Quantum Extremal Learning

TL;DR

Quantum Extremal Learning (QEL) introduces a fully quantum framework to jointly model an unknown input-output function and optimize its input to extremize the output. It leverages a trainable quantum feature map and a variational circuit to fit data, then differentiates the mapped input with respect to the model output to locate extrema, handling both continuous and discrete inputs through analytic differentiation or an extremizer circuit. The approach demonstrates data fitting, differential-equation solving, and discrete optimization (e.g., Max-Cut, molecular design) on simulated quantum hardware, showing promising accuracy and robustness under noiseless conditions. The work lays a foundation for applying quantum modeling and optimization within a single quantum processor and suggests avenues for scaling, error mitigation, and fault-tolerant enhancements.

Abstract

We propose a quantum algorithm for `extremal learning', which is the process of finding the input to a hidden function that extremizes the function output, without having direct access to the hidden function, given only partial input-output (training) data. The algorithm, called quantum extremal learning (QEL), consists of a parametric quantum circuit that is variationally trained to model data input-output relationships and where a trainable quantum feature map, that encodes the input data, is analytically differentiated in order to find the coordinate that extremizes the model. This enables the combination of established quantum machine learning modelling with established quantum optimization, on a single circuit/quantum computer. We have tested our algorithm on a range of classical datasets based on either discrete or continuous input variables, both of which are compatible with the algorithm. In case of discrete variables, we test our algorithm on synthetic problems formulated based on Max-Cut problem generators and also considering higher order correlations in the input-output relationships. In case of the continuous variables, we test our algorithm on synthetic datasets in 1D and simple ordinary differential functions. We find that the algorithm is able to successfully find the extremal value of such problems, even when the training dataset is sparse or a small fraction of the input configuration space. We additionally show how the algorithm can be used for much more general cases of higher dimensionality, complex differential equations, and with full flexibility in the choice of both modeling and optimization ansatz. We envision that due to its general framework and simple construction, the QEL algorithm will be able to solve a wide variety of applications in different fields, opening up areas of further research.

Figures (14)

• Figure 1: The workflow of QELpatent algorithm, showing both the discrete and continuous case.
• Figure 2: One epoch of Step I is shown in the gray rectangle. The part of the circuit in the inner pink box is executed for each element of the training data ${(\bm{x}_i, y_i)}$ and the results are used by a classical optimizer to suggest $\bm{\theta}$ for the next epoch.
• Figure 3: Step II in the continuous case. The part of the circuit retained from Step I is shown in orange and the new elements are shown in blue. Notice that $\bm{\theta}$ is no longer updated; the optimization is performed on $\bm{x}$.
• Figure 4: First part of Step II in the discrete case. The part of the circuit retained from Step I is shown in orange and the new elements are shown in blue. Notice that $\bm{\theta}$ is no longer updated; the optimization is performed on $\bm{\mathscr{X}}$.
• Figure 5: The final sampling in Step II of the discrete case. Due to the previous steps of the algorithm, the bitstring $\widetilde{\mathcal{F}}(\ket{{ \bm{\mathscr{X}}}})$ we measure has an enhanced probability of being mapped to $\bm{x}_\mathrm{opt}$.