# Sample-efficient Quantum Born Machine through Coding Rate Reduction

arXiv: https://arxiv.org/abs/2211.10418 · HTML: https://mlsanity.com/paper/2211.10418/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 62 references that are in the map.
- First posted on arXiv 14 Nov 2022.

## Builds on (most cited first, 27 of 27)

- [A Style-Based Generator Architecture for Generative Adversarial Networks](https://mlsanity.com/paper/1812.04948/) (2018)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Nets](https://mlsanity.com/paper/1606.03657/) (2016)
- [Autoencoding beyond pixels using a learned similarity metric](https://mlsanity.com/paper/1512.09300/) (2015)
- [SeqGAN: Sequence Generative Adversarial Nets with Policy Gradient](https://mlsanity.com/paper/1609.05473/) (2016)
- [A Learned Representation For Artistic Style](https://mlsanity.com/paper/1610.07629/) (2016)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning](https://mlsanity.com/paper/1705.07761/) (2017)
- [Learning Diverse and Discriminative Representations via the Principle of Maximal Coding Rate Reduction](https://mlsanity.com/paper/2006.08558/) (2020)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [Mode Regularized Generative Adversarial Networks](https://mlsanity.com/paper/1612.02136/) (2016)
- [A Survey on Generative Adversarial Networks: Variants, Applications, and Training](https://mlsanity.com/paper/2006.05132/) (2020)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [ReduNet: A White-box Deep Network from the Principle of Maximizing Rate Reduction](https://mlsanity.com/paper/2105.10446/) (2021)
- [The Born Supremacy: Quantum Advantage and Training of an Ising Born Machine](https://mlsanity.com/paper/1904.02214/) (2019)
- [GANS for Sequences of Discrete Elements with the Gumbel-softmax Distribution](https://mlsanity.com/paper/1611.04051/) (2016)
- [Information Perspective to Probabilistic Modeling: Boltzmann Machines versus Born Machines](https://mlsanity.com/paper/1712.04144/) (2017)
- [Learning and Inference on Generative Adversarial Quantum Circuits](https://mlsanity.com/paper/1808.03425/) (2018)
- [GANs May Have No Nash Equilibria](https://mlsanity.com/paper/2002.09124/) (2020)
- [Variational inference with a quantum computer](https://mlsanity.com/paper/2103.06720/) (2021)
- [Closed-Loop Data Transcription to an LDR via Minimaxing Rate Reduction](https://mlsanity.com/paper/2111.06636/) (2021)
- [Deep Networks from the Principle of Rate Reduction](https://mlsanity.com/paper/2010.14765/) (2020)
- [Many-body localized hidden generative models](https://mlsanity.com/paper/2207.02346/) (2022)
- [Text Analytics for Resilience-Enabled Extreme Events Reconnaissance](https://mlsanity.com/paper/2011.13087/) (2020)

## Abstract (arXiv)

The quantum circuit Born machine (QCBM) is a quantum physics inspired implicit generative model naturally suitable for learning binary images, with a potential advantage of modeling discrete distributions that are hard to simulate classically. As data samples are generated quantum-mechanically, QCBMs encompass a unique optimization landscape. However, pioneering works on QCBMs do not consider the practical scenario where only small batch sizes are allowed during training. QCBMs trained with a statistical two-sample test objective in the image space require large amounts of projective measurements to approximate the model distribution well, unpractical for large-scale quantum systems due to the exponential scaling of the probability space. QCBMs trained adversarially against a deep neural network discriminator are proof-of-concept models that face mode collapse. In this work we investigate practical learning of QCBMs. We use the information-theoretic textitMaximal Coding Rate Reduction (MCR2) metric as a second moment matching tool and study its effect on mode collapse in QCBMs. We compute the sampling based gradient of MCR2 with respect to quantum circuit parameters with or without an explicit feature mapping. We experimentally show that matching up to the second moment alone is not sufficient for training the quantum generator, but when combined with the class probability estimation loss, MCR2 is able to resist mode collapse. In addition, we show that adversarially trained neural network kernel for infinite moment matching is also effective against mode collapse. On the Bars and Stripes dataset, our proposed techniques alleviate mode collapse to a larger degree than previous QCBM training schemes, moving one step closer towards practicality and scalability.