Multi-Objective and Quality-Diversity Optimization of Quantum Circuits

Automatically discovers quantum algorithms using evolutionary optimization, multi-objective search, and quality-diversity methods -- among the first applications of MAP-Elites to quantum circuit design.

This project benchmarks 6 optimization methods across 6 quantum problems (360 experiments), comparing evolutionary algorithms, deep learning (REINFORCE), gradient-based optimization, multi-objective search (NSGA-II), and quality-diversity (MAP-Elites). A custom pure-numpy statevector simulator provides 100--1000x faster evaluation than Qiskit-based alternatives, enabling large-scale experimentation on a single machine.

Paper: paper/paper.pdf -- Multi-Objective and Quality-Diversity Optimization of Quantum Circuits via Statevector Simulation

Literature Review: literature_review/literature_review.pdf -- Survey of 35+ papers across 8 sub-fields

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Key Results

3 qubits, 10,000 evaluations, 10 trials per condition.

| Problem | RS | EA | Gradient | DL | NSGA-II | MAP-Elites | |---------|:--:|:--:|:--------:|:--:|:-------:|:----------:| | Grover | 0.83 | 0.97 | 0.00 | 0.94 | 0.77 | 0.89 | | Flip | 0.75 | 0.50 | 0.05 | 0.99 | 0.54 | 0.72 | | Inverse | 0.88 | 0.67 | 0.30 | 1.00 | 0.65 | 1.00 | | Fourier | 1.00 | 1.00 | 0.52 | 1.00 | 0.99 | 1.00 | | Deutsch-Jozsa | 0.93 | 0.95 | 0.13 | 1.00 | 0.80 | 0.98 | | Bernstein-Vazirani | 0.91 | 0.72 | 0.00 | 0.98 | 0.69 | 0.99 |

Key Findings

1. REINFORCE achieves the highest fitness on most problems, reaching perfect 1.00 on Inverse, Fourier, and Deutsch-Jozsa.
2. MAP-Elites matches or exceeds single-objective methods on 3 of 6 problems while simultaneously discovering diverse circuit families (79--87% archive coverage). This is among the first demonstrations of quality-diversity optimization for quantum circuit design.
3. NSGA-II produces the most compact circuits (2.6--11.1 non-identity gates vs 14--22 for other methods), revealing fidelity-complexity trade-offs via Pareto fronts.
4. Gradient-based optimization consistently fails for discrete quantum circuit synthesis, due to the discretization gap between soft gate mixtures and hard gate selections.
5. The EA is 3--10x faster than other methods by wall-clock time but gets trapped in local optima on harder problems.

Quality-Diversity (MAP-Elites)

MAP-Elites maintains a 2D archive indexed by circuit depth and entanglement density, illuminating the landscape of viable circuits for each problem:

| Problem | Archive Coverage | QD-Score | Best Fitness | |---------|:----------------:|:--------:|:------------:| | Grover | 87% | 36.5 | 0.85 | | Flip | 79% | 60.1 | 1.00 | | Inverse | 83% | 46.0 | 1.00 | | Fourier | 84% | 65.5 | 1.00 | | Deutsch-Jozsa | 84% | 48.9 | 1.00 | | Bernstein-Vazirani | 86% | 43.9 | 1.00 |

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Why This Matters

• Novel contribution: One of the first applications of MAP-Elites quality-diversity optimization to quantum circuit discovery. Prior work (Zorn et al., 2025) applied CMA-MAE to variational circuits for combinatorial optimization; this project applies MAP-Elites to algorithmic circuit synthesis (Grover, QFT, Deutsch-Jozsa, Bernstein-Vazirani).
• Comprehensive comparison: Six fundamentally different optimization paradigms compared head-to-head on the same problems with the same evaluation budget -- EA, random search, gradient, deep learning, multi-objective, and quality-diversity.
• Fast simulation: Pure numpy statevector simulator (~microseconds per evaluation) enables 100--1000x more evaluations than Qiskit-based competitors, making large-scale studies tractable on commodity hardware.
• Reproducible: Deterministic seeds, configurable trial counts, and automated study scripts. All results regenerable with a single command.

