Quantized-assignment Strategy

• Quantized-assignment strategy is a method that converts continuous optimization tasks into discrete allocations, enabling efficient problem-solving under hardware constraints.
• It leverages techniques such as QUBO mapping, mixed-precision DNN deployment, and integer shot allocation to transform and simplify complex optimization spaces.
• The approach balances approximation errors and resource limitations, ensuring optimal trade-offs between computational scalability and solution fidelity in various domains.

A quantized-assignment strategy denotes any method where discrete allocation (over finite or countable sets) is used to solve optimization, inference, or resource distribution tasks, and where the granularity or quantization of assignment—whether integer, bitwidth, region, or codeword—is a central design variable. Theoretical and algorithmic frameworks in recent literature span quantum optimization (QUBO mapping, VQE-encoding), neural mixed-precision deployment, resource allocation under lossy quantization, and measurement shot-allocation in variational quantum algorithms. This field is characterized by explicit transposition of continuous optimization or assignment problems into quantized domains, either for computational tractability, compatibility with hardware constraints, or for optimization within practical limits of memory, energy, and parallelism.

1. Formal Problem Settings and Canonical Objectives

The quantized-assignment paradigm appears in several mathematically distinct, but structurally analogous, optimization settings:

• QUBO-based assignment for job/resource allocation: A set of jobs and agents, with binary variables $x_{ij} \in \{0,1\}$ indicating assignment, subject to capacity, exclusivity, and affinity constraints. Objective functions are quadratic in $x_{ij}$, e.g., maximizing $\sum_{i,j} S_{ij}x_{ij}$ penalized by assignment conflicts or constraint violations (Delgado et al., 2023, Mastroianni et al., 4 Nov 2025).
• Mixed-precision channel allocation: DNN weights are quantized per-channel, each assigned a bitwidth $b_{l,c}\in\{2,4,8\}$, yielding a discrete assignment $\{b,a\}$ for weight and activation tensors, optimized for accuracy, memory and energy constraints (Risso et al., 2022).
• Quantized measurement allocation: In VQE, measurement shots are distributed in integer batch units $m_i\in\mathbb{N}$ across grouped Hamiltonian cliques, optimizing variance under a global shot budget (Zhu et al., 2023).
• Goal-oriented quantization for parametric inference/control: Quantizers $Q: G\rightarrow\{z_1,\ldots,z_M\}$ partition source parameters $g$ by minimizing the loss incurred in task execution $f(x;g)$ with quantized parameter $Q(g)$ (Zou et al., 2022).

These approaches universally map continuous or combinatorially large assignment spaces onto finite, discrete representations for tractable computation and implementation.

2. Heuristic Partitioning, Subproblemation, and Assignment Structure

Exponential scaling in assignment problem size motivates the use of partitioning heuristics and subproblem decomposition:

• Subproblemation in QUBO job reassignment: By discarding non-positive $S_{ij}$ pairs and partitioning vacant jobs into $D$ priority buckets, each subproblem spans only $O(K^2/(2D))$ variables rather than $O(K^2)$. This approach exponentially reduces the runtime and qubit requirements, from $O(2^{K^2})$ to $O(D\,2^{K^2/(2D)})$, at the cost of heuristic optimality (Delgado et al., 2023).
• Fine-grained linear assignment for quantum circuit mapping: The Hungarian Qubit Assignment (HQA) algorithm maps unfeasible two-qubit gates to cores using a cost matrix $C_t(op_i,c_j)$, solved iteratively by the Hungarian method. Dummy variables are introduced to pad the assignment matrix for non-square configurations, and attraction terms forecast future circuit topology (Escofet et al., 2023).
• Channel-wise bitwidth assignment in DNNs: Search and assignment are expanded from layer-level to channel-level (per filter/neuron), handled by differentiable NAS with “softmax-mixing” and annealing to converge to discrete, per-channel assignments (Risso et al., 2022).

This division of assignment space enables scaling to hardware with limited qubits/channels, and supports parallel evaluation of smaller, independent optimization instances.

3. Quantization Algorithms and Analytical Foundations

Distinct quantized-assignment strategies employ specialized algorithmic frameworks and analytical principles:

• Integer quantization with round-and-greedy: Measurement shots are first allocated as real optimals $m_i^*$, then quantized via floor rounding and greedy correction to restore constraints (e.g., total shot count $M$). The variance penalty due to integer quantization has provably tight bounds: for $m_i^*\gtrsim1$, $\mathrm{Var}_q=\mathrm{Var}^*[1+O(1/m_i^*)]$ (Zhu et al., 2023).
• Task-oriented Lloyd quantizers: For resource allocation, regions and representatives are alternately updated: region boundaries are set by minimal loss $\ell_f(g;z_m)$, and codepoints $z_m$ via gradient steps minimizing expected assignment loss. HR analysis for large $M$ yields an optimal codepoint density $$\lambda^*(g)\propto[\det A(g)]^{1/(p+2)}[\phi(g)]^{p/(p+2)}$$ (Zou et al., 2022).
• Encoding/decoding in quantum algorithms: Assignment variables $x_{ij}$ are mapped to “one-hot” codewords in registers of $\lceil\log_2(A+1)\rceil$ bits per task; post-processing reconstructs feasible assignments and computes penalty-augmented costs. Circuit depth and qubit requirements are dramatically reduced without loss of solution accuracy for the Generalized Assignment Problem (Mastroianni et al., 4 Nov 2025).
• Differentiable NAS for bitwidth assignment: Probability vectors representing discrete bitwidths are annealed towards one-hot distributions via temperature-controlled softmax, after alternating weight and architecture parameter updates; argmax selection finalizes the quantized assignment (Risso et al., 2022).

