# Verifying Fairness in Quantum Machine Learning

arXiv: https://arxiv.org/abs/2207.11173 · HTML: https://mlsanity.com/paper/2207.11173/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 53 references that are in the map.
- First posted on arXiv 22 Jul 2022.

## Builds on (most cited first, 14 of 14)

- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations](https://mlsanity.com/paper/1905.07697/) (2019)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [On the Apparent Conflict Between Individual and Group Fairness](https://mlsanity.com/paper/1912.06883/) (2019)
- [Quantum Recommendation Systems](https://mlsanity.com/paper/1603.08675/) (2016)
- [Quantum noise protects quantum classifiers against adversaries](https://mlsanity.com/paper/2003.09416/) (2020)
- [Verifying Individual Fairness in Machine Learning Models](https://mlsanity.com/paper/2006.11737/) (2020)
- [Probabilistic Verification of Fairness Properties via Concentration](https://mlsanity.com/paper/1812.02573/) (2018)
- [Justicia: A Stochastic SAT Approach to Formally Verify Fairness](https://mlsanity.com/paper/2009.06516/) (2020)
- [Optimal Provable Robustness of Quantum Classification via Quantum Hypothesis Testing](https://mlsanity.com/paper/2009.10064/) (2020)
- [TensorNetwork: A Library for Physics and Machine Learning](https://mlsanity.com/paper/1905.01330/) (2019)
- [Robustness Verification of Quantum Classifiers](https://mlsanity.com/paper/2008.07230/) (2020)

## Built on by (most cited first, 7 of 7)

- [Quantum Pufferfish Privacy: A Flexible Privacy Framework for Quantum Systems](https://mlsanity.com/paper/2306.13054/) (2023)
- [Contraction of Private Quantum Channels and Private Quantum Hypothesis Testing](https://mlsanity.com/paper/2406.18651/) (2024)
- [JustQ: Automated Deployment of Fair and Accurate Quantum Neural Networks](https://mlsanity.com/paper/2403.11048/) (2024)
- [Quadratic Advantage with Quantum Randomized Smoothing Applied to Time-Series Analysis](https://mlsanity.com/paper/2407.18021/) (2024)
- [On the Generalization of Adversarially Trained Quantum Classifiers](https://mlsanity.com/paper/2504.17690/) (2025)
- [Efficient and Accurate Estimation of Lipschitz Constants for Hybrid Quantum-Classical Decision Models](https://mlsanity.com/paper/2503.07992/) (2025)
- [Detecting underdetermination in parameterized quantum circuits](https://mlsanity.com/paper/2504.03315/) (2025)

## Abstract (arXiv)

Due to the beyond-classical capability of quantum computing, quantum machine learning is applied independently or embedded in classical models for decision making, especially in the field of finance. Fairness and other ethical issues are often one of the main concerns in decision making. In this work, we define a formal framework for the fairness verification and analysis of quantum machine learning decision models, where we adopt one of the most popular notions of fairness in the literature based on the intuition -- any two similar individuals must be treated similarly and are thus unbiased. We show that quantum noise can improve fairness and develop an algorithm to check whether a (noisy) quantum machine learning model is fair. In particular, this algorithm can find bias kernels of quantum data (encoding individuals) during checking. These bias kernels generate infinitely many bias pairs for investigating the unfairness of the model. Our algorithm is designed based on a highly efficient data structure -- Tensor Networks -- and implemented on Google's TensorFlow Quantum. The utility and effectiveness of our algorithm are confirmed by the experimental results, including income prediction and credit scoring on real-world data, for a class of random (noisy) quantum decision models with 27 qubits (227-dimensional state space) tripling (218 times more than) that of the state-of-the-art algorithms for verifying quantum machine learning models.