Automatic Design of Quantum Feature Maps: Auto-generated Quantum-Inspired Kernels by using Multi-Objective Genetic Algorithms (AQUIK)

This is the official code of the paper published on August 19, 2021: S. Altares-López, A. Ribeiro, J.J. García-Ripoll, Automatic design of quantum feature maps, Quantum Science and Technology, vol. 6, no. 4, 2021 [1]. Registered Software 2023 - CSIC.

• DOI: https://doi.org/10.1088/2058-9565/ac1ab1

Abstract

We propose a new technique for the automatic generation of optimal ad-hoc ansätze for classification by using quantum support vector machine -kernel methods-. This efficient method is based on NSGA-II multiobjective genetic algorithms which allow both maximize the accuracy and minimize the ansatz size. It is demonstrated the validity of the technique by a practical example with a non-linear dataset, interpreting the resulting circuit and its outputs. We also show other application fields of the technique that reinforce the validity of the method, and a comparison with classical classifiers to understand the advantages of using quantum machine learning.

• This work has also been presented in the 2nd European Quantum Technologies Conference (Virtual Conference, Dublin, Ireland) in Poster format: https://az659834.vo.msecnd.net/eventsairwesteuprod/production-abbey-public/9641064fd4a342ab9c4d81cf221ac4dd

0. Goals of the Technique

• Maximize the accuracy on unseen data.
• Minimize the quantum classifier size, in terms of quantum gates, layers and number of qubits, thus, reducing the expressivity of the quantum circuits.
• Optimization of the circuit structure, gate types and its parameters θ.
• Generate an automatic and optimized system for data encoding of classical information into the quantum feature maps.
• Take into account the use case, generating ad-hoc classifiers for each data set.
• Find robustness classifiers with a high generalization power.
• Search of quantum-inspired solutions that can be implemented on classical computers.
• Provide interpretability of the predicted results.
• Capacity to include many variables in few qubits.
• Quantum advantage taking into account the simplicity of the quantum classifiers and their results compared to classical models.

1. Genetic Quantum Feature Maps

In this paper we propose a novel technique for quantum machine learning (QML) which allows for tabular datasets the automatic generation of quantum-inspired kernels for classification by using Quantum Support Vector Machine (QSVM), based on Multi-Objective Genetic Algorithms (MO-GA).

The goal of the technique is to achieve the quantum circuit that provides the best accuracy on test data, as well as the smallest ansatz size. Since the objective of the fitness function is the test accuracy, we force the circuits-solution to be robust and to avoid overfitting effects, being quantum classifiers with a high generalization power.

Taking into account the ansatz size, our goal is to minimize it as much as possible in order to have solutions that avoid expressivity problems. This is possible because we code identity gates, which allows the possibility of eliminating gates, layers and even reducing the number of qubits in the circuits.

In addition, we penalize the occurrence of CNOT (entangling gates), in order to achieve solutions with lower computational cost and quantum-inspired machine learning solutions, by using the following expression.

2. Multi-Objective Genetic Algorithm (MO-GA)

Evolutionary algorithms allow the exploration of large solution spaces in order to find the most optimal or closest solution since the methodology is meta-heuristic [2]. Since we have two objectives, we use NSGA-II algorithm and Pareto Front, in order to find and save the non-dominated solutions [3]. Those solutions that, improve one of the two objectives without getting worse results in the other effort metric are saved. In order to provide a higher degree of elitism to the technique, we use the μ+λ algorithm, which face parents against their offspring, keeping the best individuals for the following generations.

2.1 Evolutionary Quantum Classifiers Searching Algorithm

• Step 1: Firstly, quantum gates H, CNOT and parameterized in the X,Y,Z axes with four associated angles are pre-coded to binary code. Each gate is coded into five bits, being the first three bits for gate selection and the last two bits for angle if necessary. During the process, binary strings (individuals) are created, which will encode for a specific ansatz.

• Step 2: A starting population is created -Initial population.

• Step 3: These individuals are evaluated in the evaluation function or fitness. The output of this function will determine whether the individual is accurate for the given problem or not. In the proposed technique, the binary strings are converted into quantum circuits which will act as feature maps into QSVM. Firstly, the classifier is fitted with training set and then we make predictions over test set (data not previously seen by the model) -seeking generalization power-, getting the objective of the fitness function. At the same time, we calculate the number of gates penalizing doubly the entangling operators due to a higher computational cost. We calculate a metric -Weight Control- in order to find a balance between both metrics, the accuracy and the reduction of number of gates. It is important since a high weight on the reducing circuit size objective can lead less accuracy because of information loss.

• Step 4: We select the best individuals. We apply genetic operators of crossover (Two-points) and mutation (Flipbit), generating new individuals (offspring) for the next generation. These operators are applied with a probability Pm and Pc respectively. The mutation operator allows us to reach other points in the search space since it allows us to avoid local minima, making the search for the best solution more efficient.

