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npx versuz@latest install hiyenwong-ai-collection-collection-skills-constraint-preserving-quantum-optimizationgit clone https://github.com/hiyenwong/ai_collection.gitcp ai_collection/SKILL.MD ~/.claude/skills/hiyenwong-ai-collection-collection-skills-constraint-preserving-quantum-optimization/SKILL.md---
name: constraint-preserving-quantum-optimization
description: >
Constraint-preserving quantum optimization methodology using XY-mixers and
Trotterized Adiabatic Evolution (TAE). Covers constraint handling in quantum
combinatorial optimization, mixer selection criteria, Trotter error analysis,
and structure-aware quantum circuit design. Use when working with quantum
optimization algorithms (QAOA, adiabatic evolution, VQE), constraint handling
in quantum computing, Trotterization error analysis, XY-mixer implementation,
or designing quantum circuits for constrained optimization problems.
Trigger: quantum optimization constraints, XY-mixer, Trotterized adiabatic,
constraint-preserving quantum, QAOA constraints, quantum mixer design.
---
# Constraint-Preserving Quantum Optimization
Methodology from arXiv:2605.02465 "Constraint Preserving XY-Mixers under Trotterized Adiabatic Evolution" by Awasthi et al.
## Core Insight
Constraint-preserving mixers (e.g., XY-mixers) restrict quantum evolution to feasible subspaces, avoiding penalty-based approaches that increase problem size and distort energy landscapes. However, Trotterization on gate-based hardware introduces approximation errors whose impact depends on **constraint locality**, not total problem size.
## Key Decision Rule: Mixer Selection
| Constraint Structure | Recommended Mixer | Reason |
|---------------------|-------------------|--------|
| Single global equality (all variables) | Pauli-X mixer | Trotter errors in XY-mixer significantly impair performance |
| Multiple disjoint local blocks | XY-mixer | Outperforms X-mixer by orders of magnitude even under Trotterization |
| TSP-like 2-way-1-hot | Dedicated mixer Hamiltonian | See dedicated TSP mixer below |
**Key criterion**: Constraint locality determines XY-mixer effectiveness.
## Trotter Error Analysis
### Error Scaling
- Dominant Trotter error contribution depends on **individual constraint size and structure**
- NOT dependent on total problem size
- For global constraints spanning all n variables: error scales with O(n)
- For k disjoint local constraints of size m: error scales with O(m), independent of n
### Error Mitigation Strategies
1. **Decompose global constraints** into local blocks where possible
2. **Reduce Trotter step count** by using larger time steps with fewer repetitions
3. **Use structure-aware mixer design** matching constraint topology
4. **Combine TAE with variational layers** for hybrid optimization
## Implementation Patterns
### XY-Mixer for Partition Constraints
For constraints where variables partition into groups with fixed sum:
```
H_XY = Σ_{(i,j)∈feasible} (X_i X_j + Y_i Y_j)
```
This preserves Hamming weight within each constraint block.
### TSP 2-Way-1-Hot Mixer
For traveling salesman problem constraints (each city visited exactly once, each time step has exactly one city):
```
H_TSP_mixer = Σ_{city} H_XY(row_city) + Σ_{time} H_XY(col_time)
```
Each row and column forms an independent XY-mixer block.
### Circuit Implementation
1. Decompose XY-mixer into two-qubit gates via Trotterization
2. Apply each constraint block's XY-mixer sequentially
3. Interleave with problem Hamiltonian evolution
4. Use adiabatic schedule: H(t) = (1-t/T)·H_mixer + (t/T)·H_problem
## Workflow for Constrained Quantum Optimization
1. **Identify constraint structure**: Global vs. local, equality vs. inequality
2. **Decompose constraints**: Break global constraints into local blocks if possible
3. **Select mixer**: Use decision rule above
4. **Design Trotter schedule**: Balance step size vs. accuracy
5. **Validate**: Compare with penalty-based baseline on small instances
6. **Scale**: Test on hardware with increasing qubit counts
## Common Pitfalls
- **Penalty method trap**: Adding penalty terms increases circuit depth and distains energy landscapes
- **Global XY-mixer on global constraints**: Trotter errors compound and destroy constraint preservation
- **Ignoring constraint topology**: Mixer must match constraint structure for effectiveness
- **Over-Trotterization**: Too many small steps increase gate count without proportional accuracy gain
## Activation Keywords
- quantum optimization constraints
- XY-mixer
- Trotterized adiabatic evolution
- constraint-preserving quantum
- QAOA constraints
- quantum mixer design
- quantum combinatorial optimization
- Trotter error analysis
- quantum constraint handling