Quantum Zak Transform & Applications

• Quantum Zak Transform is a unitary mapping that generalizes the classical Zak transform to quantum settings, encoding periodicity and symmetry in noncommutative phase spaces.
• It reorganizes Hilbert spaces into fibers via Plancherel isometry, serving as a rigorous foundation for techniques like Bloch wave decomposition and GKP code implementation.
• Its applications span quantum signal processing, topological quantum computing, and error correction by leveraging nonabelian group properties and operator module frameworks.

The Quantum Zak Transform (QZT) generalizes the classical Zak transform to function spaces, wavefunctions, or operator modules over compact and noncommutative phase spaces, providing a unitary mapping that encodes both periodicity and symmetry in quantum systems. Initially prominent in solid-state quantum theory (e.g., for Bloch wave decomposition in periodic crystals), the QZT is now a foundational instrument in quantum error correction, noncommutative geometry, and quantum signal processing. Its rigorous formulations naturally extend to semidirect product groups, Weyl–Heisenberg groups, quantum walks, finite fields, and phase-space codes such as the Gottesman–Kitaev–Preskill (GKP) code. Modern definitions in noncommutative geometry (especially via the Rieffel–Heisenberg module) regularize QZT to yield physically realizable quantum states and operator actions.

1. Foundational Definitions and Unitary Structure

The QZT extends the classical Zak transform (on $L^2(K)$ for LCA group $K$ and lattice $L \subset K$) to settings where the symmetry group is nonabelian, the phase space is compact (e.g., a quantum torus), or the function spaces are operator modules rather than Hilbert spaces.

For a locally compact group $H$, an LCA group $K$, a continuous homomorphism $T:H \to \mathrm{Aut}(K)$, and $G_\tau = H\ltimes_\tau K$, the QZT (often referred to as the T–Zak transform) is formulated for $f \in L^2(G_\tau)$ via: $$Z_L f(h, k, \omega) = \delta_K(h)^{1/2} \sum_{l \in L} f(h, T_h(k) + l) \omega(T_h^{-1} l),$$ with $\delta_K(h)$ the modular function, $k^h = T_h(k)$ the group action, and $\omega^h = \omega \circ T_h^{-1}$ characterizes the push-forward on the dual group. This mapping yields a function on a periodized space $$G_{\times, L} = H \ltimes_{\times, L} (K/L \times \widehat{K}/L^\perp)$$.

The QZT preserves $L^2$-norms, satisfying a Plancherel-type isometry: $$\|Z_L f\|_{L^2(G_{\times,L})} = \|f\|_{L^2(G_\tau)},$$ implying unitary equivalence between the original space and the direct integral of fibers—essential for probability and information conservation in quantum theory (Farashahi et al., 2012).

For noncommutative/quantum tori (see GGKP codes), the QZT is defined by mapping the Rieffel–Heisenberg module inner product: $$\langle \phi, \psi \rangle_{T^2_{\theta_0 \infty}} = \sum_{m,n \in \mathbb{Z}} \langle \phi | \mathcal{D}(m \alpha_0, n \beta_0) | \psi \rangle \, U^n V^m,$$ with $U,V$ acting as torus generators, to a function on the torus via character evaluation, i.e.,

$$\langle \phi, \psi \rangle_{T^2_{\theta_0 \infty}}(x, k) = \sum_{m,n \in \mathbb{Z}} \langle \phi| \mathcal{D}(m\alpha_0, n\beta_0) | \psi \rangle \exp[2\pi i(n\beta_0 x - m\alpha_0 k)],$$

embedding the noncommutative geometry directly in the phase-space function (Joseph et al., 20 Sep 2025).

2. Core Properties: Plancherel Isometry, Fiberization, and Symmetry

The QZT, whether defined on representations of compact/noncompact groups or finite fields, exhibits essential structural features:

• Plancherel Isometry: The QZT implements a unitary transformation so that the $L^2$-scalar product is preserved. This is the nonabelian/quantum analog of the classical Plancherel theorem and undergirds reliable quantum information processing (Farashahi et al., 2012, Jüstel, 2016).
• Fiber Decomposition: In the presence of periodicity or lattice structure, the QZT reorganizes the Hilbert space into "fibers" indexed by cosets in $K/L$ and characters in $\widehat{K}/L^\perp$. In modulated group actions, this is extended to direct integrals over representation spaces of the group or modules of the quantum torus algebra (Farashahi et al., 2012, Joseph et al., 20 Sep 2025).
• Equivariance and Symmetry: The QZT intertwines group actions with character/representation modulations:

$$Z\big[I_o(z) v\big](\alpha)(x) = X_z(\alpha) Z[v](\alpha)(x),$$

ensuring that group symmetries are explicitly encoded and diagonalized, refining Bloch–Floquet theory to nonabelian and quantum settings (Barbieri et al., 2014, Jüstel, 2016).

