Curve fitting on a quantum annealer for an advanced navigation method

TL;DR

The paper investigates applying quantum annealing to curve fitting by formulating it as a QUBO and comparing QA results with classical least-squares and tabu-search solutions, while also applying a QUBO-based approximation within dynamic programming to estimate a vessel speed-profile value function using fitted value iteration. It shows that, for small QUBOs described by a finite basis f(x) = ∑_j c_j φ_j(x) with c_j encoded in fixed-point binary, quantum annealing can approach classical accuracy, but performance degrades for larger or highly connected bases such as Chebyshev polynomials due to hardware embedding and landscape challenges. In the speed-profile context, the fitted-value-iteration approach enables smoother approximations of the value function, yet current hardware limitations prevent QA from consistently outperforming classical or tabu-based methods as problem size grows. Overall, the study demonstrates promise for quantum annealing in one-dimensional, sparsely connected curve-fitting tasks and outlines concrete directions—such as basis choice, QUBO sparsification, and quantum-inspired solvers—for scalable quantum-accelerated optimization in dynamic programming.

Abstract

We explore the applicability of quantum annealing to the approximation task of curve fitting. To this end, we consider a function that shall approximate a given set of data points and is written as a finite linear combination of standardized functions, e.g., orthogonal polynomials. Consequently, the decision variables subject to optimization are the coefficients of that expansion. Although this task can be accomplished classically, it can also be formulated as a quadratic unconstrained binary optimization problem, which is suited to be solved with quantum annealing. Given the size of the problem stays below a certain threshold, we find that quantum annealing yields comparable results to the classical solution. Regarding a real-world use case, we discuss the problem to find an optimized speed profile for a vessel using the framework of dynamic programming and outline how the aforementioned approximation task can be put into play. Similar to the curve fitting task, our findings indicate that quantum annealing is currently only feasible if the routing problem is modeled sufficiently small and sparse.

Figures (6)

• Figure 1: RMSE of the tabu search (red squares, $f_{\textup{tb}}^{\ast}$) and quantum annealing (green diamonds, $f_{\textup{qa}}^{\ast}$) solutions as a function of $d$. The RMSE of the exact, $d$-independent classical solution $f_{\textup{cl}}^{\ast}$ is shown as a gray dashed line. Lines between data points are shown to guide the eye. The underlying sample data set is the cubic one (\ref{['eq:sample_data_02c']}), and triangular functions with $m=4$ are used for the approximation.
• Figure 2: RMSE of the tabu search (red squares, $f_{\textup{tb}}^{\ast}$), quantum annealing (green diamonds, $f_{\textup{qa}}^{\ast}$) and classical (gray circles, $f_{\textup{cl}}^{\ast}$) solutions as a function of $m$. The underlying sample data are the trigonometric (left diagram) and quadratic (right diagram) ones, respectively.
• Figure 3: RMSE of the tabu search (red squares, $f_{\textup{tb}}^{\ast}$) and quantum annealing (green diamonds, $f_{\textup{qa}}^{\ast}$) solutions as a function of $d$. The RMSE of the exact, $d$-independent classical solution $f_{\textup{cl}}^{\ast}$ is shown as a gray dashed line. Lines between data points are shown to guide the eye. The underlying sample data set is the cubic one (\ref{['eq:sample_data_02c']}), and Chebyshev polynomials with $m=4$ are used for the approximation.
• Figure 4: Heat map plots of exemplary QUBO matrices $Q$ of piecewise-linear approximation with triangular functions (left) and Chebyshev polynomial approximation (right). Here, we display the upper-triangularized versions of these matrices since every QUBO problem can be written in a way such that $Q$ is an upper-triangular matrix.
• Figure 5: Visualization of the value function approximation using different solution strategies (upper row) and their corresponding mean squared errors (MSE) (lower row). The left column shows the results for time slice $t=3$, while the right column presents the same analysis for the first time slice $t=0$. The upper panels depict the approximated value functions, whereas the lower panels show the associated approximation errors. Here and in the following figures, we use the MSE in contrast to the RMSE as defined in \ref{['eq:sample_data_04']}.