Quantum Optimization for NMPC

• Quantum optimization for NMPC is the integration of quantum computing methods into NMPC to recast and solve nonlinear control problems via approaches like QUBO encoding.
• It employs quantum annealing and hybrid SQP techniques to accelerate bottlenecks in solving large-scale, constrained optimization tasks by efficiently handling nonlinear programming and quadratic programming subproblems.
• Emerging quantum algorithms promise favorable scaling, robust error bounds, and constant runtime performance, potentially revolutionizing real-time control in industrial applications.

Quantum optimization for Nonlinear Model Predictive Control (NMPC) refers to the integration of quantum computing paradigms—principally quantum annealing and gate-based quantum subroutines—into the online solution of constrained, finite-horizon nonlinear optimal control problems. NMPC is widely used in industrial settings but incurs prohibitive computational cost when problem size or nonconvexity grows, especially for applications demanding real-time operation. Quantum approaches aim to accelerate subproblems such as nonlinear programming (NLP) and large-scale quadratic programming (QP), either by recasting NMPC as Quadratic Unconstrained Binary Optimization (QUBO) for quantum annealers or by hybridizing classical sequential quadratic programming (SQP) with quantum routines for critical linear algebraic steps. Quantum algorithms promise run-times scaling favorably in the number of decision variables, with error and stability properties characterized in recent literature (Dehaghani et al., 20 Oct 2025, Novara et al., 2024).

1. NMPC Formulation and Quantum Suitability

NMPC solves, at each time step, an optimal control problem for a discrete-time nonlinear system: $x_{k+1} = f(x_k, u_k),\quad k = 0, \ldots, N-1$ where $x_k \in \mathbb{R}^n$ and $u_k \in \mathbb{R}^m$. The goal is to minimize the cost

$J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$

subject to dynamic constraints, algebraic path constraints $c(x_k, u_k) \leq 0$, and terminal constraints $c_N(x_N) \leq 0$, given initial state $x_0$ (Dehaghani et al., 20 Oct 2025, Novara et al., 2024). In practice, these problems are formulated as nonlinear programs with stacked variables $$z = [z_0^\top, \ldots, z_{N-1}^\top, z_N^\top]^\top$$, with $z_k = [x_k^\top, u_k^\top]^\top$.

Quantum optimization methods seek to address the most expensive steps—solving large, often nonconvex, optimization problems—via quantum computational primitives. Quantum annealing maps (after discretization and encoding) continuous control variables and objective functions into QUBO, while quantum-assisted classical optimization uses quantum subroutines to accelerate the linear algebraic bottlenecks in iterative algorithms such as SQP.

2. Quantum Annealing Approaches to NMPC

The direct quantum optimization approach begins with encoding the NMPC decision variables into binary representations suitable for quantum annealers. A finite-dimensional control sequence parametrization $u = (u_0, \ldots, u_{N-1}) = \Gamma c$, with $x_k \in \mathbb{R}^n$0, is introduced (Novara et al., 2024). Multi-step predictions are expanded in a truncated Maclaurin series yielding the cost function as a polynomial in $x_k \in \mathbb{R}^n$1, leading, after encoding, to a QUBO: $x_k \in \mathbb{R}^n$2 where the QUBO matrix $x_k \in \mathbb{R}^n$3 encapsulates information from system dynamics, cost structure, and input constraint encoding.

The QUBO is then mapped to a spin Hamiltonian: $x_k \in \mathbb{R}^n$4 with annealing performed from the ground state of a driver Hamiltonian to the problem Hamiltonian $x_k \in \mathbb{R}^n$5, using quantum annealers (e.g., D-Wave). The annealing time $x_k \in \mathbb{R}^n$6 is determined by the minimum spectral gap. Empirical studies report solution times nearly constant in problem size for QUBOs with up to $x_k \in \mathbb{R}^n$7, a favorable scaling compared to the classical $x_k \in \mathbb{R}^n$8 (Novara et al., 2024). Tracking performance matches or exceeds classical NMPC in illustrative tasks, attributable in part to global exploration via quantum tunneling.

3. Quantum-Assisted Barrier SQP for NMPC

An alternative paradigm applies quantum subroutines within classical iterative optimization frameworks such as barrier-SQP (Dehaghani et al., 20 Oct 2025). The nonlinear program is transcribed with a barrier-augmented objective: $x_k \in \mathbb{R}^n$9 where $u_k \in \mathbb{R}^m$0 is a convex barrier (e.g., $u_k \in \mathbb{R}^m$1), and $u_k \in \mathbb{R}^m$2 is the total scalar inequality count. Each SQP iterate solves a QP of the form: $u_k \in \mathbb{R}^m$3 with explicit expressions for $u_k \in \mathbb{R}^m$4, $u_k \in \mathbb{R}^m$5, $u_k \in \mathbb{R}^m$6, and $u_k \in \mathbb{R}^m$7. The KKT system leads to the Schur complement system $u_k \in \mathbb{R}^m$8, where

$u_k \in \mathbb{R}^m$9

Solving this linear system (the "Schur step") constitutes the computational bottleneck.