3. Applications: Quantum Error Correction, Quantum Walks, and Signal Analysis

The QZT is pivotal in several advanced quantum information and signal-processing protocols:

• GKP and Generalized GKP Codes: The QZT enables a regularized, normalizable representation (via Riemann Theta functions) of GKP codewords on the compact quantum torus, resolving pathologies such as infinite energy and non-normalizability. The algebra-valued module inner product established by the QZT allows for algebraic orthogonality and robust logical state discrimination critical for error correction (Joseph et al., 20 Sep 2025).
• Twisted Translates and Weyl–Heisenberg Structures: A variant of QZT adapted to the Weyl transform "fiberizes" $L^2(\mathbb{R}^{2n})$ as $$L^2(\mathbb{T}^n \times \mathbb{T}^n \times \mathbb{R}^n)$$, with the bracket map and modular symmetries characterizing frame, Riesz, and biorthogonality structures. This aligns the QZT with noncommutative harmonic analysis and quantum operator frame theory (Ramakrishnan et al., 2023).
• Quantum Walks and Zak Phase Landscapes: In photonic DTQWs, the QZT characterizes topological phases, geometric invariants (e.g., $\pi$-quantized Zak phase jumps), and Berry connections—even in the absence of net Berry curvature. Controlled symmetry-breaking maneuvers (such as inverting coin parameters) finely regulate geometric phase landscapes, supporting topological information encoding (Puentes et al., 2015, Puentes, 2023).
• Time–Frequency Signal Processing and Communications: The QZT unifies Zak-OTFS modulation for delay–Doppler domain representation in communication channels, allowing direct, periodic, and quasi-periodic mappings that facilitate efficient channel estimation and reliable coding strategies (e.g., LDPC codes mapped to the most reliable DD bins) (Dabak et al., 14 Feb 2024).

4. Mathematical Framework: Noncommutative Geometry and Fiberwise Harmonic Analysis

Advanced applications of the QZT are underpinned by a rich mathematical framework:

• Noncommutative Torus and Rieffel–Heisenberg Modules: The QZT leverages modules over $T^2_{\theta_0}$ with operators $U, V$ obeying $UV = e^{2\pi i \theta_0} VU$, embedding deformation parameters that regularize the state. The characteristic lattice spacings $\alpha_0, \beta_0$ relate to phase-space compactification ($L, P$), controlling physical realizability (Joseph et al., 20 Sep 2025).
• Riemann Theta Functions: Evaluating the QZT on squeezed coherent states yields genus-2 Riemann Theta functions, with parameters directly encoding the underlying squeezing and compactification. The resulting GGKP states are normalizable and exhibit uncertainty that is independent of squeezing, crucial for error correction (Joseph et al., 20 Sep 2025).
• Frame and Basis Structure: The QZT, when applied to twisted shift-invariant spaces or finite group settings, characterizes when systems of modulated translates form frames, Riesz sequences, or orthonormal or Schauder bases, with criterion controlled by weight/bracket functions or Muckenhoupt $\mathcal{A}_2$ weights (Ramakrishnan et al., 2023).

5. Impact on Quantum Information Processing

The QZT alters foundational aspects of quantum encoding, computation, and error correction:

• Fault-Tolerant Encoding: The QZT-based GGKP states' finite-energy, algebraically orthogonal logical bases mitigate leakage and enforce reliable syndrome extraction, enabling scalable error correction for bosonic codes (Joseph et al., 20 Sep 2025).
• Algorithmic Applications: In communication, the QZT provides an efficient (and invertible) mapping between time–domain and delay–Doppler representations, simplifying channel estimation and demodulation. Its explicit lattice structure matches well with code design and pilot-based channel prediction (Dabak et al., 14 Feb 2024).
• Topological Quantum Computing: The QZT's encoding of symmetry and geometric phases directly supports robust topological information storage and manipulation, particularly in photonic platforms or systems with protected edge modes (Puentes et al., 2015, Puentes, 2023).

6. Connections, Variations, and Outlook

The QZT encompasses and extends a spectrum of transforms:

• Classical Zak transforms for LCA groups as special cases.
• Generalizations to semidirect products (e.g., SL(2,ℤ)⋉ℝ²) and nonabelian, noncommutative, or finite settings.
• Variants realizing operator isomorphisms, as fiberizations for $G$-spaces, or via modular character sums over the quantum torus.
• Regularizations for continuous-variable codes, unifying noncommutative geometry with quantum error correction.

A plausible implication is that QZT frameworks will further integrate nonlinear and non-Gaussian resource states as experimental constraints (such as phase-space compactification) become more prominent, especially in photonic and continuous-variable quantum computing platforms.

• "Zak Transform for Semidirect Product of Locally Compact Groups" (Farashahi et al., 2012)
• "Generalized Gottesman-Kitaev-Preskill States on a Quantum Torus" (Joseph et al., 20 Sep 2025)
• "Zak transform associated with the Weyl transform and the system of twisted translates on R^{2n}" (Ramakrishnan et al., 2023)
• "Zak Phase in Discrete-Time Quantum Walks" (Puentes et al., 2015)
• "2D Zak Phase Landscape in Photonic Discrete-Time Quantum Walks" (Puentes, 2023)
• "Zak-OTFS and LDPC Codes" (Dabak et al., 14 Feb 2024)