Measurement-Efficient Variational Quantum Linear Solver for Carleman-Linearized Nonlinear Dynamics

Abstract: We present hybrid quantum-classical pipelines for solving the Duffing equation that leverage Carleman linearization and the Variational Quantum Linear Solver (VQLS). First, we demonstrate that Carleman linearization accurately approximates the weakly nonlinear Duffing equation, with errors diminishing as the truncation order increases. Next, across IBM and Xanadu platforms, we deploy VQLS with symmetry-grouped Hadamard Test evaluations under both global and local cost formulations, compare distinct Hermitianization within a common cost framework, and benchmark hardware-efficient ansatz architectures under a fixed Hermitianization. Across block-banded test cases, each method achieves near-unity fidelity and vanishing relative residuals. These results show that topology-agnostic ansatz, optimized Hermitianization, and efficient cost formulation enable VQLS to recover quantum states proportional to classical solutions for Carleman-structured systems, providing a portable recipe for quantum-in-the-loop simulation of nonlinear dynamics.

• The paper introduces a novel integration of Carleman linearization with VQLS to solve cubic nonlinear dynamics efficiently.
• It demonstrates convergence of the Carleman-linearized Duffing equation to classical Runge-Kutta solutions, validating the method’s accuracy with reduced measurement overhead.
• The study employs Hermitianization and symmetry-grouped measurements, paving the way for scalable quantum algorithms in simulating complex nonlinear systems.

Measurement-Efficient VQLS for Carleman-Linearized Nonlinear Dynamics

Background and Motivation

Solving high-dimensional nonlinear differential equations (DEs) remains fundamental in computational engineering and physics, with applications spanning structural mechanics, fluid dynamics, heat transfer, and control systems. The canonical Duffing oscillator serves as a representative testbed, encapsulating weak nonlinearity via cubic stiffness and exhibiting complex phenomena such as amplitude-dependent frequency shifts, jumps, hysteresis, and chaos.

Traditional strategies for DEs include direct discretization (Euler, Runge-Kutta), harmonic balance, perturbation theory, and polynomial lifting via Carleman linearization or Koopman embedding. Carleman linearization is particularly noteworthy: it maps polynomial ODEs into exact infinite-dimensional linear flows, yielding block-banded, structured sparse matrices upon truncation. These matrices are computationally appealing for both classical and quantum algorithms due to their sparsity and hierarchical coupling properties.

Quantum computing algorithms, such as the Quantum Linear Systems Algorithm (QLSA), offer asymptotic speedups for solving such linearized systems, but require fault-tolerant hardware that remains out of reach. Variational Quantum Algorithms (VQAs), notably the Variational Quantum Linear Solver (VQLS), operate on noisy intermediate-scale quantum (NISQ) devices, using shallow parameterized circuits with hybrid quantum-classical optimization.

Prior integration of Carleman linearization with quantum solvers has mostly covered quadratic ODEs, leaving cubic systems—exemplified by the Duffing equation—largely unaddressed. This work closes the gap, delivering a full pipeline coupling Carleman linearization for cubic ODEs with measurement-efficient VQLS.

Carleman Linearization of the Duffing Equation

The Duffing equation,

$$\ddot z + \delta \dot z + \alpha z + \beta z^3 = f(t),$$

is recast in first-order form and subjected to Carleman linearization, truncating at order $N$ to obtain a block-banded linear time-dependent system (Eq. 2.4). The linearization procedure promotes tensor powers of the state as independent variables, producing a generator with clear block structure—sub-blocks corresponding to linear, cubic, and forcing-induced couplings.

Time discretization via forward Euler yields a block-bidiagonal linear system, $LY = B$, with dimensionality scaling exponentially in $N$. The block structure aims for amenability both to quantum mapping and circuit synthesis.

Verification experiments show that Carleman-linearized solutions for the Duffing equation converge to classical Runge-Kutta solutions as $N$ increases, with errors reduced at each order. The convergence is empirically observed even outside dissipative parameter regimes where theoretical bounds hold.

Figure 1: Verification of Carleman linearization on the Duffing equation; left: $z(t)$ agreement with RK4, right: error decay with increasing $N$.

