).

3. Equivalence of CSS decoding formulations

For a CSS code with binary parity-check matrices

$X$8

a Pauli error is represented by $X$9, with local prior $\sigma$0. The joint posterior used in the factor-graph formulation of joint BP is

$\sigma$1

This factorization consists of two Tanner graphs, one for $\sigma$2 and one for $\sigma$3, coupled only through the local qubit priors $\sigma$4. It therefore retains local $\sigma$5 correlation whenever $\sigma$6 does not factor as $\sigma$7 (Kasai, 6 May 2026).

The same posterior can be written in a four-state Pauli-label representation. With

$\sigma$8

and $\sigma$9, the four-state posterior is

$I$0

The theorem in the paper proves exact equality under relabeling: $I$1 It also proves that, for every BP iteration, check-to-variable messages agree after relabeling, binary variable-to-check messages are obtained by marginalizing over the irrelevant component of the four-state messages, and the local beliefs satisfy

$I$2

With consistent tie-breaking, the hard decisions agree as well (Kasai, 6 May 2026).

Within that framework, the paper makes explicit a CSS-preserving equivalence notion between decoding representations. Two decoding formulations are equivalent when there exists a bijective local relabeling of per-qubit states and, for each edge type, a fixed marginalization making the sum-product recursions consistent, such that their posteriors agree under the global relabeling and their BP messages and beliefs correspond at every iteration. Concretely, the relabeling is $I$3 via $I$4, while the required marginalization sums over $I$5 on an $I$6-check edge and over $I$7 on a $I$8-check edge. The paper also specifies the conditions under which the equivalence is exact: CSS structure, the posterior factorizations above, the local product prior $I$9, exact probability-domain sum-product, and compatible initialization (Kasai, 6 May 2026).

The contrast with separate BP is decisive. Separate BP replaces each $$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$0 by its marginals $$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$1 and $$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$2, disconnecting the two Tanner graphs and losing all local $$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$3 correlation. Because this changes the posterior rather than merely relabeling or marginalizing it, separate BP is not CSS-preserving equivalent to joint BP or four-state BP (Kasai, 6 May 2026).

4. Geometric and homological equivalence for CSS codes

Every CSS code with parity-check matrices $$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$4 and $$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$5 defines a 3-term chain complex

$$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$6

and, from its Tanner graph, a canonical 2-dimensional simplicial complex called the Tanner cone-complex. If $$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$7, if

$$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$8

and if

$$\mathbb P_I=\{B\subseteq X: B \text{ is Borel and } B\notin I\}, \qquad \leq = \subseteq,$$9

then the Tanner cone-complex $\Lambda \rightarrow_I \Gamma$0 has vertex set $\Lambda \rightarrow_I \Gamma$1, 1-skeleton $\Lambda \rightarrow_I \Gamma$2, and 2-simplices $\Lambda \rightarrow_I \Gamma$3. The code can be recovered from $\Lambda \rightarrow_I \Gamma$4 by taking the 1-skeleton, removing the $\Lambda \rightarrow_I \Gamma$5–$\Lambda \rightarrow_I \Gamma$6 edges, and interpreting vertices according to their type. This makes $\Lambda \rightarrow_I \Gamma$7 a canonical geometric invariant of the CSS code up to simplicial isomorphism preserving the partition $\Lambda \rightarrow_I \Gamma$8 (Guemard, 2024).

This canonical complex supports a natural geometric equivalence relation. A simplicial isomorphism

$\Lambda \rightarrow_I \Gamma$9

with

$X$00

induces an isomorphism of Tanner graphs and a chain-complex isomorphism $X$01. Such an isomorphism preserves code length, the numbers of $X$02-checks, LDPC structure, homology and cohomology $X$03, and CSS distances $X$04. The paper further defines lifts of a CSS code as finite coverings $X$05 and proves that any such cover yields a valid CSS code $X$06 with $X$07. Connected covers are classified by subgroups of $X$08, and the corresponding lifted code $X$09 has length scaled by the degree $X$10, preserved maximum row and column weights, and a regular deck-group action when the cover is normal (Guemard, 2024).

The same paper makes this equivalence explicit for hypergraph product codes. If $X$11 is a hypergraph product code built from classical Tanner graphs $X$12, then

$X$13

and connected lifts are classified by Goursat quintuples for finite-index subgroups of that direct product. The paper then proves that the resulting lifts are equal, as CSS codes, to the Panteleev–Kalachev lifted product codes. In that setting, lift-equivalence is not merely heuristic: it is realized by equality of Tanner graphs together with equality of the partition into $X$14-checks, qubits, and $X$15-checks (Guemard, 2024).

A related but distinct structural passage appears for subsystem stabilizer codes. Any subsystem stabilizer code given by a subspace $X$16 can be mapped by the doubling construction

$X$17

to a subsystem CSS code with parameters

$X$18

The same paper also proves, via Goursat’s Lemma, that every subsystem stabilizer code is determined by two nested subsystem CSS codes

$X$19

together with an isomorphism

$X$20

A subsystem code is CSS exactly when $X$21, equivalently $X$22. This suggests a CSS-preserving equivalence framework in which internal and external CSS codes serve as invariants, while the deviation from CSS form is measured by the quotient isomorphism $X$23 (Liu et al., 2023).

