Clifford Group Action & Modular Transformations

• Clifford group action and modular transformations are foundational structures linking quantum computation, algebraic geometry, and number theory through automorphisms and symplectic operations.
• They enable precise implementation of quantum error correction, modular invariance, and geometric transformations via unitary actions and reflection group representations.
• This interplay informs designs for quantum gates, neural network architectures, and arithmetic protocols, driving research in TQFT, cryptography, and advanced representation theory.

The Clifford group action and modular transformations together form a foundational nexus connecting classical and quantum group theory, representation theory, algebraic geometry, mathematical physics, and quantum information science. The Clifford group, originally arising as the group of automorphisms of the Heisenberg/Weyl–Heisenberg group, has deep interrelations with modular groups, finite and infinite, and their actions on arithmetic, geometric, and quantum data. The concept of modular transformations, both in the sense of Möbius/linearfractional transformations and in the sense of finite modular/symplectic transformations (e.g., SL(2, ℤₙ)), is a recurrent theme: they appear in the structure of real and complex quadratic fields, the automorphism groups of Pauli gradings, the symmetries of error-correcting codes and quantum circuits, the structure of root systems and reflection groups, the representation theory of finite groups, and even the topological realization of quantum gates in Chern–Simons theory. Below, major facets of this landscape are detailed, spanning number theory, representation theory, geometric group theory, quantum computing, mathematical physics, and topological quantum computation.

1. Actions in Number Theory: Modular Group, Quadratic Fields, and Orbits

The modular group $$G = PSL(2, \mathbb{Z}) = \langle x, y : x^2 = y^3 = 1 \rangle$$ acts via linear fractional transformations $z \mapsto \frac{az + b}{cz + d}$ on the extended complex plane and, more generally, on subsets of real quadratic fields. When acting on the set

$$Q^*(\sqrt{n}) := \left\{ \frac{a + \sqrt{n}}{c} : a, b, c \in \mathbb{Z},~ a^2 - n = b c,~ \gcd(a, b, c) = 1 \right\}$$

with $n = k^2 m$, $m$ square-free, the modular group partitions the quadratic irrational numbers into $G$-orbits and $G$-subsets, where orbits correspond to equivalence classes under the group action (Malik et al., 2010). The stabilizers, ambiguous numbers, and orbit structure (often described via coset diagrams) provide a rigorous description of how modular symmetries organize arithmetic data. Furthermore, $s$-equivalence relations—two elements $\alpha(a, b, c)$ and $\alpha'(a', b', c')$ are $s$-equivalent if $a \equiv a'$, $b \equiv b'$, $c \equiv c' \pmod s$—stratify $Q^*(\sqrt n)$ and correspond to congruence classes preserved under $G$.

In broader terms, this explicit partitioning of quadratic numbers by modular group action not only elucidates algebraic number theory and class numbers, but also underpins arithmetic applications such as lattice theory and cryptographic protocols.

2. Clifford Group Automorphisms and Modular/Symplectic Transformations

The Clifford group $C(N)$ is defined as the normalizer of the Weyl–Heisenberg group $H(N)$ inside the unitary group $U(N)$, modulo global phases:

$C(N) = N_{U(N)}(H(N)) / U(1)$

This group precisely consists of unitary operators whose adjoint action preserves the Pauli group structure (the Pauli gradings of the operator algebra) (Tolar, 2018). The central structure theorem is that the induced action on "quantum phase space" $$H(N)/Z(H(N)) \simeq \mathbb{Z}_N \times \mathbb{Z}_N$$ is identified with the finite symplectic group $SL(2, \mathbb{Z}_N)$:

$$C(N) \simeq \left( \mathbb{Z}_N \times \mathbb{Z}_N \right) \rtimes SL(2, \mathbb{Z}_N)$$

This symplectic (modular) group acts via

$$(i', j')^T = M (i, j)^T, \qquad M \in SL(2, \mathbb{Z}_N)$$

preserving the symplectic (alternating) form $q((i, j), (i', j')) = i' j - i j' \pmod N$.

This modular action is the algebraic underpinning for why the Clifford group arises in quantum error correction, quantum computation (e.g., the Gottesman–Knill theorem), and the structure of quantum codes, and forms the backbone for modular-invariant transform operations in finite-dimensional quantum systems.