Variational Quantum Linear Solver (VQLS): Architecture and Measurement

VQLS solves $L\,|Y\rangle = |B\rangle$ on a $Q$-qubit register, leveraging an efficient state preparation and a linear combination of unitaries (LCU) decomposition. The variational circuit $V(\alpha)$ is optimized to minimize a cost function—either global or local—measuring the proportionality between $N$0 and $N$1.

Major measurement overhead arises in cost evaluation due to the need for multiple inner product overlaps, which are estimated using Hadamard Test circuits. Three distinct overlap families—$N$2, $N$3, and $N$4—are required.

Figure 2: Hadamard Test circuit variants for the VQLS overlap families, illustrating measurement protocol.

To suppress barren plateau effects and improve optimization landscape, local cost functions are used, replacing the global projectors with qubit-wise operators. Symmetry-grouped measurement strategies, enabled by Hermitianization and real LCU coefficients, reduce the Hadamard Test count by approximately a factor of two.

Operator Reformulation: Hermitianization and Pauli Decomposition

The block-banded operator $N$5 is generally non-Hermitian. Two Hermitianization approaches are considered:

1. Regularized Normal Equations: $N$6, preserving Hilbert space dimension but worsening conditioning as $N$7.
2. Augmented-System Dilation: A Hermitian embedding with doubled Hilbert space, achieving condition number $N$8 at the cost of an extra qubit and post-selection.

Both admit Pauli LCU decomposition, with symmetry in overlap measurements further exploited for measurement efficiency.

VQLS Benchmark Results

Benchmarks are performed across block-banded systems on both Qiskit (global cost, COBYLA) and PennyLane (local cost, gradient descent) pipelines, using hardware-efficient ansätze (HEA and RING).

Convergence diagnostics demonstrate that Qiskit/global cost pipeline suffers from barren plateaus, yielding low cost but poor fidelities for certain instances. In contrast, PennyLane/local cost consistently produces high direction fidelity, solution fidelity, Bhattacharyya coefficient, and low residual error, even for challenging cases.

Figure 3: Global cost-function history on two- and three-qubit block-banded systems.

Figure 4: Regularized Carleman matrices and quantum/classical solution distributions; high probability similarity.

Augmented-system dilation (PennyLane Method B) further enhances direction fidelity and distribution similarity, though post-selection is required for recovering the actual solution from the augmented state.

Figure 5: Heatmap and distribution for augmented-system dilation; block structure exploited for efficient VQLS mapping.

Results on Carleman-linearized Duffing systems (subharmonic, hardening-spring, superharmonic) confirm high solution and direction fidelity across all tested regimes.

Implications and Future Directions

This work establishes a portable quantum-in-the-loop pipeline for simulating nonlinear dynamics via Carleman linearization and VQLS. Practical implications include:

• Structured block-banded matrices resulting from Carleman linearization enable efficient quantum mapping and sparsity-aware measurement, reducing quantum circuit depth requirements.
• Symmetry-grouped Hadamard Test strategies substantially lower measurement overhead, supporting scalability for larger NISQ systems.
• The local cost formulation coupled with Hermitianization—especially regularized normal equations—provides robust optimization landscape and suppresses cost concentration artifacts.
• Augmented-system dilation offers alternative conditioning at the expense of additional quantum resources.

Theoretically, the extension of Carleman quantum algorithms to cubic regimes, and empirical convergence observed beyond dissipative bounds, signal potential for broader classes of nonlinear ODEs. This lays groundwork for future quantum algorithms targeting nonlinear PDEs, especially in fluid and wave dynamics.

Prospective directions include extension of symmetry-aware measurement protocols, advanced block encoding for non-Hermitian operators, reduction of circuit depth via operator dilation, and noise-resilience studies in the context of dissipative PDEs at larger qubit scales.

Conclusion

Measurement-efficient VQLS built on Carleman linearization for cubic nonlinear systems delivers accurate, scalable quantum solutions for structured block-banded matrices, verified against classical references. Exploiting operator symmetries, efficient Hermitianization, and local cost functions, the pipeline achieves high fidelity under realistic quantum resource constraints, supporting future advancements in quantum computational simulation of nonlinear dynamics.