Optimization Techniques in Quantum Machine Learning: SPSA, COBYLA, and Beyond

Table of Contents

1. Introduction
2. Role of Optimization in Quantum Machine Learning
3. Gradient-Based vs Gradient-Free Methods
4. Stochastic Gradient Descent (SGD)
5. Adam Optimizer
6. Simultaneous Perturbation Stochastic Approximation (SPSA)
7. SPSA: Algorithm and Use Cases
8. SPSA for Noisy Quantum Environments
9. Constrained Optimization BY Linear Approximation (COBYLA)
10. COBYLA in Qiskit and PennyLane
11. Nelder-Mead Method
12. Powell’s Method
13. Conjugate Gradient Descent
14. BFGS and L-BFGS-B
15. SPSA vs COBYLA: Strengths and Weaknesses
16. Choosing the Right Optimizer for NISQ Devices
17. Optimization Under Measurement Noise
18. Layer-Wise Optimization Strategy
19. Combining Classical and Quantum Optimizers
20. Conclusion

1. Introduction

Optimization techniques are at the heart of training quantum machine learning models, especially those based on parameterized quantum circuits. These methods adjust gate parameters to minimize a loss function, using either exact gradients or approximations.

2. Role of Optimization in Quantum Machine Learning

• Guides training of Variational Quantum Circuits (VQCs)
• Minimizes cost functions (e.g., classification loss, energy in VQE)
• Must handle noise, hardware constraints, and quantum randomness

3. Gradient-Based vs Gradient-Free Methods

• Gradient-Based: require partial derivatives (e.g., parameter-shift rule)
• Gradient-Free: rely on function evaluations (e.g., SPSA, COBYLA)

4. Stochastic Gradient Descent (SGD)

• Uses a small batch of data to compute approximate gradients
• Simple, but sensitive to learning rate and noise

5. Adam Optimizer

• Combines momentum and adaptive learning rate
• Well-suited for differentiable hybrid quantum-classical models

6. Simultaneous Perturbation Stochastic Approximation (SPSA)

• Estimates gradients by perturbing all parameters simultaneously
• Only requires two function evaluations per step:
  \[
  g_k = rac{f( heta_k + c_k \Delta_k) – f( heta_k – c_k \Delta_k)}{2 c_k \Delta_k}
  \]

7. SPSA: Algorithm and Use Cases

• Efficient for high-dimensional or noisy cost landscapes
• Popular in QAOA, QNN training on real quantum devices

8. SPSA for Noisy Quantum Environments

• Naturally robust to shot noise
• Performs well even with low-fidelity measurements

9. Constrained Optimization BY Linear Approximation (COBYLA)

• Gradient-free, constraint-respecting algorithm
• Approximates local linear models for optimization
• Good for small parameter spaces

10. COBYLA in Qiskit and PennyLane

• Qiskit: qiskit.algorithms.optimizers.COBYLA
• PennyLane: qml.optimize.COBYLAOptimizer()

11. Nelder-Mead Method

• Uses simplex-based optimization
• Sensitive to local minima
• Performs well in low-dimensional, smooth landscapes

12. Powell’s Method

• Performs line searches along conjugate directions
• No gradient required
• Effective when parameters are weakly correlated

13. Conjugate Gradient Descent

• Assumes differentiable cost function
• Optimizes along conjugate directions
• Requires Hessian approximation

14. BFGS and L-BFGS-B

• Quasi-Newton methods
• Use approximate second-order information
• Suitable for simulator-based training

15. SPSA vs COBYLA: Strengths and Weaknesses

16. Choosing the Right Optimizer for NISQ Devices

• Use SPSA or COBYLA for noisy, real-device training
• Use Adam, BFGS for clean, simulator environments

17. Optimization Under Measurement Noise

• Use averaging over multiple shots
• Apply learning rate decay
• Employ variance reduction techniques

18. Layer-Wise Optimization Strategy

• Optimize circuit layers sequentially
• Reduces barren plateau effects
• Similar to greedy layer-wise pretraining

19. Combining Classical and Quantum Optimizers

• Classical layers use Adam/SGD
• Quantum layers use SPSA/COBYLA
• Unified hybrid optimization pipelines

20. Conclusion

Optimization is a central component of quantum model training. Techniques like SPSA and COBYLA enable effective learning even on noisy, real-world quantum hardware. Understanding the landscape of optimizers helps practitioners design robust, efficient, and scalable quantum learning workflows.