Diagonal Clifford Symmetries Explained

• Diagonal Clifford symmetries are a family of operators that exploit computational-basis diagonal subgroups of the Clifford hierarchy to achieve efficient, near-diagonal representations.
• They enable square-dimensional phase-permutation representations and transversal operations in stabilizer codes, offering practical advantages in simulation and fault-tolerant quantum computing.
• This framework informs optimization in matrix-extent reduction and unitary design theory by preserving discrete symmetry constraints that simplify operator decomposition.

Searching arXiv for recent and foundational papers on diagonal Clifford symmetries and closely related formulations. Found a set of relevant papers spanning square-dimensional Clifford representations, diagonal hierarchy structure, transversal/code-theoretic classifications, and symmetry-reduced simulation. I’ll ground the article in these results and cite them by arXiv ID. Diagonal Clifford symmetries denote a family of closely related structures rather than a single universal construction. In the works surveyed here, the term covers computational-basis diagonal subgroups of the Clifford hierarchy, basis choices in which Clifford operators become phase-permutation rather than fully diagonal, and code- or symmetry-constrained Clifford operations that preserve a preferred diagonal decomposition (Cui et al., 2016, Bengtsson, 2012, Dasu et al., 14 Jul 2025). The common theme is that the full Clifford group is nonabelian and therefore not simultaneously diagonalizable in general, yet substantial parts of its action can often be reduced to diagonal, block-diagonal, or monomial form. That reduction controls hierarchy structure, transversal gate classification, unitary-design behavior, and symmetry-reduced classical simulation (Rengaswamy et al., 2019, Camillo et al., 21 Oct 2025, Mitsuhashi et al., 2023).

1. Basic notions and scope

The Clifford hierarchy is defined recursively from the Pauli group. In the qudit formulation, with first level $\mathcal C^{(1)}$ equal to the Pauli group up to phase, higher levels are

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$

and the diagonal part is denoted $\mathcal C_d^{(k)}$ (Cui et al., 2016). A basic structural fact is that, although $\mathcal C^{(k)}$ is not a group for $k\ge 3$ in general, the diagonal subset $\mathcal C_d^{(k)}$ is a group (Cui et al., 2016).

In the qubit simulation literature, a particularly important diagonal symmetry is the diagonal Pauli subgroup

$$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$

acting by conjugation. Its invariant Clifford subgroup is

$D_n=\mathcal C_n^{Z_n},$

identified there as the diagonal Clifford group and generated by $S$ and $CZ$ (Camillo et al., 21 Oct 2025).

The main senses of the subject can be organized as follows.

Sense	Defining structure	Representative source
Diagonal hierarchy gates	Computational-basis diagonal $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$0	(Cui et al., 2016)
Diagonal Clifford group	$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$1, generated by $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$2 and $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$3	(Camillo et al., 21 Oct 2025)
Nearly diagonal Clifford representation	Entire Clifford group becomes monomial in a special basis	(Bengtsson, 2012)
Transversal diagonal Clifford symmetry	$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$4 acting on $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$5 code blocks	(Dasu et al., 14 Jul 2025)

This taxonomy already excludes a common misconception. “Diagonal Clifford symmetry” does not usually mean that the whole Clifford group is diagonal in one basis. In one major construction, only a preferred commuting subgroup is genuinely diagonal, while the full Clifford group is merely phase-permutation (Bengtsson, 2012).

2. Square-dimensional phase-permutation representations

A decisive result of Bengtsson is that when the Hilbert-space dimension is a perfect square,

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$6

the finite Heisenberg group admits a basis in which the full Clifford group is represented by monomial unitary matrices, i.e. phase-permutation matrices (Bengtsson, 2012). The construction begins from generators $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$7 of the Heisenberg group satisfying

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$8

and exploits the square-dimensional identity

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$9

This produces a distinguished maximal abelian subgroup generated by $\mathcal C_d^{(k)}$0 and $\mathcal C_d^{(k)}$1.

