# Adaptive directional gradients for parameterised quantum circuits

arXiv: https://arxiv.org/abs/2606.09734 · HTML: https://mlsanity.com/paper/2606.09734/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 73 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [The Forward-Forward Algorithm: Some Preliminary Investigations](https://mlsanity.com/paper/2212.13345/) (2022)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [Theory of overparametrization in quantum neural networks](https://mlsanity.com/paper/2109.11676/) (2021)
- [Scaling Forward Gradient With Local Losses](https://mlsanity.com/paper/2210.03310/) (2022)
- [Stochastic gradient descent for hybrid quantum-classical optimization](https://mlsanity.com/paper/1910.01155/) (2019)
- [On quantum backpropagation, information reuse, and cheating measurement collapse](https://mlsanity.com/paper/2305.13362/) (2023)
- [Coupling Adaptive Batch Sizes with Learning Rates](https://mlsanity.com/paper/1612.05086/) (2016)
- [SEGA: Variance Reduction via Gradient Sketching](https://mlsanity.com/paper/1809.03054/) (2018)
- [Can Forward Gradient Match Backpropagation?](https://mlsanity.com/paper/2306.06968/) (2023)
- [Graph neural network initialisation of quantum approximate optimisation](https://mlsanity.com/paper/2111.03016/) (2021)
- [Adaptive shot allocation for fast convergence in variational quantum algorithms](https://mlsanity.com/paper/2108.10434/) (2021)
- [Training-efficient density quantum machine learning](https://mlsanity.com/paper/2405.20237/) (2024)
- [Effects of noise on the overparametrization of quantum neural networks](https://mlsanity.com/paper/2302.05059/) (2023)
- [Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks](https://mlsanity.com/paper/2406.18316/) (2024)
- [Beyond Backpropagation: Optimization with Multi-Tangent Forward Gradients](https://mlsanity.com/paper/2410.17764/) (2024)
- [Second-Order Forward-Mode Automatic Differentiation for Optimization](https://mlsanity.com/paper/2408.10419/) (2024)
- [Resource frugal optimizer for quantum machine learning](https://mlsanity.com/paper/2211.04965/) (2022)
- [Convergence guarantees for forward gradient descent in the linear regression model](https://mlsanity.com/paper/2309.15001/) (2023)
- [Random coordinate descent: a simple alternative for optimizing parameterized quantum circuits](https://mlsanity.com/paper/2311.00088/) (2023)
- [Improving the Convergence Rates of Forward Gradient Descent with Repeated Sampling](https://mlsanity.com/paper/2411.17567/) (2024)

## Abstract (arXiv)

Training parameterised quantum circuits (PQCs) on quantum hardware is bottlenecked by the measurement cost of gradient estimation, which under the parameter-shift rule scales linearly in the number of trainable parameters and dominates the total shot budget of training at scale. In this work, we propose a framework of forward gradient estimators for PQCs, based on the forward mode of automatic differentiation, that yields an unbiased estimator of the gradient by averaging a freely tunable number of random directional derivatives and recovers SPSA, random coordinate descent, and the parameter-shift rule as limiting cases, with no ancilla qubits or controlled-gate overhead. We prove that stochastic quantum forward gradient descent converges under standard assumptions, with an explicit second-moment expansion that interpolates between the single-direction extreme of SPSA and the full-gradient extreme of parameter-shift. Within this framework we derive QUIVER (Quantum Iterative V-adaptive Estimator Rule), an adaptive optimiser for parameterised circuits whose update rule follows from a closed-form minimum measurement-cost allocation. We show numerically that forward gradients train Hamming-weight-preserving orthogonal quantum neural networks with up to 60 qubits and 1770 parameters on the ECG5000 and MNIST datasets orders of magnitude more efficiently than the parameter-shift rule. We also demonstrate that our proposed QUIVER optimiser can outperform iCANS and gCANS measurement-frugal optimisers on optimisation problems using the quantum approximate optimisation algorithm and quantum simulation with the variational quantum eigensolver.