quacod-quantum-coordinate-descent

name

quacod-quantum-coordinate-descent

description

Quantum Optimization via Coordinate Descent (QUACOD) methodology. Decomposes large-scale combinatorial optimization problems into quantum-solvable subproblems using coordinate descent, enabling NISQ-era hardware to handle problems 5-35x larger than direct quantum approaches. Use when: (1) optimization problems exceed available qubits, (2) scaling quantum optimization to practical problem sizes, (3) drone scheduling/logistics optimization, (4) iterative quantum-classical hybrid optimization workflows.

QUACOD: Quantum Optimization via Coordinate Descent

Decomposes large-scale combinatorial optimization problems into quantum-solvable subproblems using coordinate descent, enabling NISQ-era quantum hardware to handle problems significantly larger than direct quantum approaches.

Core Idea

Instead of encoding the entire problem onto a quantum circuit (limited by qubit count), QUACOD:

1. Formulates the full problem as a QUBO/Ising model
2. Identifies variable subsets (coordinates) within qubit budget
3. Solves each subproblem using quantum optimization (QAOA/VQE)
4. Iterates coordinate descent until convergence
5. Uses hardware-efficient circuits for practical execution

Workflow

Step 1: Problem Formulation

Express the optimization problem as QUBO:

$$\min_x x^T Q x + c^T x, \quad x \in {0, 1}^n$$

Where $n$ is the total number of binary variables (can be >> available qubits).

Step 2: Coordinate Partitioning

Partition variables into blocks $B_1, B_2, \ldots, B_k$ where each $|B_i| \leq q$ (available qubits).

Strategy: Group correlated variables together, or use random/sequential partitioning.

Step 3: Quantum Subproblem Solving

For each block $B_i$, fix all other variables and solve:

$$\min_{x_{B_i}} x_{B_i}^T Q_{B_i} x_{B_i} + (Q_{B_i, \text{rest}} x_{\text{rest}} + c_{B_i})^T x_{B_i}$$

Using quantum optimization:

• QAOA with hardware-efficient ansatz
• VQE with problem-specific ansatz
• Quantum annealing (if available)

Step 4: Iteration & Convergence

Cycle through all blocks, updating the solution after each quantum solve. Stop when solution converges or max iterations reached.

Implementation Pattern

import numpy as np
from qiskit.algorithms import QAOA
from qiskit.primitives import Sampler

def quacod(Q, c, qubit_budget, max_iter=10, block_size=None):
    """QUACOD: Quantum Optimization via Coordinate Descent.
    
    Args:
        Q: QUBO matrix (n x n)
        c: Linear coefficients (n,)
        qubit_budget: Maximum qubits per subproblem
        max_iter: Maximum coordinate descent iterations
        block_size: Size of coordinate blocks (default: qubit_budget)
    
    Returns:
        x: Optimal binary solution
        history: Solution trajectory
    """
    n = len(c)
    x = np.random.randint(0, 2, n)
    history = [x.copy()]
    
    if block_size is None:
        block_size = qubit_budget
    
    for iteration in range(max_iter):
        # Partition variables into blocks
        blocks = [list(range(i, min(i + block_size, n))) 
                  for i in range(0, n, block_size)]
        
        converged = True
        for block in blocks:
            rest = [j for j in range(n) if j not in block]
            
            # Build subproblem QUBO for this block
            Q_block = Q[np.ix_(block, block)]
            # Effective linear term from fixed variables
            c_block = c[block] +  * Q[np.ix_(block, rest)] @ x[rest]
            
            
            x_new_block = quantum_solve_subproblem(Q_block, c_block)
            
              np.array_equal(x[block], x_new_block):
                x[block] = x_new_block
                converged = 
        
        history.append(x.copy())
         converged:
            
    
     x, history


 ():
    
    
    
    

Key Advantages

Metric	Direct Quantum	QUACOD
Max variables	~100 (qubit limit)	500+ (5x more drones)
Max routes	~50	1,750+ (35x more)
Circuit depth	Deep (full problem)	Shallow (subproblems)
NISQ feasible	Limited	Practical

Use Cases

1. Drone Scheduling: Route optimization with limited qubits
2. Portfolio Optimization: Large-scale financial problems
3. Logistics: Vehicle routing, job shop scheduling
4. Graph Problems: Max-Cut, vertex coloring on large graphs
5. Any QUBO problem exceeding available quantum resources

Activation Keywords

• quacod
• quantum coordinate descent
• quantum optimization scaling
• large-scale qubo
• iterative quantum optimization
• drone scheduling quantum
• 坐标下降量子优化
• 大规模量子优化

References

• Paper: arXiv:2605.14001
• Authors: Nguyen et al.
• Key finding: Hardware-efficient circuits effective for coordinate descent subproblems