# Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry

arXiv: https://arxiv.org/abs/2206.04663 · HTML: https://mlsanity.com/paper/2206.04663/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 252 references that are in the map.
- First posted on arXiv 9 Jun 2022.

## Builds on (most cited first, 30 of 33)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [On First-Order Meta-Learning Algorithms](https://mlsanity.com/paper/1803.02999/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [Oops I Took A Gradient: Scalable Sampling for Discrete Distributions](https://mlsanity.com/paper/2102.04509/) (2021)
- [Provably efficient machine learning for quantum many-body problems](https://mlsanity.com/paper/2106.12627/) (2021)
- [Quantum circuit architecture search for variational quantum algorithms](https://mlsanity.com/paper/2010.10217/) (2020)
- [Fast Convergence of Natural Gradient Descent for Overparameterized Neural Networks](https://mlsanity.com/paper/1905.10961/) (2019)
- [TensorFlow Quantum: A Software Framework for Quantum Machine Learning](https://mlsanity.com/paper/2003.02989/) (2020)
- [A quantum-inspired classical algorithm for recommendation systems](https://mlsanity.com/paper/1807.04271/) (2018)
- [Exploiting symmetry in variational quantum machine learning](https://mlsanity.com/paper/2205.06217/) (2022)
- [Learning to learn with quantum neural networks via classical neural networks](https://mlsanity.com/paper/1907.05415/) (2019)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [The Information Geometry of Mirror Descent](https://mlsanity.com/paper/1310.7780/) (2013)
- [Equivalence of quantum barren plateaus to cost concentration and narrow gorges](https://mlsanity.com/paper/2104.05868/) (2021)
- [Efficiently learning Ising models on arbitrary graphs](https://mlsanity.com/paper/1411.6156/) (2014)
- [Learning Graphical Models Using Multiplicative Weights](https://mlsanity.com/paper/1706.06274/) (2017)
- [Gram-Gauss-Newton Method: Learning Overparameterized Neural Networks for Regression Problems](https://mlsanity.com/paper/1905.11675/) (2019)
- [Sample-efficient learning of quantum many-body systems](https://mlsanity.com/paper/2004.07266/) (2020)
- [Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions](https://mlsanity.com/paper/1811.00414/) (2018)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [Quantum Hamiltonian-Based Models and the Variational Quantum Thermalizer Algorithm](https://mlsanity.com/paper/1910.02071/) (2019)

## Built on by (most cited first, 3 of 3)

- [Practical Black Box Hamiltonian Learning](https://mlsanity.com/paper/2206.15464/) (2022)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)

## Abstract (arXiv)

The dual tasks of quantum Hamiltonian learning and quantum Gibbs sampling are relevant to many important problems in physics and chemistry. In the low temperature regime, algorithms for these tasks often suffer from intractabilities, for example from poor sample- or time-complexity. With the aim of addressing such intractabilities, we introduce a generalization of quantum natural gradient descent to parameterized mixed states, as well as provide a robust first-order approximating algorithm, Quantum-Probabilistic Mirror Descent. We prove data sample efficiency for the dual tasks using tools from information geometry and quantum metrology, thus generalizing the seminal result of classical Fisher efficiency to a variational quantum algorithm for the first time. Our approaches extend previously sample-efficient techniques to allow for flexibility in model choice, including to spectrally-decomposed models like Quantum Hamiltonian-Based Models, which may circumvent intractable time complexities. Our first-order algorithm is derived using a novel quantum generalization of the classical mirror descent duality. Both results require a special choice of metric, namely, the Bogoliubov-Kubo-Mori metric. To test our proposed algorithms numerically, we compare their performance to existing baselines on the task of quantum Gibbs sampling for the transverse field Ising model. Finally, we propose an initialization strategy leveraging geometric locality for the modelling of sequences of states such as those arising from quantum-stochastic processes. We demonstrate its effectiveness empirically for both real and imaginary time evolution while defining a broader class of potential applications.