In the adapted basis $\mathcal C_d^{(k)}$2, with $\mathcal C_d^{(k)}$3, the generators act by

$\mathcal C_d^{(k)}$4

and

$\mathcal C_d^{(k)}$5

Consequently,

$\mathcal C_d^{(k)}$6

Thus the subgroup generated by $\mathcal C_d^{(k)}$7 and $\mathcal C_d^{(k)}$8 is literally diagonal in this basis, while $\mathcal C_d^{(k)}$9, $\mathcal C^{(k)}$0, and in fact every Clifford unitary are phase-permutation operators (Bengtsson, 2012).

The conceptual mechanism is group-theoretic. In square dimension, the subgroup generated by $\mathcal C^{(k)}$1 and $\mathcal C^{(k)}$2 is the unique maximal abelian subgroup consisting solely of elements of order $\mathcal C^{(k)}$3. Because Clifford unitaries act by automorphisms of the Heisenberg group and preserve element order, they must permute this subgroup among itself, hence permute its joint eigenbasis up to phase. The resulting representation is therefore “almost diagonal”: diagonal on a canonical commuting Heisenberg subgroup, monomial on the full Clifford group (Bengtsson, 2012).

The same paper emphasizes that this is not full diagonalization. Even the Zauner unitary, a central symmetry in SIC-POVM studies, is not generically diagonal in the phase-permutation basis; after suitable rephasings and permutations it decomposes into $\mathcal C^{(k)}$4 cyclic blocks

$\mathcal C^{(k)}$5

plus some diagonal entries, with eigenvalues $\mathcal C^{(k)}$6 on each $\mathcal C^{(k)}$7 block (Bengtsson, 2012). The representation therefore realizes maximal sparsity compatible with noncommutativity, not simultaneous diagonalization.

3. Diagonal subgroups of the Clifford hierarchy

For prime-dimensional qudits, diagonal hierarchy gates admit a complete polynomial-phase classification. A one-qudit diagonal gate of the form

$\mathcal C^{(k)}$8

lies exactly at diagonal hierarchy level

$\mathcal C^{(k)}$9

and in the multiqudit case the degree parameter is replaced by the total weight $k\ge 3$0, so that

$k\ge 3$1

determines the level (Cui et al., 2016). In this language, diagonal Clifford gates are the special case $k\ge 3$2. For odd prime $k\ge 3$3, they are precisely quadratic polynomial phases modulo $k\ge 3$4; for qubits, the familiar diagonal Clifford generator $k\ge 3$5 appears already at level $k\ge 3$6 because the qubit hierarchy introduces $k\ge 3$7-th roots of unity at lower levels than the odd-prime case (Cui et al., 2016).

For qubits, a complementary algebraic model is provided by symmetric matrices over residue rings. Given $k\ge 3$8 and a symmetric $k\ge 3$9 matrix $\mathcal C_d^{(k)}$0 over $\mathcal C_d^{(k)}$1, one defines

$\mathcal C_d^{(k)}$2

Every such $\mathcal C_d^{(k)}$3 lies in $\mathcal C_d^{(k)}$4, and all two-local diagonal hierarchy unitaries arise in this form up to global phase (Rengaswamy et al., 2019). The associated ring-valued symplectic matrix is

$\mathcal C_d^{(k)}$5

so diagonal hierarchy gates retain the upper-triangular symplectic pattern familiar from diagonal Cliffords, but now over $\mathcal C_d^{(k)}$6 rather than $\mathcal C_d^{(k)}$7 (Rengaswamy et al., 2019).

The same framework yields an exact recursion under Pauli conjugation: $\mathcal C_d^{(k)}$8 At level $\mathcal C_d^{(k)}$9, the lower-level diagonal factor collapses and one recovers ordinary diagonal Clifford action. Beyond the Clifford group, the action no longer closes on Paulis but closes recursively on “Pauli times lower-level diagonal” (Rengaswamy et al., 2019). This is the precise algebraic sense in which diagonal Clifford symmetry extends to higher hierarchy levels.

