qaoa-feasibility-penalty-scheduling

name	qaoa-feasibility-penalty-scheduling
description	Feasibility-driven QAOA with penalty scheduling. Introduces Λ-lr-QAOA (per-penalty linear-ramp schedules) and piecewise-ramp QAOA (two-segment piecewise schedules) for constrained optimization. Promotes penalty weights from external hyperparameters to internal variational parameters. Activation: feasibility-driven QAOA, penalty scheduling, constrained QAOA, Λ-lr-QAOA, piecewise-ramp QAOA, MWIS optimization
metadata	{"arxiv_id":"2606.25117","published":"2026-06-23","authors":"Francesco Ferrari, Matteo Vandelli, Daniele Dragoni","tags":["quantum","optimization","QAOA","constraints","penalty-method","scheduling"]}

QAOA with Feasibility Penalty Scheduling

Description

Extends standard linear-ramp QAOA (lr-QAOA) for problems with multiple heterogeneous constraints. Introduces Λ-lr-QAOA (per-penalty linear-ramp) and piecewise-ramp QAOA (two-segment schedules) that promote penalty weights from external hyperparameters to internal variational parameters.

Activation Keywords

• feasibility-driven QAOA
• penalty scheduling
• constrained QAOA
• Λ-lr-QAOA
• piecewise-ramp QAOA
• MWIS quantum optimization
• 可行性驱动QAOA
• 惩罚调度量子优化

Core Concepts

Λ-lr-QAOA (Lambda Linear-Ramp QAOA)

• Each penalty term gets its own linear-ramp schedule
• Penalty weights become internal variational parameters (not external hyperparameters)
• Joint optimization of all schedules eliminates nested penalty tuning
• Scales efficiently to multiple constraints

Piecewise-Ramp QAOA

• Linear ramps replaced by two-segment piecewise schedules
• Enhances Ansatz expressiveness at cost of small parameter overhead (independent of circuit depth)
• Consistently outperforms lr-QAOA and Λ-lr-QAOA across depths and system sizes

Feasibility-Driven Loss Function

• Pushes quantum state towards high-quality feasible solutions
• Filtered variant provides single hyperparameter to tune feasibility-optimality trade-off

Methodology

Step 1: Formulate Constrained Problem

Express as QUBO with multiple heterogeneous constraints: H = H_obj + Σ λ_i H_constraint_i

Step 2: Construct Λ-lr-QAOA Ansatz

• For each constraint i: define linear ramp schedule γ_i(t) = α_i · t
• Promote α_i to variational parameter alongside γ, β

Step 3: Piecewise-Ramp Enhancement (Optional)

• Replace linear ramps with two-segment piecewise: γ_i(t) = {α_i1·t for t < t*; α_i2·t + β_i for t ≥ t*}

Step 4: Joint Optimization

• Optimize all parameters simultaneously: {γ, β, α_1, ..., α_k}
• Use feasibility-driven loss: L = -<H_obj> + λ_violation · (constraint violation penalty)

Step 5: Filtered Loss for Feasibility-Optimality Balance

• Apply filtered variant to tune balance with single hyperparameter
• Higher filter → prioritize feasibility; Lower filter → prioritize optimality

Usage Patterns

Pattern 1: Constrained Portfolio Optimization

Apply to portfolio problems with budget, risk, and sector constraints.

Pattern 2: Satellite Mission Planning

Benchmarked on Earth-observation satellite mission planning (budget-constrained MWIS).

Pattern 3: General Constrained QUBO

Use for any QUBO with multiple heterogeneous constraints where penalty tuning is challenging.

Pitfalls

Parameter Scaling

Piecewise-ramp adds ~2× parameters vs. linear-ramp, but overhead is independent of circuit depth → acceptable trade-off for most applications.

Feasibility-Optimality Trade-off

Inherent trade-off between solution feasibility and quality. No free lunch — must tune via filtered loss hyperparameter.

High Feasibility Rate

Both Λ-lr-QAOA and piecewise-ramp exhibit high feasibility rates — crucial for industrial applications but may sacrifice some solution quality.

Circuit Depth Requirements

Performance improvements are most pronounced at moderate-to-high circuit depths. At very low depths (p=1), benefits are limited.

References

• arXiv: 2606.25117 - "Feasibility-driven QAOA with penalty scheduling"
• Related: qaoa-manifold-optimization (Riemannian manifold optimization for QAOA)
• Related: distributed-qaoa-simulator (distributed QAOA execution)