A Hybrid Quantum–Classical Spectral Solver for Nonlinear Differential Equations

Abstract

We investigate hybrid quantum–classical solvers for nonlinear boundary value problems using Chebyshev spectral collocation. Unlike prior methods such as H–DES, which repeatedly recompile circuits and encode the entire spectral basis on the quantum processor, our framework offloads only the residual minimisation to a quantum backend while retaining classical enforcement of boundary conditions. Two paradigms are considered: (i) gate-based residual minimisation on CUDA-Q using variational circuits to evaluate a Cubic Unconstrained Binary Optimisation (CUBO) cost, which naturally arises from the discretisation, and (ii) a Quadratic Unconstrained Binary Optimisation (QUBO) reformulation, which is required for execution on a quantum annealer, executed via a classical–quantum mapping. We further explore a CUBO extension on CUDA-Q and direct residual-to-energy mapping on annealers. Benchmarks confirm that the classical solver reproduces the analytic solution with spectral accuracy; among quantum-enhanced methods, the annealer-based QUBO yields the closest approximation. The gate-based CUBO solver improves upon a legacy variational baseline but exhibits a small interior bias due to limited circuit depth and precision. These findings underscore the complementary roles of annealers and gate-based devices in hybrid scientific computing and demonstrate a feasible workflow for the NISQ era rather than a speedup over classical methods. Recent progress in quantum algorithms for differential equations signals a rapidly maturing field with significant potential for practical quantum advantage.

1. Introduction and Motivation

Numerical solutions of boundary value problems (BVPs) are indispensable across physics, engineering, and finance [1]. Classical discretisation techniques—finite differences, finite elements, and spectral methods—have been refined over decades to deliver high accuracy and scalability [2,3]. Nonetheless, some nonlinear partial differential equations (PDEs) pose challenges for classical solvers because the cost of assembling and inverting dense Jacobian matrices scales poorly with problem size or nonlinearity [4]. Quantum computing promises polynomial or exponential speedups for particular linear algebraic tasks [3,5,6], motivating interest in quantum-accelerated numerical methods.

The present quantum hardware landscape is described as the noisy intermediate-scale quantum (NISQ) era [7]. Devices in this class contain hundreds to roughly a thousand qubits and are not yet fault-tolerant; they are prone to decoherence and cannot perform continuous error correction [7]. Practical algorithms for NISQ systems therefore need to minimise circuit depth and shot count [8]. Hybrid quantum–classical algorithms, notably the variational quantum eigensolver (VQE) [8] and quantum approximate optimisation algorithm (QAOA) [9], couple classical optimisation with quantum state preparation and measurement. Differential equation solvers built on the variational paradigm are a logical extension: they rely on parameterised quantum circuits (ansätze) and iterative classical feedback to minimise a cost function derived from the governing PDE [10,11].

Dual Paradigm Comparison. In this study we contrast two quantum paradigms—gate-based variational circuits and quantum annealing—within a common hybrid framework that offloads only the residual minimisation. Each paradigm is evaluated in its natural encoding regime: CUBO-style cost functions on gate-based devices and QUBO formulations on annealers. This unified perspective highlights complementary strengths and guides the design of practical hybrid solvers.

Recent advances in quantum algorithms for differential equations have demonstrated various approaches. Kyriienko et al. [10] pioneered quantum spectral methods using Chebyshev quantum feature maps for nonlinear differential equations. Leong et al. [11] developed variational quantum evolution equation solvers using implicit time-stepping schemes. More recently, Leong et al. provided rigorous error analysis for VQA-based differential equation solvers using Runge–Kutta methods, while Arora et al. [12] introduced the Quantum Finite Element Method (Q-FEM) using VQLS algorithms. We also cite very recent works by Tennie and Magri [13], Zhang and Shao [14], Berry and Costa [15], and Mizuno and Komatsuzaki [16], which further develop quantum approaches to differential equations in 2024–2025.