Quantum routines accelerate this step by using block-encoding to construct unitary embeddings of $J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$0, $J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$1, and associated vectors, and Quantum Singular Value Transformation (QSVT) to invert these via polynomial singular value transformations. The overall gate complexity for the Schur step is

$J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$2

where $J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$3 is the relevant condition number, and $J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$4 are accuracy parameters. Hybridization consists of classical assembly and preparation steps, the quantum Schur step, and classical post-processing (e.g., line search, feasibility checks).

4. Error, Stability, and Convergence Analysis

Convergence analysis of quantum-assisted optimization must address the effect of quantum computational error. With quantum error bounded by $J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$5, the perturbed algorithm exhibits local input-to-state stability: $J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$6 for suitable class-$J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$7 functions $J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$8, where $J = \sum_{k=0}^{N-1} \ell(x_k, u_k) + \ell_N(x_N)$9 is a stationary point (Dehaghani et al., 20 Oct 2025). Provided $c(x_k, u_k) \leq 0$0 and $c(x_k, u_k) \leq 0$1, the scheme achieves full convergence. Explicit geometric error bounds are established: $c(x_k, u_k) \leq 0$2 Limit point proximity to the true solution set is

$c(x_k, u_k) \leq 0$3

illustrating that quantum- and barrier-induced errors dictate convergence to a controlled neighborhood.

5. Computational Complexity and Performance

Quantum optimization yields asymptotic complexity improvements for the most costly NMPC subproblems. Under standard sparse-access and QRAM assumptions, the gate complexity for the quantum Schur step is polylogarithmic in problem dimension and logarithmic in error tolerance, dominated by QSVT inversions. For quantum annealing approaches, circuit runtimes are nearly constant for moderate variable counts, and precision error can be made arbitrarily small by increasing encoding depth $c(x_k, u_k) \leq 0$4, with error decaying as $c(x_k, u_k) \leq 0$5 (Novara et al., 2024). However, overheads arise from embedding, chain lengths, and penalty parameters on current hardware.

Reported benchmarks show that, for prediction horizons of up to $c(x_k, u_k) \leq 0$6 and moderate decision variable dimensions, quantum annealers achieve practical runtimes and control performance on par with or better than classical solvers, owing to enhanced global search capability.

6. Practical Considerations and Implementation Challenges

Practical deployment of quantum optimization for NMPC faces challenges related to hardware capabilities and algorithmic integration. Quantum annealing hardware is currently limited by qubit count (≈5,000), connectivity, and control fidelity. Embedding QUBOs with general coupling graphs leads to physical qubit overheads. Noise, precision in coupler strengths, and stochasticity of quantum outcomes introduce practical uncertainty, and real-time control requirements necessitate robust scheduler design, with classical fallback strategies for worst-case timing (Novara et al., 2024).

Quantum-assisted hybrid schemes are contingent on efficient block-encoding (often via QRAM) and error-tolerant quantum circuits. Levenberg–Marquardt damping can regularize condition numbers, improving algorithmic robustness, and careful tuning of quantum accuracy ($c(x_k, u_k) \leq 0$7), block-encoding tolerance ($c(x_k, u_k) \leq 0$8), and barrier parameter ($c(x_k, u_k) \leq 0$9) is necessary to retain theoretical guarantees (Dehaghani et al., 20 Oct 2025). Maintaining error below the contraction threshold is critical to convergence and stability.

7. Outlook, Open Problems, and Research Directions

Both quantum annealing and quantum-assisted hybrid SQP demonstrate theoretical and numerical feasibility for accelerating NMPC for large, nonlinear systems. However, major open challenges persist:

• Reducing binary encoding overhead for high-precision and large-scale problems.
• Extending beyond QUBO (e.g., to higher-degree polynomials or combinatorially constrained NMPC) without deleterious increases in auxiliary qubit count and penalty scaling.
• Developing hybrid quantum-classical routines—such as QAOA or VQE—that can exploit near-term gate-model devices for more general constraints and finer Hamiltonian control.
• Stabilizing annealing/protocol timing and integrating warm-starting, or solution reuse across control cycles.
• Formalizing closed-loop stability and convergence properties under stochastic quantum optimizer outputs.

Future advances in quantum hardware—coherence time, error rates, connectivity—and efficient hybrid algorithm design are expected to determine the onset of real-time, quantum-accelerated NMPC in industrial and safety-critical applications. Existing mathematical mappings, complexity bounds, and pilot empirical studies establish a foundation for these developments (Dehaghani et al., 20 Oct 2025, Novara et al., 2024).