A further refinement concerns square roots of Hermitian Clifford gates. If

$$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$0

then for Hermitian diagonal Clifford gates one obtains a sharp criterion for when $$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$1 lies in the third level. Writing a diagonal Clifford as

$$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$2

the relevant conditions are that the associated symplectic matrix $$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$3 be a hyperbolic involution and that

$$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$4

In that case $$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$5, and all such Hermitian Clifford gates are Clifford-conjugate to the canonical diagonal form

$$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$6

If $$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$7, then $$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$8 (Bastioni et al., 12 Mar 2026). Thus even within the diagonal Clifford subgroup, square-root lifting is rigid rather than automatic.

4. Transversal and code-theoretic diagonal Clifford symmetries

In stabilizer-code theory, the phrase “diagonal Clifford symmetries” acquires a distinct but closely related meaning. For an $$Z_n=\{Z(z):=Z^{z_1}\otimes\dots\otimes Z^{z_n}\mid z\in\mathbb Z_2^n\},$$9-qubit stabilizer code $D_n=\mathcal C_n^{Z_n},$0 and $D_n=\mathcal C_n^{Z_n},$1 code blocks, one studies $D_n=\mathcal C_n^{Z_n},$2-qubit Clifford operators $D_n=\mathcal C_n^{Z_n},$3 applied identically to each physical qubit position, i.e. gates of the form $D_n=\mathcal C_n^{Z_n},$4. Modulo phases, their tableaux lie in

$D_n=\mathcal C_n^{Z_n},$5

and the one-block endomorphism algebra

$D_n=\mathcal C_n^{Z_n},$6

determines the full $D_n=\mathcal C_n^{Z_n},$7-block structure through

$D_n=\mathcal C_n^{Z_n},$8

Up to local-diagonal Clifford equivalence, exactly six families occur: $D_n=\mathcal C_n^{Z_n},$9 corresponding respectively to self-dual CSS, $S$0-linear, non-self-dual CSS, self-dual non-CSS, semi-self-dual/semi-CSS, and generic stabilizer codes (Dasu et al., 14 Jul 2025). In this code-theoretic sense, diagonal Clifford symmetry is a transversal symmetry class fixed by a small matrix algebra, not merely a computational-basis diagonal gate set.

For CSS codes, diagonal logical operators built from single-qubit phase gates admit an explicit XP-operator description. At precision $S$1, a diagonal XP operator has the form

$S$2

and acts logically precisely when it satisfies explicit commutator constraints with the $S$3-checks. If $S$4 is an $S$5-check and $S$6 generates the diagonal logical identity group at level $S$7, then $S$8 is logical iff for every such $S$9,

$CZ$0

At level $CZ$1, $CZ$2, this is exactly the diagonal Clifford case; the lower-level identities are $CZ$3-type logical identities, and the method detects all transversal $CZ$4- and $CZ$5-type logical symmetries (Webster et al., 2023).

The same XP formalism extends to controlled-phase and phase-rotation gates via

$CZ$6

with duality relations that express each family through the other. This yields algorithms for: computing the diagonal logical identity group, testing whether a given diagonal XP operator is logical, generating all diagonal logical XP operators of a fixed precision, extracting their logical action as products of logical controlled-phase gates, and searching depth-one realizations with multi-qubit diagonal gates (Webster et al., 2023). The framework recovers, for example, logical $CZ$7 on the $CZ$8 code and a level-3 diagonal structure on the 3D hypercube code, where $CZ$9 acts as a logical identity and logical $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$00 and $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$01 operators appear in the same diagonal hierarchy (Webster et al., 2023).

5. Symmetry-restricted optimization, design theory, and operational consequences

For diagonal target unitaries, diagonal Clifford symmetry can be exploited exactly in matrix-extent optimization. If

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$02

denotes the matrix stabilizer extent, then for a diagonal unitary $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$03,

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$04

so optimization over the full Clifford group can be restricted without loss of optimality to the diagonal Clifford subgroup $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$05. For real-diagonal targets one can reduce further to

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$06

generated by $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$07 and $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$08 (Camillo et al., 21 Oct 2025). This strong symmetry reduction, together with weak reduction under additional invariances such as permutation symmetry, enables exact decompositions of diagonal and real-diagonal unitaries on up to seven qubits on a standard laptop and yields the reported exponential simulation improvements for controlled-phase families, QFT phase blocks, hypergraph-state generators, and Union Jack MBQC blocks (Camillo et al., 21 Oct 2025).

