# Low-variance estimators overcome the phase-gradient bottleneck in complex-valued neural quantum states

arXiv: https://arxiv.org/abs/2606.13912 · HTML: https://mlsanity.com/paper/2606.13912/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 58 references that are in the map.
- First posted on arXiv 11 Jun 2026.

## Builds on (most cited first, 13 of 13)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Backpropagation through the Void: Optimizing control variates for black-box gradient estimation](https://mlsanity.com/paper/1711.00123/) (2017)
- [Quantum Natural Gradient](https://mlsanity.com/paper/1909.02108/) (2019)
- [REBAR: Low-variance, unbiased gradient estimates for discrete latent variable models](https://mlsanity.com/paper/1703.07370/) (2017)
- [Deep autoregressive models for the efficient variational simulation of many-body quantum systems](https://mlsanity.com/paper/1902.04057/) (2019)
- [Neural network wave functions and the sign problem](https://mlsanity.com/paper/2002.04613/) (2020)
- [Pairing-based graph neural network for simulating quantum materials](https://mlsanity.com/paper/2311.02143/) (2023)
- [Transferable Neural Wavefunctions for Solids](https://mlsanity.com/paper/2405.07599/) (2024)
- [Large Electron Model: A Universal Ground State Predictor](https://mlsanity.com/paper/2603.02346/) (2026)
- [QERNEL: a Scalable Large Electron Model](https://mlsanity.com/paper/2604.26018/) (2026)

## Built on by (most cited first, 1 of 1)

- [Apparent Compression, Real Stability: The Intrinsic Dimension of Learning a Quantum Wavefunction](https://mlsanity.com/paper/2609.33193/) (2026)

## Abstract (arXiv)

Complex neural quantum states are difficult to optimize when their wavefunction phase carries gauge, chiral, fermionic, or topological structure. We show that the major failure mode is not only ansatz expressivity, but the Monte Carlo estimator used to learn this phase. For separated amplitude-phase states, differentiating the local energy at fixed samples gives a different unbiased estimator of the same variational Monte Carlo phase force, without changing the objective. We further extend the construction to coupled two-head networks by keeping the amplitude-gradient contribution and applying the direct derivative only to the phase path. An adaptive minimum-variance mixture interpolates between standard and direct estimators during training. Across flux ladders, chiral chains, two-dimensional flux cylinders, an interacting fermion ladder, shared-network controls, and a fractional quantum Hall benchmark, the resulting estimators reduce phase-gradient variance, suppress seed failures, and often move multi-percent standard-gradient plateaus to sub-percent accuracy.