logq-quantum-inspired-optimization

name: logq-quantum-inspired-optimization description: "LogQ algorithm reformulated as classical non-linear continuous relaxation for QUBO problems. Use when: solving portfolio optimization, fleet optimization, charging station placement, or any QUBO combinatorial problem; implementing quantum-inspired classical algorithms; reducing qubit requirements from quantum formulations; eliminating Pauli decomposition overhead; applying gradient-inspired methods to discrete optimization. Keywords: quantum-inspired, logq, qubo, portfolio optimization, continuous relaxation, binary optimization"

LogQ Quantum-Inspired Optimization

Core Concept

The LogQ algorithm was originally developed for quantum combinatorial optimization, drastically reducing qubit count and circuit depth. It has been reformulated as a classical heuristic using non-linear continuous relaxation of binary variables, eliminating the need for Pauli decomposition and quantum measurement overhead.

When to Use

• QUBO problems in finance (portfolio optimization), logistics (fleet optimization), or infrastructure (charging stations)
• When quantum circuit depth or qubit count is prohibitive
• When gradient-inspired parameter optimization is beneficial
• As a classical baseline before deploying quantum hardware

Algorithm Steps

1. Formulate as QUBO: Express the problem as minimize x^T Q x where x ∈ {0,1}^n
2. Continuous Relaxation: Relax binary constraints x ∈ {0,1} to x ∈ [0,1]
3. LogQ Encoding: Apply logarithmic encoding to reduce variable dimensionality
4. Gradient-Inspired Optimization: Use gradient-like methods on the relaxed continuous variables
5. Rounding: Map continuous solution back to binary via thresholding or randomized rounding

Key Patterns

Pattern 1: Non-linear Continuous Relaxation

Instead of standard linear relaxation, LogQ uses non-linear transformation:

f(x) = log(x + ε) / log(2)  # maps [0,1] → [-∞, 0]

This creates sharper gradients near boundaries, accelerating convergence to binary solutions.

Pattern 2: Gradient-Inspired Parameter Updates

After relaxation, parameters can be optimized using gradient-like methods:

• Momentum-based updates for faster convergence
• Adaptive learning rates per parameter
• No Pauli decomposition required (classical-only)

Pattern 3: Quantum-to-Classical Translation

When a quantum algorithm shows promise but hardware limitations exist:

1. Identify the quantum-specific components (Pauli terms, measurements)
2. Replace with equivalent classical mathematical operations
3. Preserve the algorithmic structure and convergence properties
4. Validate against quantum simulation results

Pitfalls

• Non-convexity: The relaxed problem may have local minima; use multiple restarts
• Rounding loss: Continuous-to-binary rounding may lose solution quality
• Parameter sensitivity: LogQ encoding parameters (ε, temperature) require tuning

Comparison

References

• arXiv: 2604.12925 - "From quantum to quantum-inspired: the LogQ algorithm"