A distinct structural result concerns unitary designs under symmetry constraints. For a subgroup $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$09, define the symmetry-preserving unitary group

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$10

and the corresponding symmetric Clifford group

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$11

Then $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$12 is a $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$13-symmetric unitary $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$14-design iff

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$15

for some Pauli subgroup $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$16 (Mitsuhashi et al., 2023). Diagonal Pauli symmetries generated by commuting $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$17-type operators therefore preserve the exact 3-design property of the symmetric Clifford group. By contrast, continuous global diagonal symmetries such as

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$18

are not equivalent to Pauli-subgroup constraints for $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$19, and the corresponding symmetric Clifford group is a symmetric unitary $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$20-design but not a $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$21-design (Mitsuhashi et al., 2023). Discrete diagonal Pauli symmetry and continuous diagonal $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$22 symmetry are therefore sharply different from the design-theoretic viewpoint.

Recent work on early fault-tolerant architectures further emphasizes that the diagonal Clifford hierarchy supplies a closed phase-polynomial normal form but not a universal ordering principle for magic generation. In the qubit case, diagonal hierarchy gates are written as

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$23

with hierarchy level

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$24

and the paper proves both that hierarchy level alone cannot universally order operational magic and that no state-independent sequence of operations can guarantee monotonic magic improvement (Lu et al., 6 May 2026). In that setting, diagonal Clifford structure is analytically powerful but not by itself decisive: graph-state preconditioning and nonlinear multi-qubit diagonal phases are needed to escape the expressibility bottleneck of architectures restricted to single-qubit $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$25-rotations (Lu et al., 6 May 2026).

6. Related usages in Clifford-algebra and topological-symmetry theory

In adjacent literature on abstract Clifford algebras, closely related but distinct notions of diagonal Clifford symmetry appear through monomial bases and sign involutions. For $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$26, the canonical bilinear form $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$27 is diagonal in the standard monomial basis $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$28, and the signature-dependent automorphism

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$29

acts diagonally on monomials by

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$30

Its $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$31 eigenspaces define a Cartan decomposition

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$32

with

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$33

(Eberlein, 2017). Here “diagonal” refers to sign-diagonal action on a monomial basis rather than to computational-basis diagonal quantum gates.

A related block-diagonal philosophy appears in the classification of quadratic-fermion systems. On real Nambu space $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$34, the unitary symmetry subgroup $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$35 yields a real isotypic decomposition

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$36

and the flattened Hamiltonian acts as

$$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$37

On each multiplicity space $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$38, the residual symmetry operators and $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$39 satisfy Clifford relations, so each sector becomes a Clifford module (Abramovici et al., 2011). In the topological-insulator literature, additional symmetries may either furnish a new anticommuting Clifford generator, merely block-diagonalize the problem, or change the real/complex character of the algebra; the effect is encoded as a shift or conversion of the relevant Clifford extension problem (Morimoto et al., 2013). These usages are conceptually adjacent to diagonal Clifford symmetry in quantum information, but they concern basis-diagonal or block-diagonal Clifford-module structure rather than the diagonal subgroup of the quantum Clifford group itself.

Taken together, the literature presents diagonal Clifford symmetries as a hierarchy of reductions. At one extreme lie genuinely diagonal gates such as the diagonal Clifford subgroup generated by $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$40 and $$\mathcal{C}^{(k)} := \{U \mid UPU^\dagger \in \mathcal{C}^{(k-1)},\ \forall P\in P_n\},$$41; at another lie square-dimensional representations in which the full Clifford group is only monomial; and in code, design, simulation, and topological settings the same theme reappears as preservation of a canonical decomposition into one-dimensional eigenspaces, multiplicity sectors, or Pauli-symmetry blocks (Bengtsson, 2012, Camillo et al., 21 Oct 2025, Mitsuhashi et al., 2023). The subject is therefore best understood not as a single subgroup, but as a family of exact and approximate diagonalization phenomena that expose how much of Clifford structure can be made sparse, phase-valued, or block-separable without destroying the underlying noncommutative symmetry.