lie-algebra-quantum-control-interpolation

name: lie-algebra-quantum-control-interpolation description: "Lie algebra-based quantum optimal control interpolation methodology. Combines Lie group theory with feed-forward neural networks to generate quantum optimal control pulses for arbitrary unitary operations, bypassing explicit optimization at inference time. Demonstrated on superconducting qubits (2-4 qubits) and applied to Trotter propagators for neutrino collective flavor oscillations. Use when: quantum control pulse generation, scalable quantum simulation, Lie group control, neural network control mapping." metadata: arxiv_id: "2606.02014" published: "2026-06-01" authors: ["Piero Luchi", "Francesco Pederiva"] categories: ["quant-ph"]

Lie Algebra-Based Quantum Optimal Control Interpolation

Paper: Lie Algebra-Based Quantum Optimal Controls Interpolation (arXiv:2606.02014, 2026-06-01)

Core Methodology

This paper presents a framework combining Lie group theory with feed-forward neural networks to efficiently generate quantum optimal control pulses for arbitrary unitary operations in superconducting qubit systems, bypassing the need for explicit optimization at inference time.

Problem Statement

The exponential scaling of Hilbert space dimension (2^N) with qubit count makes standard quantum optimal control optimization (like GRAPE/GOAT) computationally prohibitive when large ensembles of distinct propagators must be processed — particularly acute in Trotterized quantum simulation where many time-step propagators need control pulses.

Solution: Lie-NN Control Framework

Phase 1: Lie Group Pre-computation

• Parameterize the SU(2^N) unitary group via its Lie algebra su(2^N)
• Generate a representative set of target unitaries by sampling Lie algebra parameters
• Compute optimal control pulses for each sample using standard optimization (GRAPE)
• This creates a training dataset: {Lie parameters → optimal control pulses}

Phase 2: Neural Network Training

• Train feed-forward neural network to map Lie algebra parameters → control pulse sequences
• The network learns the complex nonlinear relationship between target propagators and their optimal controls
• Key insight: Lie algebra provides a compact, structured parameterization of the target space

Phase 3: Inference (Zero Optimization)

• For any new target unitary: compute its Lie algebra decomposition → feed to network → get control pulses
• No optimization needed at inference time — just forward pass through the trained network
• One model trained once serves as universal control-pulse generator for any compatible Hilbert space dimension

Benchmark Results

Physical benchmark: Successfully reconstructed control pulses for Trotter propagators of a neutrino system undergoing collective flavor oscillations — demonstrating generalization across system types.

Key Technical Details

Lie Algebra Parameterization:

• SU(d) has d²-1 generators (Pauli strings for qubit systems)
• Any unitary U = exp(i Σ θ_k G_k) where G_k are Lie algebra generators
• The coefficients θ_k form a compact representation of the target

Network Architecture:

• Input: Lie algebra parameters (θ₁, ..., θ_{d²-1})
• Output: Control pulse amplitudes at each time step
• Feed-forward with sufficient depth to capture nonlinear mapping

Hardware Independence:

• Model trained on hardware-specific random propagators
• Single model serves as universal generator for any target quantum system of compatible dimension
• Hardware-specific control constraints baked into training data

Reusable Patterns

1. Lie-NN Decoupling: Separate the mathematical structure (Lie algebra) from the learning problem (neural mapping). Use Lie theory for target parameterization, not for pulse generation.

2. Offline Optimization → Online Inference: Move all expensive computation to offline training phase. Inference is just forward pass.

3. Universal Control Generator: One model trained on random propagators generalizes to physically meaningful targets (Trotter steps, Hamiltonian evolution, etc.)

4. Structured Parameterization: Lie algebra provides optimal compact representation — no redundant parameters, covers all reachable unitaries.

Comparison with Related Methods

Application Scenarios

• Trotterized Hamiltonian simulation: Generate control pulses for all Trotter steps in one shot
• Quantum algorithm compilation: Map circuit gates to hardware pulses efficiently
• Pulse library generation: Build comprehensive pulse libraries for quantum compilers
• Adaptive quantum simulation: Rapidly switch between different simulated Hamiltonians

Pitfalls

• Hilbert space dimension limit: Currently demonstrated up to 4 qubits. SU(2^N) has 4^N - 1 parameters — input dimension grows exponentially.
• Hardware-specific: Training data must match target hardware's control constraints and noise model.
• Fidelity varies by region: Some Lie algebra parameter combinations yield higher reconstruction fidelity than others — may need adaptive sampling.
• Not a replacement for optimal control: Still needs initial optimization to generate training data. Benefits accrue when many propagators need pulses.

Activation

quantum control, Lie algebra control, quantum optimal control interpolation, GRAPE neural network, scalable quantum simulation, Trotter propagator control, feed-forward quantum control

Cross-references

• [[quantum-optimal-control-irrep-distillation]] - IRD method for Dicke manifold control in Rydberg atom arrays (complementary: handles leakage, uses GrAPE optimization)
• [[drl-quantum-optimal-control]] - Deep reinforcement learning for quantum optimal control
• [[quantum-control-engineering]] - Engineering patterns for reliable quantum control
• [[pinn-quantum-pulse-optimization]] - Physics-informed neural networks for quantum pulse optimization