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Optimizers

| Method | Type | Approach | |--------|------|----------| | Evolutionary Algorithm | Single-objective | DEAP-based EA with two-point crossover and bit-flip mutation | | Random Search | Baseline | Uniform random circuit generation | | Gradient-Based | Single-objective | Continuous relaxation with softmax weighting, L-BFGS-B | | Deep Learning | Single-objective | REINFORCE policy gradient with feedforward network | | NSGA-II | Multi-objective | Pareto-front selection on (fidelity, depth, gate count) | | MAP-Elites | Quality-diversity | 2D archive indexed by (active depth, entanglement density) |

Problems

| Problem | Target | Classical Complexity | |---------|--------|---------------------| | Grover's Search | P(marked_item) = 1.0 | O(N) linear search | | Flip | Bitwise NOT | O(N) direct | | Inverse | 1/x | O(N) direct | | Fourier | QFT | O(N log N) | | Deutsch-Jozsa | P(|0...0>) = 0 for balanced function | O(N/2+1) queries | | Bernstein-Vazirani | P(|s>) = 1.0 for hidden string s | O(N) queries |

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How It Works

Quantum circuits are represented as integer matrices where each cell encodes a gate type:

| Symbol | Gate | |--------|------| | T | T gate | | H | Hadamard gate | | . - (+) | CNOT (control above target) | | (+) - . | CNOT (target above control) | | \| | Identity (no-op) |

Fitness is the probability of measuring the correct target state from the circuit output. Optimizers evolve these gate arrays to maximize fitness (and optionally minimize depth/complexity).

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Getting Started

Installation

python -m venv venv
source venv/bin/activate
pip install -r requirements.txt

For the comparative study (requires PyTorch and matplotlib):

pip install -r requirements-study.txt

Quick Start

# Evolve a Grover's search circuit
python -m examples.grover_demonstration

# Run the full 6-optimizer comparative study
python -m examples.run_qd_study

# Full study with scaling analysis (2-5 qubits, 25K budget, 20 trials)
python -m examples.run_qd_study --full

Results are saved to study_results/qd_study/ including JSON data and PNG charts.

Tests

pytest

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Project Structure

quantum_ea/                        # Main package
    gates.py                       # GateType enum, CNOT handling, redundancy removal
    circuit.py                     # Pure numpy statevector simulation
    fitness.py                     # Fitness evaluation with caching
    visualization.py               # ASCII circuit output
    evolutionary_algorithm.py      # DEAP EA with injectable evaluation
    config.py                      # Typed EAConfig from config.ini
    target_generation.py           # Problem target state generation
    optimizers/                    # All 6 optimizer implementations
        base.py                    # OptimizerBase ABC + OptimizationResult
        ea_optimizer.py            # Evolutionary algorithm
        random_search.py           # Random search baseline
        gradient_optimizer.py      # Continuous relaxation + L-BFGS-B
        dl_optimizer.py            # REINFORCE policy gradient
        nsga2_optimizer.py         # NSGA-II multi-objective
        mapelites_optimizer.py     # MAP-Elites quality-diversity
    problems/                      # 6 benchmark problem definitions
    classical/                     # Classical algorithm baselines
    study/                         # Experiment runner, metrics, plotting
examples/                          # Entry points and demonstrations
tests/                             # pytest test suite
paper/                             # Research paper (LaTeX + PDF)
literature_review/                 # Literature review (LaTeX + PDF)
study_results/                     # Generated results and charts

Configuration

EA hyperparameters are configured in config.ini:

[EvolutionaryAlgorithm]
INDIVIDUAL_DNA_SIZE = 30
INDIVIDUAL_SWAP_PROBABILITY = 0.1
TOURNAMENT_SIZE = 3
POPULATION = 100
BREEDING_PROBABILITY = 0.5
MUTATION_PROBABILITY = 0.2
GENERATIONS = 200

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Citation

If you use this software, its results, or algorithms in a publication or official document, you are required to cite the original work (see NOTICE and CITATION.cff):

Snell, T. (2026). Multi-Objective and Quality-Diversity Optimization
of Quantum Circuits via Statevector Simulation. Quantum Computing
Evolutionary Algorithm Design.

License

This project is licensed under the GNU General Public License v3.0 with an additional academic citation requirement under Section 7(b). See LICENSE, NOTICE, and CITATION.cff for details.