Benchmarking Quantum Surrogate Models on Scarce and Noisy Data

TL;DR

The paper tackles the challenge of surrogate modelling with scarce and noisy data by evaluating quantum neural networks (QNNs) as surrogate models and comparing them to compact classical ANNs. It presents a practical QNN architecture that uses angle encoding, data reuploading, and carefully chosen ansätze, tested on standard high-dimensional benchmark functions and a real-world dataset. The authors demonstrate that QNNs can achieve higher predictive accuracy than similarly parameterized ANNs in noisy, data-scarce settings and provide empirical analysis for NISQ hardware performance, including gate-fidelity requirements to replicate simulation results. The work suggests a realistic near-term path to quantum advantage in surrogate modelling, contingent on hardware improvements and scalable circuit design.

Abstract

Surrogate models are ubiquitously used in industry and academia to efficiently approximate given black box functions. As state-of-the-art methods from classical machine learning frequently struggle to solve this problem accurately for the often scarce and noisy data sets in practical applications, investigating novel approaches is of great interest. Motivated by recent theoretical results indicating that quantum neural networks (QNNs) have the potential to outperform their classical analogs in the presence of scarce and noisy data, we benchmark their qualitative performance for this scenario empirically. Our contribution displays the first application-centered approach of using QNNs as surrogate models on higher dimensional, real world data. When compared to a classical artificial neural network with a similar number of parameters, our QNN demonstrates significantly better results for noisy and scarce data, and thus motivates future work to explore this potential quantum advantage in surrogate modelling. Finally, we demonstrate the performance of current NISQ hardware experimentally and estimate the gate fidelities necessary to replicate our simulation results.

Figures (5)

• Figure 1: The general QNN architecture used in this paper, exemplarily showing two layers, each comprised of a data encoding and a parameterized layer. It combines data reuploading by inserting a feature map ("FM") in each layer with parallel encoding by using two qubits per input data point dimension. For the parameterized part of the circuit, two different circuits from established literature have been combined: Circuit 11 is alternating with circuit 9 to create the required minimum number of parameters https://doi.org/10.1002/qute.201900070. CNOT, Hadamard and CZ gates create superposition and entanglement, while trainable parameters $\theta_i$ allow the approximation of the surrogate model. After repeated layers, the standard measurement (denoted with "MSMT") is applied to all qubits.
• Figure 2: Qualitative performance of the quantum surrogate for the Schwefel function with two-dimensional input.
• Figure 3: Delta_R2 score obtained by subtracting the classical ANN R2 score from the QNN R2 score for different noise levels ($x$-axis) and sample sizes ($y$-axis) for the Griewank function with two-dimensional input data. Positive values indicate a performance advantage of the QNN (as can be seen for higher noise levels and smaller simple sizes), while negative values represent a disadvantage of the QNN.
• Figure 4: Surface plots of the Griewank function when introducing noise and sample scarcity. The input to the QNN and classical ANN was modified by multiplication of standard-normal noise with a factor of 0.5 on the 400 individual input data points (which corresponds to a sample size of 20).
• Figure 5: Original data points (sample size of 20) for a Griewank function with one-dimensional input data and the quantum surrogate function that has been obtained by running 100 iterations of 6 layers of our ansatz described in figure \ref{['fig_vqc_example']} on the ibmq_belem QPU.