It is important to emphasise at the outset that the present work does not aim to outperform well-established classical solvers. On today’s NISQ hardware, communication overhead between quantum and classical processors, limited qubit counts, and the need for repeated sampling typically negate any potential speedup for small-to-medium problems [7]. Rather, the goal is to demonstrate a feasible hybrid workflow—combining classical spectral discretisation with quantum evaluation of a simple observable—and to identify the challenges that must be overcome before hybrid quantum solvers can deliver practical benefits. The current solver thus serves as a prototype: it highlights the roles of classical and quantum components, underscores the importance of efficient ansatz design, and provides a platform for investigating how future advances in hardware and algorithms might enable quantum assistance for computationally intensive problems.

In addition to advances in quantum–classical algorithms, prior work by the present authors has emphasised reproducibility, benchmarking, and performance optimisation of spectral and FFT-based applications in high-performance computing [17,18,19,20,21]. Such efforts underscore the importance of connecting algorithmic advances with practical performance evaluation, and they motivate our present exploration of hybrid spectral solvers that integrate quantum backends while maintaining classical efficiency.

Prior Work

Lubasch et al. demonstrated that nonlinear PDEs can be addressed by variational quantum algorithms that use multiple copies of quantum states and tensor networks [22]. Their proof-of-concept implementation treated the nonlinear Schrödinger equation and showed that the variational ansatz can outperform classical matrix product state representations. More recently, Sarma et al. extended these ideas to nonlinear and multidimensional PDEs such as the Black–Scholes and Kardar–Parisi–Zhang equations, using up to twelve ansatz qubits and testing on a trapped-ion processor [23].

Jaffali et al. proposed H–DES, a hybrid differential equation solver that encodes the entire solution function into the amplitudes of a quantum state through a spectral decomposition and optimises circuit parameters variationally [24]. Their method includes pseudocode and a complexity analysis, and it demonstrates applicability to several PDEs. H–DES performs the spectral expansion on the quantum processor, which can lead to significant circuit depth and recompilation overhead for each iteration.

On the classical side, Chebyshev spectral methods offer exponential convergence for smooth solutions [2]. Collocation at Chebyshev–Gauss–Lobatto (CGL) points clusters nodes near the boundaries, where solution gradients are often largest [25]. In the collocation approach, the continuous residual is forced to zero at discrete nodes, converting a differential equation into a system of algebraic equations. This technique can achieve high accuracy with a modest number of points [2].

While these lines of research highlight the potential of variational quantum algorithms [10,11] and spectral methods [14], there remains a gap between fully quantum-encoded solvers and purely classical schemes. Our goal is therefore to explore a middle ground that leverages classical spectral discretisation and offloads only the computationally intensive component—the evaluation of a scalar loss function—to a quantum backend. By doing so, we aim to retain the maturity of classical algorithms while meeting the constraints of NISQ devices.

Most recently, Endo and Takahashi proposed a variational-principle-based algorithm tailored for fault-tolerant quantum computers (FTQCs), addressing nonlinear equilibrium equations with linearised time-evolution formulations [26]. While outside the NISQ scope of this work, such approaches highlight future directions for scaling nonlinear solvers once large-scale quantum devices become available.

2. Problem Statement and Classical Spectral Solver

We consider the second-order nonlinear boundary–value problem

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}\left(a(x,u)u^{\prime }\right)=f\left(x\right),u\left(0\right)=\alpha ,u\left(2\right)=\beta .$$

(1)

Three representative forms of the coefficient function $a(x,u)$ are examined:

• Constant coefficient: $a(x,u)=1$.
• Variable coefficient: $a(x,u)=x+1$.
• Nonlinear coefficient: $a(x,u)=u+1$.

