adaptive-directional-gradient-qc

name: adaptive-directional-gradient-qc description: "Forward gradient estimation methodology for training parameterised quantum circuits (PQCs). Introduces QUIVER adaptive optimiser that recovers SPSA, random coordinate descent, and parameter-shift rule as limiting cases. Enables efficient training of 60-qubit quantum neural networks." tags: ["quantum", "optimization", "machine-learning", "gradient-descent", "parameter-shift", "QUIVER"] related_skills: ["quantum-neural-architecture", "qml-framework-agnostic-design"]

Adaptive Directional Gradient Estimation for Parameterised Quantum Circuits

Context

Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation. Under the parameter-shift rule, gradient cost scales linearly with the number of trainable parameters, dominating the total shot budget at scale.

Core Methodology

1. Forward Gradient Estimator Framework

Based on the forward mode of automatic differentiation, this framework yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives.

2. Unified Gradient Framework

The framework recovers three existing methods as limiting cases:

• SPSA (Simultaneous Perturbation Stochastic Approximation) — single-direction extreme
• Random coordinate descent — intermediate case
• Parameter-shift rule — full-gradient extreme

All achieved with no ancilla qubits or controlled-gate overhead.

3. Convergence Proof

Stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the SPSA extreme and the parameter-shift extreme.

4. QUIVER Optimiser

Quantum Iterative V-adapter Rule:

• Adaptive optimiser for parameterised circuits
• Update rule derived from closed-form minimum measurement-cost allocation
• Outperforms iCANS and gCANS measurement-frugal optimisers on QAOA and VQE problems

5. Practical Results

• Trains Hamming-weight-preserving orthogonal quantum neural networks
• Up to 60 qubits and 1770 parameters
• Tested on ECG5000 and MNIST datasets
• Orders of magnitude more efficient than parameter-shift rule

Implementation Steps

1. Implement forward gradient estimator for your PQC
2. Configure number of random directional derivatives (tunable parameter)
3. Use QUIVER update rule for minimum measurement-cost allocation
4. For small parameter counts: approach parameter-shift rule for precision
5. For large parameter counts: approach SPSA-like behaviour for efficiency

Pitfalls

• Parameter-shift overhead: Linear scaling in number of parameters makes it infeasible for large circuits (>100 parameters)
• SPSA noise: Single-direction SPSA has high variance; use intermediate number of directions for best trade-off
• No ancilla advantage: The method works without ancilla qubits or controlled gates — simpler hardware requirements but may miss opportunities where ancillae are available
• Convergence assumptions: Standard convergence proofs assume bounded gradients and appropriate learning rate schedules

Verification

1. Implement forward gradient estimator on a small PQC (2-4 qubits)
2. Compare gradient estimates against parameter-shift rule (ground truth)
3. Verify convergence on MNIST/ECG5000 benchmarks
4. Measure shot efficiency vs iCANS/gCANS baselines

Activation

quantum gradient estimation, parameter-shift rule, SPSA, QUIVER, forward gradient, quantum neural network training, variational quantum circuits, PQC optimization, measurement-efficient training, arxiv:2606.09734