# Differentiable Learning of Quantum Circuit Born Machine

arXiv: https://arxiv.org/abs/1804.04168 · HTML: https://mlsanity.com/paper/1804.04168/

## Key facts

- Cited by 93 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Apr 2018.

## Built on by (most cited first, 30 of 93)

- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
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- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
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- [Recent Advances for Quantum Neural Networks in Generative Learning](https://mlsanity.com/paper/2206.03066/) (2022)
- [Subtractive Mixture Models via Squaring: Representation and Learning](https://mlsanity.com/paper/2310.00724/) (2023)
- [Trainability barriers and opportunities in quantum generative modeling](https://mlsanity.com/paper/2305.02881/) (2023)
- [Recent advances for quantum classifiers](https://mlsanity.com/paper/2108.13421/) (2021)
- [Shadows of quantum machine learning](https://mlsanity.com/paper/2306.00061/) (2023)
- [Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus](https://mlsanity.com/paper/2102.01828/) (2021)
- [QTN-VQC: An End-to-End Learning framework for Quantum Neural Networks](https://mlsanity.com/paper/2110.03861/) (2021)
- [Universal Adversarial Examples and Perturbations for Quantum Classifiers](https://mlsanity.com/paper/2102.07788/) (2021)
- [QuanTA: Efficient High-Rank Fine-Tuning of LLMs with Quantum-Informed Tensor Adaptation](https://mlsanity.com/paper/2406.00132/) (2024)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [High Dimensional Quantum Machine Learning With Small Quantum Computers](https://mlsanity.com/paper/2203.13739/) (2022)
- [A Framework for Demonstrating Practical Quantum Advantage: Racing Quantum against Classical Generative Models](https://mlsanity.com/paper/2303.15626/) (2023)
- [Non-IID Quantum Federated Learning with One-shot Communication Complexity](https://mlsanity.com/paper/2209.00768/) (2022)
- [Learning quantum data with the quantum Earth Mover's distance](https://mlsanity.com/paper/2101.03037/) (2021)
- [Generalization in Quantum Machine Learning: a Quantum Information Perspective](https://mlsanity.com/paper/2102.08991/) (2021)
- [Training-efficient density quantum machine learning](https://mlsanity.com/paper/2405.20237/) (2024)
- [Matching Point Sets with Quantum Circuit Learning](https://mlsanity.com/paper/2102.06697/) (2021)
- [Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series](https://mlsanity.com/paper/2108.03190/) (2021)
- [Splitting and Parallelizing of Quantum Convolutional Neural Networks for Learning Translationally Symmetric Data](https://mlsanity.com/paper/2306.07331/) (2023)
- [Quantum Generative Diffusion Model: A Fully Quantum-Mechanical Model for Generating Quantum State Ensemble](https://mlsanity.com/paper/2401.07039/) (2024)
- [Generalization Metrics for Practical Quantum Advantage in Generative Models](https://mlsanity.com/paper/2201.08770/) (2022)
- [Trade-off between Gradient Measurement Efficiency and Expressivity in Deep Quantum Neural Networks](https://mlsanity.com/paper/2406.18316/) (2024)
- [Hyperparameter Importance of Quantum Neural Networks Across Small Datasets](https://mlsanity.com/paper/2206.09992/) (2022)
- [Variational measurement-based quantum computation for generative modeling](https://mlsanity.com/paper/2310.13524/) (2023)
- [Quantum-Informed Machine Learning for Predicting Spatiotemporal Chaos with Practical Quantum Advantage](https://mlsanity.com/paper/2507.19861/) (2025)
- [Power of Quantum Generative Learning](https://mlsanity.com/paper/2205.04730/) (2022)

## Abstract (arXiv)

Quantum circuit Born machines are generative models which represent the probability distribution of classical dataset as quantum pure states. Computational complexity considerations of the quantum sampling problem suggest that the quantum circuits exhibit stronger expressibility compared to classical neural networks. One can efficiently draw samples from the quantum circuits via projective measurements on qubits. However, similar to the leading implicit generative models in deep learning, such as the generative adversarial networks, the quantum circuits cannot provide the likelihood of the generated samples, which poses a challenge to the training. We devise an efficient gradient-based learning algorithm for the quantum circuit Born machine by minimizing the kerneled maximum mean discrepancy loss. We simulated generative modeling of the Bars-and-Stripes dataset and Gaussian mixture distributions using deep quantum circuits. Our experiments show the importance of circuit depth and gradient-based optimization algorithm. The proposed learning algorithm is runnable on near-term quantum device and can exhibit quantum advantages for generative modeling.