To discretise the PDE we adopt Chebyshev–Gauss–Lobatto collocation. Given an integer N, we define nodes $x_j=cos(\pi j/N)$ for $j=0,\dots ,N$, which map the physical domain $[0,2]$ to $[-1,1]$ via an affine transformation. These points cluster at the boundaries $x=0$ and $x=2$, enhancing resolution where the solution may vary rapidly. Let D denote the CGL differentiation matrix and $diag\left(a\right(u\left)\right)$ the diagonal matrix formed by evaluating a at a vector u. For a candidate solution vector u, we define the residual vector

$$R\left(u\right)=D\left(a\left(u\right)\circ \left(Du\right)\right)-f,$$

(2)

where ∘ denotes element-wise multiplication and f is sampled at the collocation nodes. In the classical solver we form the scalar loss function $\mathcal{L}\left(u\right)={\parallel R\left(u\right)\parallel }^2$ and minimise it with respect to u using standard optimisation routines such as the BFGS quasi-Newton method.

3. Hybrid Quantum–Classical Strategy

3.1. Motivation for Hybridisation

Evaluating the nonlinear residual vector $R\left(u\right)$ and its squared norm $\mathcal{L}\left(u\right)$ constitutes the most computationally intensive part of the classical solver when a depends nonlinearly on u. Unlike linear problems, the nonlinear residual couples all degrees of freedom through pointwise multiplication and differentiation. We therefore propose to offload the computation of $\mathcal{L}\left(u\right)$ to a quantum backend. By encoding the residual norm as an observable expectation value, we exploit the ability of a quantum processor to estimate inner products via measurement while leaving the remainder of the algorithm—discretisation, ansatz initialisation, and optimisation—on the classical processor.

3.2. Quantum Kernel for Residual Evaluation

Let $\left|\psi \right(\theta )\rangle$ denote a parameterised quantum state prepared by an ansatz with parameters $\theta$. We choose an ansatz depth sufficient to represent the functional dependence of u on x but keep the circuit shallow to accommodate NISQ restrictions. The residual norm is encoded as

$$\mathcal{L}\left(\theta \right)=\langle \psi \left(\theta \right)\left|H\right|\psi \left(\theta \right)\rangle ,$$

(3)

where H is constructed such that its expectation reproduces ${\parallel R\left(u\left(\theta \right)\right)\parallel }^2$. In practice we embed the evaluation into frameworks such as CUDA–Q, which allow the programmer to specify a classical function for $\mathcal{L}$ that internally dispatches quantum kernels. A pseudocode sketch of the hybrid evaluation is given below:

• from scipy.optimize import minimize
• import~cudaq

• def loss_function(theta):
•     result = cudaq.observe(my_ansatz, H, theta)
•     return result.expectation()

• theta0 = [0.1] ∗ n_qubits
• opt_result = minimize(
•     loss_function,
•     theta0,
•     method = ‘BFGS’
• )

Here cudaq.observe evaluates the expectation value of H for the current parameters on a selected backend (emulator, GPU, or real hardware). Only the scalar $\mathcal{L}\left(\theta \right)$ is returned to the classical optimiser, thereby reducing quantum resource requirements compared with approaches that encode the entire residual or solution vector.

3.3. Division of Labour

The hybrid solver separates tasks between classical and quantum processors:

• Classical: This processor constructs CGL nodes, differentiation matrices, and the nonlinear residual $R\left(u\right)$ symbolically. It performs the outer optimisation loop (e.g., BFGS) and updates the ansatz parameters.
• Quantum: This processor implements the variational circuit and estimates $\mathcal{L}\left(\theta \right)$ via repeated measurements. No Jacobian or gradient information is required at this stage, although quantum gradient evaluation could be incorporated in future extensions.

This separation minimises the circuit depth and number of qubits, addressing the constraints of NISQ hardware. See Figure 1.

3.4. Binary Quadratic Model (BQM) Formulation and Mapping

The nonlinear term $a(x,u)·{\displaystyle \frac{\partial u}{\partial x}}$ in the differential equation, when discretised by Chebyshev spectral collocation and subsequently binarised for quantum processing, generates a squared residual loss function $\mathcal{L}\left(\mathbf{x}\right)={\parallel R\left(\mathbf{x}\right)\parallel }^2$ that contains terms up to $\mathcal{O}\left(x^3\right)$. This process yields a Cubic Unconstrained Binary Optimisation (CUBO) problem, $E_{CUBO}\left(\mathbf{x}\right)$, which serves as the minimisation target for our gate-based solver.

