Verifying Fairness in Quantum Machine Learning (2207.11173v1)

Abstract: 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 ($2^{27}$-dimensional state space) tripling ($2^{18}$ times more than) that of the state-of-the-art algorithms for verifying quantum machine learning models.

• The paper introduces a fairness framework for QML that uses trace distance between quantum states to assess individual fairness.
• It presents an algorithm that computes the Lipschitz constant using tensor networks to quantify output sensitivity and identify bias kernels.
• Experimental results on financial datasets demonstrate the framework's scalability, with up to 27 qubits, and its practical applicability in ensuring fairness.

Verifying Fairness in Quantum Machine Learning

The paper introduces a framework and algorithm for verifying the fairness of quantum machine learning (QML) models. Given the rapid advancements in quantum computing, aspects of fairness in quantum models hold significant importance. Here, the primary focus is on individual fairness, and the problem is approached using quantum information theory concepts.

Framework and Algorithm Development

Fairness Framework

The paper defines a fairness framework centered on individual fairness, where two similar individuals (quantum states) should receive similar treatment from a quantum model. The similarity between these quantum states is measured using the trace distance, a standard distance metric in quantum information theory.

A quantum decision model, denoted as $$\mathcal{A} = (\mathcal{E}, \{M_i\}_{i\in\mathcal{O}})$$, consists of a quantum operation $\mathcal{E}$ and a set of measurements $\{M_i\}$ with classical outcomes. The model's fairness is then determined by checking for a lack of bias pairs, defined as quantum state pairs with small trace distance but significant differences in their outcome distributions.

Lipschitz Constant

The fairness criteria are further translated into a problem of computing the Lipschitz constant of the quantum decision model, which quantifies the maximum change in output for a small change in input. The paper shows that determining fairness reduces to calculating this constant, a complex task given it involves optimizing over the space of quantum states.

Algorithm

An algorithm is developed to compute the Lipschitz constant, utilizing the efficiency of tensor networks to handle the scalability issues associated with large quantum state spaces. The algorithm involves:

1. Calculation of Lipschitz Constant: Involves computing eigenvalues of operators derived from the measurement matrices.
2. Bias Kernel Identification: When a model is not fair, the algorithm identifies bias kernels, pairs of quantum states that can be further used to analyze and rectify bias.

Experimental Evaluation

The algorithm is implemented using TensorFlow Quantum, and its effectiveness is demonstrated on quantum models trained on real-world financial datasets like the German Credit Data and Adult Income Dataset. The experimental results showcase:

• Scalability: The algorithm efficiently scales to verify models with up to 27 qubits.
• Impact of Quantum Noise: It was observed that certain types of quantum noise improve fairness by reducing the Lipschitz constant, aligning with the theoretical predictions.
• Practical Applicability: The framework can be applied to train QML models with embedded fairness guarantees, critical for deploying these models in sensitive areas like finance.

Conclusion

The paper establishes a principled framework for fairness verification in QML, introducing an efficient algorithm to compute the Lipschitz constant and assess fairness. Future directions include developing methods for embedding fairness guarantees during model training and further exploration of bias kernels in practical scenarios, reflecting a step towards responsible quantum AI model deployment.