# Explaining Quantum Circuits with Shapley Values: Towards Explainable Quantum Machine Learning

arXiv: https://arxiv.org/abs/2301.09138 · HTML: https://mlsanity.com/paper/2301.09138/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 126 references that are in the map.
- First posted on arXiv 22 Jan 2023.

## Builds on (most cited first, 30 of 32)

- [A Unified Approach to Interpreting Model Predictions](https://mlsanity.com/paper/1705.07874/) (2017)
- [Axiomatic Attribution for Deep Networks](https://mlsanity.com/paper/1703.01365/) (2017)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [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)
- [Quantum Machine Learning](https://mlsanity.com/paper/1611.09347/) (2016)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Generalization in quantum machine learning from few training data](https://mlsanity.com/paper/2111.05292/) (2021)
- [Problems with Shapley-value-based explanations as feature importance measures](https://mlsanity.com/paper/2002.11097/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Connecting ansatz expressibility to gradient magnitudes and barren plateaus](https://mlsanity.com/paper/2101.02138/) (2021)
- [Visual Interpretability for Deep Learning: a Survey](https://mlsanity.com/paper/1802.00614/) (2018)
- [Explainable AI (XAI): A Systematic Meta-Survey of Current Challenges and Future Opportunities](https://mlsanity.com/paper/2111.06420/) (2021)
- [The Shapley Value in Machine Learning](https://mlsanity.com/paper/2202.05594/) (2022)
- [On the Tractability of SHAP Explanations](https://mlsanity.com/paper/2009.08634/) (2020)
- [Quantum circuit architecture search for variational quantum algorithms](https://mlsanity.com/paper/2010.10217/) (2020)
- [Quantum-enhanced machine learning](https://mlsanity.com/paper/1610.08251/) (2016)
- [Generating Hierarchical Explanations on Text Classification via Feature Interaction Detection](https://mlsanity.com/paper/2004.02015/) (2020)
- [Interpretability and Explainability: A Machine Learning Zoo Mini-tour](https://mlsanity.com/paper/2012.01805/) (2020)
- [A Survey on Quantum Reinforcement Learning](https://mlsanity.com/paper/2211.03464/) (2022)
- [Capacity and quantum geometry of parametrized quantum circuits](https://mlsanity.com/paper/2102.01659/) (2021)
- [The complexity of quantum support vector machines](https://mlsanity.com/paper/2203.00031/) (2022)
- [Noise-Adaptive Compiler Mappings for Noisy Intermediate-Scale Quantum Computers](https://mlsanity.com/paper/1901.11054/) (2019)
- [Learning quantum data with the quantum Earth Mover's distance](https://mlsanity.com/paper/2101.03037/) (2021)
- [Quantum Circuit Evolution on NISQ Devices](https://mlsanity.com/paper/2012.13453/) (2020)
- [Representation of binary classification trees with binary features by quantum circuits](https://mlsanity.com/paper/2108.13207/) (2021)
- [Analysing Neural Network Topologies: a Game Theoretic Approach](https://mlsanity.com/paper/1904.08166/) (2019)

## Built on by (most cited first, 17 of 17)

- [Interpretable Machine Learning in Physics: A Review](https://mlsanity.com/paper/2503.23616/) (2025)
- [On the Interpretability of Quantum Neural Networks](https://mlsanity.com/paper/2308.11098/) (2023)
- [Quantum Gradient Class Activation Map for Model Interpretability](https://mlsanity.com/paper/2408.05899/) (2024)
- [Opportunities and limitations of explaining quantum machine learning](https://mlsanity.com/paper/2412.14753/) (2024)
- [Towards interpretable quantum machine learning via single-photon quantum walks](https://mlsanity.com/paper/2301.13669/) (2023)
- [Feature Importance and Explainability in Quantum Machine Learning](https://mlsanity.com/paper/2405.08917/) (2024)
- [Quantum Machine Unlearning: Foundations, Mechanisms, and Taxonomy](https://mlsanity.com/paper/2511.00406/) (2025)
- [Quantum Artificial Intelligence: A Brief Survey](https://mlsanity.com/paper/2408.10726/) (2024)
- [Quantum Adjoint Convolutional Layers for Effective Data Representation](https://mlsanity.com/paper/2404.17378/) (2024)
- [A Shapley Value Estimation Speedup for Efficient Explainable Quantum AI](https://mlsanity.com/paper/2412.14639/) (2024)
- [QuXAI: Explainers for Hybrid Quantum Machine Learning Models](https://mlsanity.com/paper/2505.10167/) (2025)
- [Resting-state fMRI Analysis using Quantum Time-series Transformer](https://mlsanity.com/paper/2509.00711/) (2025)
- [Quantum Artificial Intelligence for Mission-Critical Systems: Foundations, Architectural Elements, and Future Directions](https://mlsanity.com/paper/2511.09884/) (2025)
- [Quantum Circuit-Based Learning Models: Bridging Quantum Computing and Machine Learning](https://mlsanity.com/paper/2602.00048/) (2026)
- [Detecting underdetermination in parameterized quantum circuits](https://mlsanity.com/paper/2504.03315/) (2025)
- [Component Based Quantum Machine Learning Explainability](https://mlsanity.com/paper/2506.12378/) (2025)
- [QILP-0: Constructing Observational Declarative Twins of Quantum Circuits](https://mlsanity.com/paper/2609.01049/) (2026)

## Abstract (arXiv)

Methods of artificial intelligence (AI) and especially machine learning (ML) have been growing ever more complex, and at the same time have more and more impact on people's lives. This leads to explainable AI (XAI) manifesting itself as an important research field that helps humans to better comprehend ML systems. In parallel, quantum machine learning (QML) is emerging with the ongoing improvement of quantum computing hardware combined with its increasing availability via cloud services. QML enables quantum-enhanced ML in which quantum mechanics is exploited to facilitate ML tasks, typically in the form of quantum-classical hybrid algorithms that combine quantum and classical resources. Quantum gates constitute the building blocks of gate-based quantum hardware and form circuits that can be used for quantum computations. For QML applications, quantum circuits are typically parameterized and their parameters are optimized classically such that a suitably defined objective function is minimized. Inspired by XAI, we raise the question of the explainability of such circuits by quantifying the importance of (groups of) gates for specific goals. To this end, we apply the well-established concept of Shapley values. The resulting attributions can be interpreted as explanations for why a specific circuit works well for a given task, improving the understanding of how to construct parameterized (or variational) quantum circuits, and fostering their human interpretability in general. An experimental evaluation on simulators and two superconducting quantum hardware devices demonstrates the benefits of the proposed framework for classification, generative modeling, transpilation, and optimization. Furthermore, our results shed some light on the role of specific gates in popular QML approaches.