5. Support-preserving and channel-preserving equivalence

For a binary CSS pair

$X$24

the finite-field extension problem asks for matrices

$X$25

such that

$X$26

and

$X$27

In the LDPC-CSS case emphasized in the paper, each pair of rows overlaps in either $X$28 or $X$29 positions. Writing nonzero entries as powers of a primitive element $X$30,

$X$31

orthogonality becomes a sparse linear congruence system

$X$32

For arbitrary even-overlap CSS pairs in characteristic $X$33, the paper gives a canonical separable assignment

$X$34

which guarantees $X$35 because each row-pair sum collapses to $X$36, and $X$37 is even (Kasai, 29 Oct 2025).

From these constructions, the paper extracts a fixed-support CSS-equivalence notion. Two labeled pairs $X$38 and $X$39 over possibly different finite fields are CSS-equivalent, with fixed support, when one can pass from one to the other by row operations on $X$40 and on $X$41, simultaneous column permutations, permissible scaling operations that preserve orthogonality, and—in the $X$42-overlap case—relabeling by adding nullspace vectors of the congruence system $X$43. Under this equivalence, the support pattern, overlap structure, Tanner graph, block length, and orthogonality are invariant, while field labels and sometimes distance properties can vary (Kasai, 29 Oct 2025).

An analogous but more algebraic equivalence appears for CSS-preserving stabilizer circuits. The paper defines CSS-preserving rebit stabilizer circuits as those built from initialization of $X$44 and $X$45, one-rebit gates $X$46 and $X$47, two-qubit gate $X$48, Walsh–Hadamard transform $X$49 on all rebits, $X$50- and $X$51-basis measurements, discarding channels, and classical control, with no standalone $X$52 gate. On the channel level, CSS-preserving Clifford channels are exactly those whose standard quadratic-form expansion is purely linear: $X$53 The paper states that it suffices to store a triple $X$54 to represent such a channel (Yashin et al., 7 Nov 2025).

That linear form admits two equivalent operational presentations. First, every CSS-preserving circuit can be rewritten gate by gate into a classical probabilistic circuit on paired $X$55- and $X$56-bits, and the rewriting is exact: the output distribution of the classical circuit equals the output distribution of the quantum circuit for all inputs. Second, under the Walsh–Hadamard–Fourier transform, a CSS-preserving channel induces a stochastic affine hidden-variable kernel

$X$57

The paper therefore identifies three equivalent CSS-preserving equivalence notions: equality, up to gauge, of the linear data $X$58; equality of the classical rewritten circuits as classical input-output channels; and equality of the hidden-variable transition kernels. All three coincide with equality of the underlying CSS-preserving Clifford channel (Yashin et al., 7 Nov 2025).

6. Related preservation paradigms, contextuality, and limits

Several other supplied papers exhibit the same methodological pattern, even when they are not about CSS codes in the narrow Calderbank–Shor–Steane sense. In concurrency theory, enabling-preserving bisimilarity is defined on labelled transition systems with successors by enriching state equivalence to triples $X$59, where $X$60 matches enabled transitions and transports this matching through a successor relation $X$61. The resulting equivalence preserves justness exactly: if $X$62 and $X$63 are ep-bisimilar paths, then

$X$64

A related contextual program appears in reversible CCS, where strong back-and-forth barbed congruence on RCCS is shown to correspond to hereditary history-preserving bisimulation on configuration structures. These are not CSS-code constructions, but they show the same design principle: an equivalence relation is strengthened until it preserves the structural feature that ordinary behavioral equivalence forgets (Glabbeek et al., 2021, Aubert et al., 2015, Aubert et al., 2018).

Universal algebra supplies a further analogue. For an algebra $X$65, an equivalence relation is a congruence exactly when it is preserved by every basic operation in $X$66, and the clone $X$67 consists of all congruence-preserving operations. In the concrete case of $X$68, all congruences are the relations $X$69, and the unary self-maps preserving all congruences are exactly those with an expansion

$X$70

The paper frames this as preservation of an entire family of equivalence relations rather than of a single one, which is structurally close to the CSS-preserving viewpoint when “CSS” is interpreted as a designated family of similarity structures (Pouzet, 2017).

Taken together, these examples delimit both the power and the limits of CSS-preserving equivalence relations. Their power lies in converting a semantic or algebraic invariant into a transport law: Borel complexity is canonized on a large set, posterior distributions are preserved under relabeling, CSS chain complexes are preserved under covers, orthogonality survives support-preserving relabelings, and certain stabilizer channels are reduced to classical affine kernels. Their limit is equally clear. When the designated structure is genuinely altered rather than merely repackaged—by discarding $X$71 correlation in separate BP, by moving to higher projective complexity where no Borel canonization exists, or by introducing non-CSS stabilizer operations with nontrivial quadratic form $X$72—the equivalence ceases to be CSS-preserving in the strict sense (Kasai, 6 May 2026, Chan, 2015, Yashin et al., 7 Nov 2025).