These techniques are universally characterized by tight coupling of assignment granularity to algorithmic steps—quantized variables are native in the search, evaluation, and structure of constraints.

4. Trade-offs, Scaling Laws, and Theoretical Guarantees

Quantized assignment schemes entail critical trade-offs in optimality, scaling, and resource usage:

• Resource efficiency: Partitioning, quantized encoding, and per-channel assignment reduce hardware demand (qubit count, circuit depth, memory) by factors up to logarithmic or exponential relative to brute-force assignment (Delgado et al., 2023, Mastroianni et al., 4 Nov 2025, Risso et al., 2022).
• Approximation error and loss: Quantization-induced variance or task-loss scales as $M^{-\kappa}$ or $O(1/M)$ depending on assignment granularity and function regularity. For smooth, low-curvature loss functions, coarse quantization suffices; for sharply curved or sensitive goals, finer quantization is necessary (Zou et al., 2022, Zhu et al., 2023).
• Solution quality versus hardware fit: Heuristic splitting (e.g., by priority buckets in job reassignment) sacrifices global optimality, potentially excluding chains of assignments that would only be feasible in the full problem (Delgado et al., 2023). Choice of partition granularity ($D$, bucket thresholds) must balance device capacity and solution fidelity.
• Complexity and scalability: Subproblem sizes, assignment matrix dimension, and search-space per subproblem scale polynomially or logarithmically in key domain variables. O(n³) Hungarian assignment per timeslice is tractable for thousands of gates; per-channel DNAS search remains practical for edge-scale DNN deployments (Escofet et al., 2023, Risso et al., 2022).

A plausible implication is that quantized assignment offers profound scalability and hardware-compatibility at controlled loss, provided partitioning and quantizer design are matched to problem and hardware structure.

5. Empirical Performance and Application Domains

Recent empirical results demonstrate the reach and impact of quantized-assignment methodologies:

• Quantum optimization and job reassignment: Exponential runtime reductions and device-fit for near-term quantum hardware, with only minor sub-optimality. Subproblemation occasionally yields chain assignments superior to unconstrained QUBOs, but lacks absolute optimality guarantee (Delgado et al., 2023, Mastroianni et al., 4 Nov 2025).
• Neural network deployment: Pareto-optimality in accuracy-memory and accuracy-energy, with up to 63% reduction in model size and 27% less energy for edge inference at iso-accuracy, compared to layer-wise assignment (Risso et al., 2022).
• VQE shot-allocation: Integer shot quantization matches continuous-optimal variance to within a few percent, saving 14–22% shots in practice and converging to chemical accuracy on molecular benchmarks (Zhu et al., 2023).
• Resource allocation with goal-oriented quantization: Channel quantizers tailored to decision function regularity attain 2–10× group reductions to meet target loss versus uniform clustering, facilitating lossy but application-robust radio and power distribution (Zou et al., 2022).

Quantized-assignment strategies are thus directly implicated in energy-efficient deep inference on MCUs, scalable quantum resource allocation, and robust task execution under limited measurement or communication budgets.

6. Extensions, Limitations, and Design Guidelines

Extensions of quantized-assignment frameworks address adaptivity, irregular function properties, and empirical/online processing:

• Adaptive quantizer design: When $\phi(g)$ is unknown or nonstationary, quantizer learning proceeds via empirical averages over training data, preserving the same Lloyd-type algorithmic framework (Zou et al., 2022).
• Task-driven assignment density: Bits or assignments should be concentrated where loss Hessian/Jacobian is large—not merely where input density is high—providing principled allocation for critical applications.
• Hardware model-driven quantization: Energy cost for DNN inference is profiled for combinations of activation/weight bitwidths, enabling precise trade-off tuning via hardware-specific LUTs (Risso et al., 2022).
• Constraint enforcement decoupling: For generalized assignment, feasibility is guaranteed by encoding exclusivity in one-hot registers, with capacity handled classically to avoid surplus circuit overhead (Mastroianni et al., 4 Nov 2025).
• Non-smooth and nondifferentiable goals, joint end-to-end design: Ongoing extensions involve non-smooth objective quantizers, codebook structuring for complexity reduction, and integration with channel coding for communication-oriented resource allocation (Zou et al., 2022).

Limitations chiefly concern heuristic solution gaps (no global optimality guarantee), sensitivity to regularization coefficients and penalty weights, and dependence on problem-specific tuning for partitioning and quantizer count.

7. Comparative Table of Select Quantized-Assignment Strategies

Setting	Assignment Granularity	Main Algorithmic Principle
Job Reassignment QUBO (Delgado et al., 2023)	Binary $x_{ij}$, priority buckets	Subproblem partitioning, filtering
DNN Channel Mixed-Precision (Risso et al., 2022)	Per-channel bitwidths $b_{l,c}$	Differentiable NAS, annealing
VQE Shot Assignment (Zhu et al., 2023)	Integer shot counts $m_i$	Real-optimal rounding, greedy
Resource Allocation GO Quantization (Zou et al., 2022)	Regions/codepoints $z_m$	Lloyd alternation, HR analysis
Generalized Assignment Quantum (Mastroianni et al., 4 Nov 2025)	Encoded register bits per task	One-hot encoding, classical decode

This comparison underscores the breadth of quantized-assignment strategy deployment, ranging from hardware-efficient quantum and classical algorithms to energy-aware neural inference and application-optimal resource management.