• Step 5: The process is repeated until convergence or when stop conditions are achieved. The best individuals are kept in the Pareto front.

3. Interpretability of Results

Once the evolution is finished, we obtain the optimized quantum circuit with the best test accuracy - thus ensuring that there is no overfitting on the train data, being classifiers with a high generalization power and robustness - and with the lowest number of quantum gates and qubits for non-linear datasets [6].

3.1 Quantum AI-Explainability Approach

The resulting quantum circuit can be decomposed by qubits because there are no entangling gates among them. Each qubit constitutes its own kernel. We note that when we evaluate each qubit separately does not provide a high accuracy for this non-linear dataset [6], as can be seen in the decision boundaries (b-d). However, the combination of all kernels produces a prediction of 1.0 in test data (a) with the next expression:

3.2 Generalization Power

We also create 500 random data of the same distribution [6], in order to understand the classifier's generalization power to previously unseen data. After the application of the circuit-solution we can conclude that this technique produces robust quantum-inspired classifiers for tabular data, obtaining an accuracy of 94.6%.

4. Other Insights

By using this technique, we are able to include many variables in few qubits, because the genetic algorithm takes into account the possibility of combining more than one variables per qubits as in the Parkinson's example [5]. In this use case 22 variables are included in only 8 qubits with no correlations, decreasing considerably the expressivity of the circuits-solution.

5. Files Description

• circuit.py: We create the quantum operators that will composed the quantum circuit.
• fitness.py: Evaluation fuction of the genetic algorithm (we fit 2 variables to return -the objectives)
• gsvm.py: Genetic algorithm function with the genetic operators. We call the fitness function.
• qsvm.py: We create a simulated quantum support vector machine by using sklearn.
• encoding.py: In this file we create the encoding of the quantum gates and the parameters θ.
• encoding2.py: This file is used to visualize the solution after the evolution.

5.1 Notebook - Use case Folder:

• Sample_Usecase.ipynb: Notebook used to launch the quantum feature maps' evolution, and save the best individuals which appear along the evolution in an excel file, so it can be decoded into a quantum circuit.
• sample_iot_data.csv: Dataset free available on Kaggle. Used as an example [4].
• sample_iot_result_n5.csv: Output file with the best individuals in the evolution. The structure of the file is: ID | Individual (quantum classifier) to be decoded | Weight control metric | Accuracy on test set

6. How to Cite

This code has been registered by the Spanish National Council of Research (CSIC). Please note this code is the official code for article Automatic design of quantum feature maps. Authors of scientific papers including results generated using this technique or these ideas, are encouraged to cite this paper as follows.

• Altares-López, S., Ribeiro, A., & García-Ripoll, J. J. (2021). Automatic design of quantum feature maps. Quantum Science and Technology, 6(4), 045015. https://doi.org/10.1088/2058-9565/ac1ab1

@article{altares2021automatic,
  title={Automatic design of quantum feature maps},
  author={Altares-L{\'o}pez, Sergio and Ribeiro, Angela and Garc{\'\i}a-Ripoll, Juan Jos{\'e}},
  journal={Quantum Science and Technology},
  volume={6},
  number={4},
  pages={045015},
  year={2021},
  doi = {10.1088/2058-9565/ac1ab1},
  url = {https://doi.org/10.1088%2F2058-9565%2Fac1ab1},
  publisher={IOP Publishing}
}

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Date: 2022-10-04. This license has been provided through the use of blockchain.

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7. References

• [1] S. Altares-López, A. Ribeiro, J.J. García-Ripoll, Automatic design of quantum feature maps, Quantum Science and Technology, vol. 6, no. 4, 2021. https://doi.org/10.1088/2058-9565/ac1ab1
• [2] De Rainville, F. M., Fortin, F. A., Gardner, M. A., Parizeau, M., & Gagné, C. (2012, July). Deap: A python framework for evolutionary algorithms. In Proceedings of the 14th annual conference companion on Genetic and evolutionary computation (pp. 85-92). https://doi.org/10.1145/2330784.2330799
• [3] Deb, K., Pratap, A., Agarwal, S., & Meyarivan, T. A. M. T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation, 6(2), 182-197. https://doi.org/10.1109/4235.996017

7.1 Datasets

• [4] Patel H 2020 Intelligent irrigation system (by using temperature and moisture data) - Kaggle Dataset. https://www.kaggle.com/harshilpatel355/autoirrigationdata
• [5] Little, M., Mcsharry, P., Roberts, S., Costello, D., & Moroz, I. (2007). Exploiting nonlinear recurrence and fractal scaling properties for voice disorder detection. Nature Precedings, 1-1.
• [6] Thirion B, Varoquaux G, Gramfort A, Michel V, Grisel O, Louppe G and Nothman J scikit-datasets (generate samples of synthetic data sets). URL https://github.com/scikit-learn/scikit-learn