3. Clifford Algebra, Coxeter/Reflection Groups, and Modular Transformations

Clifford algebra provides a "universal" geometric and algebraic framework for representing reflections, rotations, and their compositions. For any vector space with a metric, the Clifford algebra encodes all orthogonal transformations via "sandwiching" operations:

$v' = \pm \tilde A v A$

where $A$ is a multivector (versor) constructed from products of unit vectors (reflections), and $\tilde A$ is the reversal. By the Cartan–Dieudonné theorem, any orthogonal transformation (hence any modular transformation, when suitably embedded) factors into a product of reflections (Dechant, 2016, Dechant, 2016). In conformal geometric algebra (CGA), modular transformations in 2D (generators $S: \tau \mapsto -1/\tau$, $T: \tau \mapsto \tau + 1$) are realized as versor actions; e.g.,

$\text{Inversion:} \quad F(x) \mapsto S F(x) S^{-1}$

This unifies modular, conformal, and reflection symmetries, and is the basis for Clifford-based computation of group actions in Coxeter, Moonshine, and string theory applications.

A summary of group action implementation via Clifford algebra:

Transformation	Clifford Representation	Essential Formula
Reflection	Sandwiching w/ $\alpha$	$v' = -\alpha v \alpha$
Rotation	Product of reflections	$v' = \tilde{A} v A$, $A$-even versor
Modular group	Versor in CGA	$F(x) \mapsto U F(x) U^{-1}$

This approach facilitates direct computations of group orbits, invariants, and representations for both crystallographic and non-crystallographic reflection (Coxeter) groups, leading to new insights into phenomena such as Arnold’s trinities and the McKay correspondence (at the level of spinors and automorphism groups of exceptional root systems).

4. Clifford Group Actions in Quantum Information and Representation Theory

In quantum information, the Clifford group’s action organizes and partitions quantum states via orbits and stabilizers. In particular:

• The Cayley graph structure of the $n$-qubit Clifford group $\mathcal{C}_n$ encodes the group presentation, with vertices labeled by group elements and edges by generators (Hadamard, phase, CNOT) (Keeler et al., 2023).
• The "quotient-by-stabilizer" construction produces a reduced "reachability graph" whose vertices correspond to Clifford orbits of a chosen state $|\psi\rangle$, providing a visual and algebraic map of state evolution and quantum circuit complexity.

$$\operatorname{Stab}_G(|\psi\rangle) = \{ g \in G : g|\psi\rangle = |\psi\rangle \}$$

• These methods extend beyond stabilizer states to non-stabilizer (e.g., $W_n$, Dicke) states, and are crucial for analyzing entanglement, complexity, and resource theory in fault-tolerant quantum computation.

In representation theory, Clifford theory (especially in the modular setting) describes how the representations of a finite group $G$ decompose in terms of those of a normal subgroup $N$. For a modular field $\mathbb{F}_p$ (characteristic dividing $|N|$), induced and restricted representations satisfy explicit decompositions involving the inertia/stabilizer group $I_G(\varphi)$:

$$\operatorname{Res}_N^G (\operatorname{Ind}_N^G \varphi) = \bigoplus_{t \in T} t\varphi$$

These decompositions control how modular representations "inflate" from $SL_2(\mathbb{F}_p)$ to $GL_2(\mathbb{F}_p)$ (Basu, 4 Mar 2025) and allow for precise harmonic analysis of modular transformation properties under Clifford action.

5. Clifford Group Equivariance, Neural Networks, and Grade-Preserving Actions

Clifford group equivariant neural networks implement actions of the orthogonal (or more generally Spin/Pin) group directly on the data’s Clifford algebra representations (Ruhe et al., 2023, Ruhe et al., 2023). The key operation is grade- and parity-preserving twisted conjugation:

$$\rho(w)(x) = w x^{(0)} w^{-1} + \alpha(w) x^{(1)} w^{-1}$$

where $w$ is a homogeneous, invertible element, $x^{(0)}$ and $x^{(1)}$ are the even and odd-grade components, and $\alpha$ is the automorphism. This automorphism respects the full multigrade structure and the geometric product. Thus, any polynomial in multivectors is equivariant:

$P(\rho(w)(x)) = \rho(w)(P(x))$

This property allows Clifford group equivariant layers to be designed modularly (in the Editor's term sense) as composable modules mirroring the action structure of modular transformations.

By the Cartan–Dieudonné theorem, all orthogonal transformations are implemented as finite products of reflections, which are realized algebraically by the Clifford group. Thus, these models provide a unifying architecture for constructing O(n)-, E(n)-, or Lorentz-equivariant networks across dimensions.