In contrast, the quantum annealer backend is fundamentally restricted to minimising only linear and and quadratic terms, requiring a Quadratic Unconstrained Binary Optimisation (QUBO) formulation Figure 2. To allow comparison with the annealer, the CUBO objective must be polynomially reduced to an equivalent QUBO problem, $E_{QUBO}(\mathbf{x},\mathbf{z})$. This is achieved by introducing auxiliary binary variables, $z_k\in \{0,1\}$, and a sufficiently large penalty constant $\lambda$ to enforce the cubic identity constraint $x_i{x}_j{x}_k\equiv x_i{x}_{j}z$. The final QUBO objective is $E_{QUBO}(\mathbf{x},\mathbf{z})=E_{CUBO}^{\prime }(\mathbf{x},\mathbf{z})+\lambda P(\mathbf{x},\mathbf{z})$. This distinction is crucial: the gate-based VQA directly processes the CUBO, while the annealer requires the variable-increased QUBO.

4. Implementation and Preliminary Results

We implemented the proposed solver using CUDA–Q, a high-performance framework selected for its native support of hybrid classical–quantum workflows and its ability to leverage powerful NVIDIA GPU acceleration for the classical optimisation loop and quantum circuit emulation. The classical discretisation used $N=50$ CGL nodes. For each coefficient function listed earlier, we compared the purely classical spectral solver with the hybrid solver in which the residual norm was evaluated on the CUDA–Q emulator.

4.1. Constant and Variable Coefficients

For the constant-coefficient case $a(x,u)=1$ and the forcing term $f\left(x\right)=2$, the exact solution is a parabola. Both classical and hybrid solvers reproduced this solution with spectral accuracy; the maximum absolute error at the collocation points was ∼${10}^{-12}$. When $a(x,u)=x+1$ and $f\left(x\right)$ were chosen so that the exact solution was $u\left(x\right)=sinx+x+1$, the hybrid solver again matched the classical solver.

4.2. Nonlinear Coefficient

The nonlinear test used $a(x,u)=u+1$ and a right-hand side $f\left(x\right)={\mathrm{e}}^x({\mathrm{e}}^x+x+1)$, for which the exact solution is $u\left(x\right)={\mathrm{e}}^x+1$. This case is more challenging because the residual couples the solution vector nonlinearly.

Our classical solver converged rapidly to the exact solution with spectral accuracy. Indeed, the numerical solution is essentially indistinguishable from the analytic curve, confirming that the Chebyshev collocation with $N=50$ nodes yields spectral convergence.

In contrast, the hybrid classical–quantum solver used to minimise the CUBO cost required a detailed specification of the quantum circuit. Our implementation employed an Alternating Operator Ansatz, composed of L layers. Each layer consisted of single-qubit $R_Y\left(\theta \right)$ gates followed by a linear entangling layer of $CNOT$ gates across neighbouring qubits. The study began with a six-qubit circuit and $L=3$ (depth-3), resulting in P variational parameters, which exhibited a noticeable discrepancy. While the Dirichlet boundary conditions were enforced exactly by construction, the interior solution diverged from the exponential profile (Figure 3). This deviation arises from the limited expressivity of the variational ansatz, the finite number of encoded Chebyshev modes, and the optimiser’s sensitivity to the highly non-convex residual landscape [10,12]. Potential strategies to reduce this gap include increasing the number of qubits or circuit depth, refining the ansatz to better capture Chebyshev modes, careful rescaling, and hybridizing with classical warm-starts.

