Collaborative Unitary Sequence Decoding Paradigm

• The paradigm is a unified framework combining controlled quantum secure direct communication and collaborative neural decoding using unitary and probabilistic operators.
• It employs a multi-party quantum protocol with decoy-photon authentication to achieve deterministic decoding and enhanced resistance to eavesdropping.
• In neural modeling, selective collaboration between fast and slow systems based on uncertainty metrics optimizes inference efficiency and scalability.

The collaborative unitary sequence decoding paradigm refers to a generalized framework in which sequence decoding is performed through the orchestrated interaction of independently parameterized modules or agents—whether quantum (as in quantum secure direct communication protocols) or classical neural models (as in collaborative large-small LLM decoding). This paradigm fundamentally relies on the sequential application of unitary or probabilistic operators, collaborative triggering mechanisms, and context-sensitive fusion strategies. It has advanced the fields of high-efficiency quantum communication as well as LLM deployment by optimizing both security/control (in QSDC) and inference efficiency (in LLMs). The key architectural principle centers on selective collaborative intervention, either by deterministic unitary chains or by uncertainty-triggered model fusion (Lu et al., 17 Dec 2025, Zhang et al., 18 Jun 2024).

1. Quantum Protocol: Four-Dimensional Qudit Basis and Unitary Operations

In the quantum setting, the collaborative unitary sequence decoding paradigm is exemplified in controlled quantum secure direct communication (QSDC) using 4D single particle states. The Hilbert space $\mathcal{H}_4$ is spanned by the basis $\{|0\rangle, |1\rangle, |2\rangle, |3\rangle\}$, with a generic state $|\psi\rangle = \sum_{j=0}^3\alpha_j|j\rangle$, $\sum|\alpha_j|^2=1$.

Three classes of unitary operators define the protocol:

• Oracle Operators $U_w$: $U_w = I_4 - 2|w\rangle\langle w|$, $U_w^2 = I_4$.
• Diffusion/Grover Operators $U_s$: $U_s = 2|S\rangle\langle S| - I_4$ for $|S\rangle$ taken from a set of 16 equal-amplitude states.
• Composed Decoding Sequence: The protocol exploits the result $U_s\,U_{w_c}U_{w_A}U_{w_c}|S\rangle = |w_A\rangle$ up to phase, ensuring deterministic decoding under controller permission.

This unitary chain ensures that direct quantum operations suffice for decoding, eliminating classical post-processing and securing the protocol against unauthorized interception (Lu et al., 17 Dec 2025).

2. Collaborative Protocol Workflow: Three-Party Quantum Interaction

The paradigm enables a three-party workflow:

• Preparation: The controller (Charlie) selects initial states $|S^{(j)}\rangle$, applies $U_{w_c^{(j)}}$, and inserts decoy photons according to a secret ID sequence.
• Transmission and Layered Authentication: Decoys are measured, Quantum Bit Error Rate (QBER) is computed to detect eavesdropping, and Alice encodes her message using $U_{w_A^{(j)}}$ before adding another set of random decoys.
• Collaborative Decoding: Upon controller authorization, the receiver (Bob) applies $U_{w_c^{(j)}}$, followed by $U_s^{(j)}$, enabling deterministic recovery of Alice's message $w_A^{(j)}$ with no classical computation.

The protocol is depicted by the quantum-circuit:

──|S^{(j)}⟩──[U_{w_c^{(j)}]──[U_{w_A^{(j)}]──[U_{w_c^{(j)}]──[U_s^{(j)}]──Measure Z─→ w_A^{(j)}.

This tightly orchestrated multi-unitary scheme underlines the collaborative nature and ensures protocol integrity only under authorized decoding (Lu et al., 17 Dec 2025).

3. Decoy-Photon Authentication and Security

Decoy-photon strategies contribute a multi-layered defense against both external and internal attacks:

• Decoy Variants: Z-basis decoys ($|0\rangle$–$|3\rangle$) and X-basis decoys ($|X_k\rangle$) are used. Intercept-resend or basis-mismatch incurs a QBER, enabling detection.
• Security Metrics: For $k$ decoys, undetected eavesdropper probability is $P_{\rm detect}=1-(\frac{1}{2})^k$ for this protocol, outperforming previous schemes where $P_{\rm detect}=1-(\frac{3}{4})^k$.
• Resistance: Protocol is resilient to entangle-and-measure attacks and Trojan-horse tactics via phase disturbance and photon-number filtering.

