Optimizers

Divi provides built-in support for optimizing quantum programs using three distinct methods, each suited to different problem types and user requirements.

All optimizers can be accessed through the divi.qprog.optimizers module. Scipy-based optimizers rely on the ScipyMethod enum to specify the optimizer used.

Monte Carlo Optimization

The Monte Carlo [1] method in Divi is a stochastic global optimization approach. It works by randomly sampling the parameter space and selecting configurations that minimize the target cost function. This method is particularly useful when:

  • The optimization landscape is rugged or non-convex.

  • Gradients are not available or are unreliable.

  • A rough global search is preferred before local refinement.

Monte Carlo optimization can help identify promising regions in high-dimensional parameter spaces before applying more refined methods.

In Divi, one can configure the optimizer; providing the size of the population n_param_sets, and the number of well-performing parameter sets to carry on to the subsequent iteration n_best_sets.

SciPy Optimizers

Divi provides several SciPy-based optimizers through the ScipyOptimizer class:

Nelder-Mead

Nelder-Mead [2] is a gradient-free, simplex-based optimization algorithm. It is ideal for local optimization in low to moderate dimensional spaces. The method iteratively refines a simplex (a geometrical figure defined by a set of parameter vectors) by evaluating cost function values and applying operations such as reflection, expansion, and contraction.

Use Nelder-Mead when:

  • Your problem is continuous but noisy.

  • Gradients are unavailable or expensive to compute.

  • You are tuning parameters in a relatively low-dimensional space.

from divi.qprog.optimizers import ScipyOptimizer, ScipyMethod

optimizer = ScipyOptimizer(method=ScipyMethod.NELDER_MEAD)

L-BFGS-B

L-BFGS-B (Limited-memory Broyden–Fletcher–Goldfarb–Shanno with Bound constraints) [3] is a quasi-Newton method that leverages gradient information to efficiently converge to a local minimum. In Divi, gradient calculation is performed using the parameter shift rule, a technique well-suited to quantum circuits that allows for analytical gradient computation by evaluating the function at shifted parameter values.

Divi computes these parameter shifts in parallel, significantly reducing wall-clock time for gradient evaluations.

Use L-BFGS-B when:

  • You require fast convergence to a local minimum.

  • Your cost function is smooth and differentiable.

optimizer = ScipyOptimizer(method=ScipyMethod.L_BFGS_B)

COBYLA

COBYLA (Constrained Optimization BY Linear Approximations) [4] is a gradient-free, local optimization method—like Nelder-Mead—that supports nonlinear inequality constraints. It constructs successive linear approximations of the objective function and constraints, iteratively refining the solution within a trust region.

Use COBYLA when:

  • Your optimization problem includes constraints.

  • Gradients are inaccessible or too noisy.

  • You seek a reliable optimizer for low to moderate-dimensional spaces.

COBYLA is also a good choice of optimizer when trying out QAOA for a new problem/experimenting, but your mileage may vary.

optimizer = ScipyOptimizer(method=ScipyMethod.COBYLA)

PyMOO Optimizers

Divi also supports evolutionary algorithms through PyMOO:

CMA-ES (Covariance Matrix Adaptation Evolution Strategy)

CMA-ES [5] is a stochastic, derivative-free method for numerical optimization of non-linear or non-convex continuous optimization problems.

from divi.qprog.optimizers import PymooOptimizer, PymooMethod

optimizer = PymooOptimizer(method=PymooMethod.CMAES)

Differential Evolution

Differential Evolution [6] is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality.

optimizer = PymooOptimizer(method=PymooMethod.DE)

Choosing the Right Optimizer

For VQE:

  • L-BFGS-B: Best for smooth, differentiable landscapes with good initial parameters

  • Monte Carlo: Excellent for exploration and avoiding local minima

  • COBYLA: Good for constrained problems or when gradients are unreliable

  • Nelder-Mead: Robust choice for noisy or discontinuous landscapes

For QAOA:

  • COBYLA: Often the best starting point for QAOA problems

  • Nelder-Mead: Good for noisy landscapes and parameter initialization

  • Monte Carlo: Excellent for global exploration and avoiding barren plateaus

  • L-BFGS-B: Use when you have good initial parameters and smooth landscapes

For PyMOO Optimizers:

  • CMA-ES: Excellent for high-dimensional parameter spaces and when you need robust global optimization. Particularly effective for VQE with many parameters.

  • Differential Evolution: Good for multimodal optimization landscapes and when you need to escape local minima. Works well for QAOA parameter optimization.

For Hyperparameter Sweeps:

  • Monte Carlo: Best for initial exploration across parameter ranges

  • L-BFGS-B: Use for fine-tuning after Monte Carlo exploration

  • Nelder-Mead: Robust fallback when other methods fail

  • CMA-ES: Excellent for high-dimensional sweeps with many parameters

Quantum-Specific Considerations:

  • Barren Plateaus: Use Monte Carlo or CMA-ES to avoid getting trapped in flat regions

  • Parameter Initialization: Start with small random values (typically [-0.1, 0.1]) for better convergence

  • Circuit Depth: Deeper circuits benefit from more robust optimizers like CMA-ES or Monte Carlo

  • Noise Resilience: Nelder-Mead and COBYLA are more robust to quantum noise than gradient-based methods

Next Steps

  • 🔬 Tutorials: Try the runnable examples in the tutorials/ directory.

  • ⚡ Algorithm Guides: Learn about VQE and QAOA for algorithm-specific guidance.

  • ⚡ Batching and Sweeps: See how to use optimizers in large-scale computations in the Program Batches and Workflows guide.

References