A key counterintuitive finding emerged from testing a shallower circuit (Figure 4) with $L=1$ and three qubits. This lower-complexity configuration aligned more closely with the exact and classical solutions than the deeper $L=3$ circuit. This observation is significant and likely points to a trade-off in VQA design. The shallower circuit, while having reduced expressivity, is less susceptible to Barren Plateaus—a phenomenon where the optimisation landscape becomes exponentially flat in deep circuits—and accumulates fewer hardware noise errors (decoherence and gate error) typical of NISQ devices. Therefore, the reduced noise and more stable gradient landscape, rather than increased expressivity, appear to govern the convergence quality in this particular configuration.

5. Hybrid Picard–QUBO Annealer Results

In addition to the CUDA-Q based workflow, we also tested a hybrid strategy that couples a classical Picard iteration with a quantum annealer. For this, we employed Automatski’s solver, which reformulates the nonlinear boundary value problem into a Quadratic Unconstrained Binary Optimisation (QUBO) instance and executes it on a quantum annealing backend. Figure 5 shows the results of this hybrid Picard + QUBO annealer approach. Compared against the exact solution $u\left(x\right)=e^x+1$, the hybrid annealer method exhibits visibly closer agreement than our initial CUDA-Q variational implementation, particularly in capturing the exponential growth across the domain. This suggests that annealer-based QUBO formulations may provide more stable convergence for certain classes of nonlinear ODEs, even at modest qubit counts [27].

5.1. Comparison with CUDA-Q VQE Approach

The CUDA-Q framework enabled us to encode the solution through a variational quantum circuit (VQC) and optimise its parameters with a classical optimiser. However, as illustrated in Figure 3, convergence was more challenging, and the final approximation deviated more significantly from the exact solution compared to the annealer-based QUBO method. This contrast highlights an important point: different quantum paradigms may have different strengths when embedded in hybrid classical–quantum solvers [13]. While the VQC approach offers flexibility and potential scalability on gate-based devices, the annealer-based QUBO formulation naturally encodes discrete optimisation problems and may achieve better numerical stability in boundary value problems at this stage of hardware maturity. It is worth noting that both paradigms follow the same overarching design principle: the quantum backend is tasked solely with residual minimisation, while boundary conditions and operator structure are enforced classically. In CUDA-Q, the residual norm is expressed as an observable expectation value over a parameterised ansatz, whereas in Automatski the residual equations are reformulated as QUBO problems whose energy encodes the loss. Thus, although the mechanisms differ—continuous parameter optimisation versus discrete binary encoding—both methods exemplify the “residual-only” hybrid strategy.

5.2. Additional Plots

To complement the qualitative overlays in Secs. III–IV, we present diagnostics for two complementary settings: (i) a CUBO formulation executed on a gate-based CUDA-Q backend and (ii) direct residual minimisation formulated as a QUBO and solved on the Automatski annealer within a Picard loop. These plots quantify approximation quality, convergence behaviour, and the effect of discretisation and ansatz choices (see Figure 6 and Figure 7).

6. Comparison with Alternative Approaches

Existing variational quantum solvers such as H–DES encode the entire solution function in a quantum state and perform the spectral expansion on the quantum processor [24]. While this approach is powerful and general, it incurs a large quantum overhead because the residual vector or solution coefficients must be measured or updated at each iteration. In contrast, our solver retains the spectral basis and optimisation entirely on the classical processor and passes only the scalar residual norm to the quantum backend. This reduces circuit depth, avoids recompilation of the ansatz at each step, and is more compatible with the qubit counts and noise levels of current NISQ devices.

Variational algorithms for nonlinear problems often require multiple copies of quantum states or tensor network structures to handle nonlinearities [22]. By contrast, our method handles nonlinearity on the classical side; the quantum circuit merely evaluates an observable associated with the residual norm. This design choice makes the hybrid solver extensible: advances in quantum measurement techniques or hardware can be integrated without altering the classical numerical core.

Recent developments in quantum algorithms for scientific computing [6] have demonstrated various complementary approaches. Berry and Costa [15] developed quantum algorithms for time-dependent differential equations using Dyson series, while Zhang and Shao [14] introduced quantum spectral methods for gradient estimation. Mizuno and Komatsuzaki [16] integrated quantum dynamic mode decomposition with differential equation solvers, and Liu et al. [28] applied variational quantum algorithms to fluid flow problems.

