Coupled Quantization (CQ): Concepts & Applications

• Coupled Quantization (CQ) is a framework that leverages the interplay between classical, quantum, and computational degrees of freedom to jointly quantize systems with unique geometric and topological properties.
• It applies across spin-orbit physics, quantum communications, and iterative fixed-point methods, achieving robustness via nondegenerate symplectic forms and deterministic channel behavior.
• CQ methods support practical implementations in quantum HPC and signal processing, offering efficient coding schemes, error decay performance, and resource-optimized quantum–classical workflows.

Coupled Quantization (CQ) denotes a spectrum of strategies in quantum information science, mathematical physics, machine learning, and signal processing that exploit the structured interplay between multiple degrees of freedom—classical, quantum, or computational—via joint quantization or coupling. The CQ concept encompasses both the geometric-topological distinction of quantum states, joint quantization effects in semiclassical systems, iterative fixed-point frameworks, coding structures in quantum communications, and collaborative mechanisms in neural and hardware-efficient quantizers.

1. Geometric and Topological Characterization of CQ States

A bipartite “classical–quantum” (CQ) state is defined on a Hilbert space $\mathcal{H}_A \otimes \mathcal{H}_B$, typically as

$$\rho = \sum_i p_i \, |i\rangle \langle i| \otimes \rho_i,$$

where $|i\rangle$ forms an orthonormal basis in $\mathcal{H}_A$, $p_i$ are classical probabilities, and $\rho_i$ are density matrices on $\mathcal{H}_B$.

The symplectic geometry of CQ states is analyzed via the Kirillov–Kostant–Souriau (KKS) symplectic form:

$$\omega_\rho(\tilde{X}_\rho, \tilde{Y}_\rho) = (i\rho | [Y,X]),$$

with $\tilde{X}_\rho = [X, \rho]$, for $(X,Y)$ in the Lie algebra of $SU(N_1) \times I_{N_2}$. The orbit $K \cdot \rho$ features a nondegenerate restricted form $\omega|_{K \cdot \rho}$ only for CQ states. Necessary and sufficient conditions for symplecticity require that, whenever $p_i = p_j$ for $i \neq j$, $p_i\rho_i = p_j\rho_j$.

Topologically, the Hopf–Samelson theorem is invoked to compute the Euler–Poincaré characteristic. The Euler characteristic of a CQ orbit is

$$\chi(K \cdot \rho) = \frac{N_1!}{\prod_{\sigma \in \mathcal{Q}} |\mathcal{I}_\sigma|!},$$

with index sets $\mathcal{Q}$ and $\mathcal{I}_\sigma$ tracking distinct weights and density matrices. A nonvanishing $\chi$ uniquely signifies the CQ nature in the space of mixed states; generic correlated states lack this feature.

CQ states are thus geometrically and topologically unique: they are the only mixed states with nonzero Euler characteristic and nondegenerate symplectic structure in local unitary orbits (Oszmaniec et al., 2013).

2. Semiclassical Coupled Quantization in Spin-Orbit Systems

In spin-orbit coupled 2D systems, Landau quantization is generalized by CQ through the incorporation of a matrix-valued phase. The wavefunction is

$\psi(\theta) = e^{-i\eta\nu\theta}\chi(\theta),$

and the spinor $\chi(\theta)$ evolves under a one-dimensional Schrödinger equation with a momentum-dependent effective field. After a full cyclotron orbit, spinor evolution produces a non-Abelian SU(2) phase:

$$U(2\pi) = \mathcal{P}\exp\left[i \frac{\eta}{\omega} \int_0^{2\pi} (\beta(\nu,\theta)\cdot\sigma) d\theta\right],$$

resulting in a coupled quantization condition for Landau levels

$$E_{n,\pm} = \omega\left(n + \frac{1}{2} \pm \frac{\Phi}{2\pi}\right).$$

The matrix-valued phase $\Phi$ couples orbital and spin quantizations; this mechanism explains experimentally observed oscillations and ESR responses in spintronic systems (Li et al., 2016).

3. CQ in Coding and Information Theory

CQ channels also appear in quantum communication, where polarization techniques produce synthetic cq-channels that project input symbols onto quotients of the input group. Polar codes for CQ channels are constructed by recursively merging and splitting, using the Arıkan-style transformation:

$x_1 = u_1 + u_2, \quad x_2 = u_2$

and recursively synthesized states

$$\rho^-_{u_1} = \frac{1}{q}\sum_{u_2 \in G} \rho_{u_1+u_2} \otimes \rho_{u_2}, \quad \rho^+_{u_2} = \frac{1}{q}\sum_{u_1 \in G} \rho_{u_1+u_2} \otimes \rho_{u_2} \otimes |u_1\rangle\langle u_1|.$$

As polarization advances, channels become deterministic homomorphism channels upon which reliable coding is possible. The error probability decays exponentially:

$P_e < 2^{-N^\beta}, \qquad \forall \beta < 1/2,$

under quantum successive cancellation decoding (Nasser et al., 2017).