Deterministic decoding prohibits message recovery absent all three unitary operations, thus enforcing collaborative control (Lu et al., 17 Dec 2025).

4. Efficiency Metrics and Comparative Performance

The collaborative unitary sequence decoding paradigm achieves substantial improvements in resource efficiency:

• Quidt Efficiency: For $N$ logical 4D symbols ($2N$ classical bits), total qudits transmitted are $3N$. The efficiency is $\eta = \frac{2N}{3N} = \frac{2}{3}\approx 66.7\%$.
• Comparison Table:

Scheme	Qudit Efficiency (%)
Tseng et al. (2012)	18.2
Kao & Huang (2013)	20.0
Yang et al. (2022)	≈25
Paradigm (Lu et al., 17 Dec 2025)	66.7

The pronounced performance gain is attributed to direct quantum decoding and reduced protocol overhead (Lu et al., 17 Dec 2025).

5. Collaborative Decoding in LLMs: FS-GEN Framework

The paradigm finds a direct parallel within neural sequence modeling via the FS-GEN (Fast and Slow Generating) framework. Here collaborative decoding is performed between small (System 1, fast) and large (System 2, slow) models:

• Formal Definition:
  • At each step $t$, the fused output distribution is $$p_{f}(y_{t}) = (1-I_{t})p_{s}(y_t) + I_{t}p_{l}(y_t)$$.
  • Indicator $I_{t}$ is triggered when System 1's uncertainty $U_{t}$ (top-1 confidence or entropy) exceeds the threshold $\tau$.
  • Only a subset of steps require System 2 intervention, determined by token-level uncertainty.
• Decoding Algorithm:

Input: prompt X, models Ms (fast) and Ml (slow), threshold τ for t = 1 to T_max: U_t ← UncertaintyMetric(logits_s) if U_t > τ: y_t ← argmax(logits_l) else: y_t ← argmax(logits_s) Append y_t to Y

This enables inexpensive inference while systematically invoking full-model verification only where uncertainty is high (Zhang et al., 18 Jun 2024).

6. Scaling Laws and Mapping of Collaborative Methods

Collaboration frequency adheres to a scaling principle:

• Empirical Law: Let $R=|\theta_{s}|/|\theta_{l}|$, $r$ (collaboration ratio) satisfies $r(R) = 1.10\,R^{0.85} + 0.03$, remaining below 20% for typical model sizes.
• Table of Empirical Results:

Model Pair (S1:S2)	Param Ratio ($R$)	Mean $r$ (%)
0.5B:14B	0.036	11.8
1.8B:32B	0.056	13.5
7B:72B	0.097	17.2

• Unified Mapping: Speculative, contrastive, and emulator/proxy collaborative decoders align with the generic fusion rule, differing in the definition of $p_{\rm corrected}$ and intervention logic.

This suggests that collaboration can be predictably managed via parametric ratios and token-wise uncertainty, optimizing for both accuracy and cost (Zhang et al., 18 Jun 2024).

7. Assumptions, Limitations and Extensions

Protocols built on the collaborative unitary sequence decoding paradigm assume:

• Quantum: Honest receiver, pre-shared sequences, authenticated classical channel, quantum memory.
• Neural: Calibrated uncertainty metrics, accessible paired models, hardware and latency constraints.

Limitations include susceptibility to denial-of-service (in quantum) and possible computational bottlenecks (in neural). Both fields anticipate protocol extensions: quantum, to multi-controller or higher-dimensional qudits; neural, to hierarchical or multi-model fusion beyond binary collaboration (Lu et al., 17 Dec 2025, Zhang et al., 18 Jun 2024).

A plausible implication is that future optimization of collaborative unitary sequence decoding will rely increasingly on adaptive control of collaboration frequency (scaling laws) and the integration of authentication and robustness measures in both quantum and neural settings.