7. Discussion and Outlook

The numerical experiments demonstrate that the nonlinear classical solver and the exact solution are essentially indistinguishable, as expected from the spectral Chebyshev discretisation with sufficient resolution. This agreement highlights the robustness of the classical root-based solver.

Among the hybrid quantum results, the annealer-based Automatski solver yielded remarkably accurate approximations, while the gate-based CUBO solver improved over the variational baseline but exhibited a small interior bias. The variational solver itself failed to match the nonlinear exact solution under practical depth and qubit counts, underscoring the limitations of generic VQAs for this problem. These observations suggest a practical roadmap: annealers are naturally aligned with quadratic optimisation tasks and can be leveraged today, while gate-based CUBO solvers offer greater programmability and may benefit from future hardware and algorithmic advances.

7.1. Encoding Choices Across Hardware Paradigms

Our hybrid solver deliberately uses different encodings for gate-based and annealing hardware. A direct QUBO encoding on the CUDA-Q backend would in principle be possible, but it requires mapping binary variables into qubits with additional ancilla overhead, leading to circuits of impractical depth for NISQ devices. Conversely, annealers accept only QUBO/Ising formulations and cannot implement parameterised ansätze or continuous expectation values. We therefore evaluate each paradigm in the encoding it naturally supports: CUBO-style cost functions on gate-based devices and QUBO formulations on annealers. This design emphasises complementary strengths rather than imposing a uniform formulation across incompatible hardware.

7.2. Limitations and Future Work

This proof-of-concept study intentionally focuses on a single nonlinear boundary value problem to isolate the effects of residual offloading. We did not include detailed timing tables because annealer service times depend strongly on queue delays, embedding overhead and network latency, making direct comparisons with gate-based or classical runtimes unreliable. Future work will develop controlled benchmarks across multiple problems and hardware platforms, explore deeper variational circuits and improved ansätze, and investigate error mitigation. We also plan to integrate annealers and variational circuits in a unified hybrid scheme, specifically by investigating a warm-start protocol where the annealer’s QUBO solution is used to initialise the VQE parameters, and to extend the approach to higher-dimensional PDEs.

We also propose two new specific investigations. First, we will conduct a detailed analysis of the **QUBO scaling complexity**, examining how the number of auxiliary variables (introduced during the CUBO-to-QUBO reduction) scales with the discretisation size, a critical factor for annealer feasibility. Second, we will explore the use of **meshfree discretisation methods**, such as Radial Basis Functions (RBFs) [29], as an alternative to Chebyshev spectral collocation. This is motivated by the potential for meshfree methods to provide a more versatile framework for applications involving irregular or evolving domains, which the quantum solver could then minimise.

The field is experiencing rapid growth, with significant contributions in 2024–2025 [6,12,13,14,15,16,28,30] building on foundational work from the early quantum computing era [3,5]. This demonstrates both the cutting-edge relevance and established theoretical foundations necessary for practical quantum advantage in numerical computing.

8. Conclusions

We introduced a hybrid spectral solver that integrates quantum residual evaluation into classical nonlinear boundary value problem solvers. Our results confirm that while the classical solver achieves spectral accuracy, the annealer-based QUBO approach provides the most faithful quantum-enhanced solution. The gate-based CUDA-Q CUBO solver, though less accurate at present, offers a resource-efficient pathway for future improvements with better ansätze and hardware. Together, these results highlight a practical roadmap: annealers as near-term tools for quadratic residual minimisation and gate-based devices as flexible platforms for scalable hybrid solvers. This dual approach marks a concrete step toward practical quantum advantage in scientific computing. The rapid recent progress in quantum algorithms for differential equations [13,14,15,16] indicates a maturing field with significant potential for practical quantum advantage. Additional details on the QUBO formulation and a pure annealer implementation are provided in Appendix A.