# Variational inference with a quantum computer

arXiv: https://arxiv.org/abs/2103.06720 · HTML: https://mlsanity.com/paper/2103.06720/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 59 of its 141 references that are in the map.
- First posted on arXiv 11 Mar 2021.

## Builds on (most cited first, 30 of 59)

- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Towards Principled Methods for Training Generative Adversarial Networks](https://mlsanity.com/paper/1701.04862/) (2017)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Are GANs Created Equal? A Large-Scale Study](https://mlsanity.com/paper/1711.10337/) (2017)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Adversarial Variational Bayes: Unifying Variational Autoencoders and Generative Adversarial Networks](https://mlsanity.com/paper/1701.04722/) (2017)
- [Information-theoretic bounds on quantum advantage in machine learning](https://mlsanity.com/paper/2101.02464/) (2021)
- [Quantum machine learning: a classical perspective](https://mlsanity.com/paper/1707.08561/) (2017)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Robust data encodings for quantum classifiers](https://mlsanity.com/paper/2003.01695/) (2020)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer](https://mlsanity.com/paper/2102.12846/) (2021)
- [On the Quantum versus Classical Learnability of Discrete Distributions](https://mlsanity.com/paper/2007.14451/) (2020)
- [Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hidden Markov models to quantum machine learning](https://mlsanity.com/paper/1907.03741/) (2019)
- [Foundations for Near-Term Quantum Natural Language Processing](https://mlsanity.com/paper/2012.03755/) (2020)

## Built on by (most cited first, 18 of 18)

- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [Quantum Deep Hedging](https://mlsanity.com/paper/2303.16585/) (2023)
- [Quantum Machine Learning on Near-Term Quantum Devices: Current State of Supervised and Unsupervised Techniques for Real-World Applications](https://mlsanity.com/paper/2307.00908/) (2023)
- [Quantum HyperNetworks: Training Binary Neural Networks in Quantum Superposition](https://mlsanity.com/paper/2301.08292/) (2023)
- [Conditional Born machine for Monte Carlo event generation](https://mlsanity.com/paper/2205.07674/) (2022)
- [Building Continuous Quantum-Classical Bayesian Neural Networks for a Classical Clinical Dataset](https://mlsanity.com/paper/2406.06307/) (2024)
- [QKSA: Quantum Knowledge Seeking Agent](https://mlsanity.com/paper/2107.01429/) (2021)
- [F-Divergences and Cost Function Locality in Generative Modelling with Quantum Circuits](https://mlsanity.com/paper/2110.04253/) (2021)
- [Quantum-Aided Meta-Learning for Bayesian Binary Neural Networks via Born Machines](https://mlsanity.com/paper/2203.17089/) (2022)
- [Bayesian Learning of Parameterised Quantum Circuits](https://mlsanity.com/paper/2206.07559/) (2022)
- [Tensor tree learns hidden relational structures in data to construct generative models](https://mlsanity.com/paper/2408.10669/) (2024)
- [Bayesian Quantum Orthogonal Neural Networks for Anomaly Detection](https://mlsanity.com/paper/2504.18103/) (2025)
- [Quantum Advantage in Variational Bayes Inference](https://mlsanity.com/paper/2207.03104/) (2022)
- [Sample-efficient Quantum Born Machine through Coding Rate Reduction](https://mlsanity.com/paper/2211.10418/) (2022)
- [Machine Learning for Quantum Computing Specialists](https://mlsanity.com/paper/2404.18555/) (2024)
- [Variational Inference for Quantum HyperNetworks](https://mlsanity.com/paper/2506.05888/) (2025)
- [Towards unsupervised representation learning for quantum data: quantum models with inference and generation](https://mlsanity.com/paper/2609.00372/) (2026)
- [Simulation-Based Quantum System Inference with Neural Posterior Estimation](https://mlsanity.com/paper/2609.34995/) (2026)

## Abstract (arXiv)

Inference is the task of drawing conclusions about unobserved variables given observations of related variables. Applications range from identifying diseases from symptoms to classifying economic regimes from price movements. Unfortunately, performing exact inference is intractable in general. One alternative is variational inference, where a candidate probability distribution is optimized to approximate the posterior distribution over unobserved variables. For good approximations, a flexible and highly expressive candidate distribution is desirable. In this work, we use quantum Born machines as variational distributions over discrete variables. We apply the framework of operator variational inference to achieve this goal. In particular, we adopt two specific realizations: one with an adversarial objective and one based on the kernelized Stein discrepancy. We demonstrate the approach numerically using examples of Bayesian networks, and implement an experiment on an IBM quantum computer. Our techniques enable efficient variational inference with distributions beyond those that are efficiently representable on a classical computer.