For CQ-MACs, a fusion of algebraic structured codes (nested coset codes) with unstructured coding enables function computation capacity strictly exceeding prior methods:

$$v^n(a,m) = a\,g_I \oplus_q m\,g_{O/I} \oplus_q b^n,$$

with decoding based on simultaneous joint POVMs, leveraging both random and algebraic coding layers (Sohail et al., 2022).

4. CQ Methods in Iterative and Fixed-Point Theory

In nonlinear analysis, the generalized CQ method unifies a family of iterative schemes to approximate fixed points in Hilbert spaces. The procedure constructs closed, convex sets preserving the fixed point set,

$$y_n = (1-\alpha_n)x_n + \alpha_n T x_n, \ C_n = \{z \in C : \|z-y_n\| \le \|z-x_n\|\}, \ Q_n = \{z \in C: \langle z-x_n, x_n-x_0\rangle \ge 0 \}, \ x_{n+1} = P_{C_n \cap Q_n}(x_0).$$

Variations—monotone Q, monotone C, monotone CQ—admit ordering relations: CQ method TRUE $\Rightarrow$ monotone Q method TRUE $\Rightarrow$ monotone C method TRUE $>$ monotone CQ method TRUE.

This hierarchy underlines strong convergence properties and robustness across nonexpansive and pseudo-contractive mappings (He et al., 2020).

5. CQ in Signal Processing and Machine Learning Frameworks

Collaborative Quantization in neural waveform coding enables joint learning of codebooks for LPC coefficients and residuals, integrating differentiable softmax quantization in both branches and optimizing bit allocation adaptively. The framework minimizes the loss

$$L = \lambda_1 T(y, \hat{y}) + \lambda_2 F(y, \hat{y}) + \lambda_3 Q(\mathrm{soft}) + \lambda_4 E(\mathrm{soft}),$$

where $T$ and $F$ penalize time and frequency domain reconstruction errors, $Q$ ensures hard quantization constraints, and $E$ regularizes the entropy for bitrate control. Efficient end-to-end neural codecs (less than $10^6$ parameters) outperform state-of-the-art waveform codecs at low and high bitrates (Zhen et al., 2020).

CQ in LLM inference exploits channel interdependency by jointly learning centroids for groups of key/value activations:

$$C^*_i = \arg\min_{C \subset \mathbb{R}^c, |C|=2^b} \|A_i - CQ(A_i)\|_F^2,$$

and optionally uses Fisher information guidance for weighted k-means optimization. This coupled channel quantization achieves 1-bit per activation storage, enabling memory- and latency-efficient inference in very large transformer models (Zhang et al., 2024).

6. Quantized Markov Chain Coupling

A novel approach quantizes Markov chains using classical coupling constructions (e.g. grand coupling) and produces a completely positive, trace-preserving quantum map:

$$T^*(\cdot) = D^{-\frac{1}{2}} C^*( D^{\frac{1}{2}} \cdot D^{\frac{1}{2}} ) D^{-\frac{1}{2}},$$

with $D$ the diagonal matrix of the stationary distribution, and $C^*$ the dual coupling map. The unique fixed point is the pure “qsample” state $\lvert V_T ) = D^{1/2} \lvert e )$ encoding the stationary distribution. The convergence rate of the quantum map $T$ is directly proportional to the classical coupling time, establishing an explicit bridge between classical mixing and quantum state preparation (Temme et al., 3 Apr 2025).

7. API and Implementation for CQ in Quantum HPC

CQ as a specification denotes a minimal C-like API for quantum-accelerated high-performance computing (HPC), supporting strictly-typed offloading and fine-grained classical data movement. The reference implementation (CQ-SimBE) demonstrates kernel offloading, resource management (allocation/deallocation), and synchronization routines from both C and Fortran, including support for analogue quantum operations (pulse play/capture, channel retargeting). Both the CQ specification and CQ-SimBE are open-source, fostering open development in coupled quantum–classical workflows (Brown et al., 14 Aug 2025).

Summary Table of CQ Facets

Domain	CQ Mechanism	Key Distinction
Quantum state geometry/topology	Symplectic orbits, Euler χ	Only CQ states have nondegenerate orbit
Spin-orbit semiclassical physics	Matrix-valued phase	Coupling of orbital/spin quantization
Quantum channel coding	Polarization/homomorphism	Deterministic quotient channel behavior
Fixed-point iterative methods	Convex set/projection CQ	Hierarchy of strong convergence
Neural coding / signal processing	Collaborative codebooks	Jointly-optimized multi-branch quant.
Quantum Markov sampling	CPTP map via coupling	Direct link between coupling & mixing
Quantum-accelerated HPC	C-like API and offloading	Incremental/classical–quantum workflow

Concluding Remarks

Coupled Quantization methods rigorously exploit structural dependencies within and across quantum, classical, and computational systems. By leveraging geometric, topological, algebraic, and algorithmic properties, CQ enables precise state classification, resource-efficient coding, robust fixed-point algorithms, and scalable deployments in quantum and quantum-hybrid hardware. The confluence of symplectic geometry, coding theory, Markov couplings, and hardware APIs under the CQ paradigm reflects the versatility and depth of this field across disciplines.