6. Clifford Groups in Quantum Gates, Topological QFT, and Modular Functors

Topological realizations of Clifford group actions appear in the context of topological quantum field theory (TQFT), notably in SU(2)$_1$ Chern–Simons theory (Munizzi et al., 16 Oct 2025). Identification is as follows:

• Pauli and Clifford operators are realized as path integrals over 3-manifolds with Wilson loop insertions; for instance, the Pauli $X$ is encoded via the fusion tensor $N_{a,1}^{a+1}$, and the Hadamard by the modular $S$-matrix

$$S_{ab} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \ 1 & -1 \end{bmatrix}$$

• Modular transformations on the torus (the action of mapping class group generators $S$ and $T$) are implemented as quantum gates, explicitly via Dehn twists:

$(ST)^3 = S^2$

• Clifford group orbits of states (including non-stabilizer $W_n$ and Dicke states) are encoded via 3-manifold constructions, and entanglement entropy is computed topologically by sewing (gluing) together relevant manifold pieces,

$$S_A = -\frac{1}{n} \log_d \frac{1}{n} - \frac{n-1}{n} \log_d \frac{n-1}{n}$$

• Mapping class group actions—geometrically modular transformations—are in bijection with the action of Clifford gates in TQFT, thus linking geometric quantum computation and quantum resource theory.

7. Advanced Algebraic Structures: Clifford-Bianchi Groups and Geometric Algorithms

In the paper of Clifford–Bianchi groups $\mathrm{SL}_2(\mathcal{O})$ acting on hyperbolic $(n+1)$-space $\mathcal{H}^{n+1}$, the group’s action is implemented via Clifford-valued Möbius transformations,

$x \mapsto (a x + b)(c x + d)^{-1}$

with $a, b, c, d \in \mathcal{O}$ (a maximal order in a Clifford algebra) (Dupuy et al., 26 Jul 2024). Fundamental domains are constructed, and the tessellation graph method is used to algorithmically derive generators and relations for a finite presentation of the group, relating geometric properties (faces, bubbles, boundary spheres) to algebraic structure (via gcd algorithms in the order and explicit descriptions of unimodular pairs). This framework generalizes classical Fuchsian and Kleinian group theory to high-dimensional arithmetic settings, with potential applications to automorphic forms, arithmetic geometry, and high-dimensional quantum codes.

8. Subtleties: Unitriangular Designs, Limiting Behavior, and Modular Invariants

An important subtlety is that the Clifford group, while a unitary 2-design for prime (and qubit) dimensions, fails to be a 2-design when the local dimension is composite or variable; the induced adjoint representation then decomposes into more than two irreducible components (Graydon et al., 2021). This richer representation structure implies that, in such dimensions, Clifford group–based twirling cannot fully emulate Haar-random symmetry—impacting randomization strategies in quantum information and possibly in modular-invariant constructions.

The decomposition criterion is:

$\| \chi_{\pi \otimes \overline{\pi}} \| = \sqrt{2}$

if and only if the group is a (projective) unitary 2-design.

This affects the design of quantum channels, codes, and modular-invariant objects, necessitating modification or supplementation of Clifford-based approaches in composite-dimensional systems.

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Summary Table: Key Contexts Linking Clifford Group Action and Modular Transformations

Domain	Manifestation of Clifford–Modular Link	Reference
Quadratic fields	Orbits/subsets in $Q^*(\sqrt{n})$ under $PSL(2,\mathbb{Z})$	(Malik et al., 2010)
Quantum error correction	Adjoint action as $SL(2,\mathbb{Z}_N)$ on phase space	(Tolar, 2018, Appleby et al., 2011)
Coxeter/Reflection theory	Orthogonal transformations as Clifford sandwiched reflections	(Dechant, 2016, Dechant, 2016)
Neural networks	Grade-preserving twisted conjugation/direct Clifford group equiv.	(Ruhe et al., 2023, Ruhe et al., 2023)
TQFT/Quantum gates	Modular S/T as Clifford gates, Dehn twists	(Munizzi et al., 16 Oct 2025)
Arithmetic groups	Generators/relations in $\mathrm{SL}_2(\mathcal{O})$ via domains	(Dupuy et al., 26 Jul 2024)
Representation theory	Modular Clifford theory, inertia decompositions	(Basu, 4 Mar 2025)

These interrelations enable the analysis of orbits, invariants, circuit complexity, entropy, and symmetry-protected structures in settings ranging from arithmetic geometry to quantum computation and topological field theory. The Clifford group’s compatibility with modular transformations, both in algebraic and geometric guises, underpins much of the modern architecture of symmetry in mathematical and physical systems.