In large-scale scientific computing and engineering modeling, calculus problems are widely used as basic mathematical tools in physical simulation, control systems, image processing and other fields. However, their analytical and numerical solutions in high-dimensional and complex functions often face bottlenecks such as high computational complexity and slow convergence. To this end, this paper proposes a calculus acceleration solution method based on a quantum computing framework. First, this paper constructs a quantum state encoding mechanism suitable for function integration and differentiation operations. Then, Variational Quantum Algorithm (VQA) is applied to solve the optimal solution path in structured calculus expressions, and combined with Quantum Fourier Transform (QFT) to improve function boundary processing and analysis speed. Finally, a quantum-classical hybrid iterator is designed through a hybrid architecture with classical numerical methods. Experimental results show that in high-dimensional integration scenarios, compared with the classical Monte Carlo method, Quantum Amplitude Estimation (QAE) can reduce the number of queries from thousands to hundreds with the same error accuracy (such as); in the simulation experiment of ordinary differential equations, the simulated VQA shows a higher convergence efficiency with the same number of iterations. The above results fully demonstrate the acceleration potential and solution accuracy advantages of quantum algorithms in high dimensions and complex boundary conditions, and provide new ideas for the application of calculus problems in large-scale scientific and engineering calculations.
In large-scale scientific computing and engineering modeling, calculus problems are widely used as basic mathematical tools in physical simulation, control systems, image processing and other fields. However, their analytical and numerical solutions in high-dimensional and complex functions often face bottlenecks such as high computational complexity and slow convergence. To this end, this paper proposes a calculus acceleration solution method based on a quantum computing framework. First, this paper constructs a quantum state encoding mechanism suitable for function integration and differentiation operations. Then, Variational Quantum Algorithm (VQA) is applied to solve the optimal solution path in structured calculus expressions, and combined with Quantum Fourier Transform (QFT) to improve function boundary processing and analysis speed. Finally, a quantum-classical hybrid iterator is designed through a hybrid architecture with classical numerical methods. Experimental results show that in high-dimensional integration scenarios, compared with the classical Monte Carlo method, Quantum Amplitude Estimation (QAE) can reduce the number of queries from thousands to hundreds with the same error accuracy (such as); in the simulation experiment of ordinary differential equations, the simulated VQA shows a higher convergence efficiency with the same number of iterations. The above results fully demonstrate the acceleration potential and solution accuracy advantages of quantum algorithms in high dimensions and complex boundary conditions, and provide new ideas for the application of calculus problems in large-scale scientific and engineering calculations.
Quantum computation is a promising approach to solving problems in physics, optimization, and numerical simulation. Quantum algorithms leverage superposition and entanglement in an exponentially large Hilbert space to encode and solve problems that are intractable for classical computers. However, these algorithms require error-corrected quantum hardware, which is not yet available. This thesis explores two alternative quantum-based approaches that promise to achieve better scalability than traditional algorithms in the near-term: Hybrid variational quantum algorithms (VQAs) and fully classical, quantum-inspired algorithms. In the first part, we review and benchmark optimization methods well suited for VQAs, specifically the Simultaneous Perturbation Stochastic Approximation (SPSA). We introduce complex-variable extensions and improvements to the second-order and quantum natural implementations. Finally, we assess their performance across quantum control, quantum tomography, and the variational quantum eigensolver, providing practical guidelines for common VQA scenarios. In the second part, we apply Matrix Product States (MPS) to solve time-dependent partial differential equations (PDEs). By integrating Hermite Distributed Approximating Functionals (HDAF) into the MPS framework, we develop a novel encoding to efficiently approximate differential operators with exponential precision. We benchmark this approach against traditional schemes for simulating the expansion of a levitated nanoparticle. The quantum-inspired numerical integration schemes deliver large memory savings and adequate runtimes even in presence of moderate chirping.
Differential equations (DEs) serve as the cornerstone for a wide range of scientific endeavors, their solutions weaving through the core of diverse fields such as structural engineering, fluid dynamics, and financial modeling. DEs are notoriously hard to solve, due to their intricate nature, and finding solutions to DEs often exceeds the capabilities of traditional computational approaches. Recent advances in quantum computing have triggered a growing interest from researchers for the design of quantum algorithms for solving DEs. In this work, we introduce two different architectures of a novel variational quantum algorithm (VQA) with Lagrange polynomial encoding in combination with derivative quantum circuits using the Hadamard test differentiation to approximate the solution of DEs. To demonstrate the potential of our new VQA, two well-known ordinary differential equations are used: the damped mass-spring system from a given initial condition and the Poisson equation for periodic, Dirichlet, and Neumann boundary conditions. It is shown that the proposed new VQA has a reduced gate complexity compared to previous variational quantum algorithms, for a similar or better quality of the solution.
Differential equations underpin a vast range of scientific and engineering domains, including structural mechanics, fluid dynamics, and financial modelling. Their intricate nature often renders classical numerical methods computationally demanding, requiring fine discretization or large basis expansions, which can lead to prohibitive costs. Quantum computing offers a fundamentally different paradigm, leveraging superposition and entanglement with the promise of computational speedup. However, in the current Noisy Intermediate-Scale Quantum (NISQ) era, fully fault-tolerant quantum computers remain decades away, motivating the development of hybrid strategies such as Variational Quantum Algorithms (VQAs), which train parametrised quantum circuits using classical optimizers. This work investigates spectral VQAs for solving differential equations, extending the Chebyshev-encoded VQA of Kyriienko et al. and introducing a novel quantum spectral approach: the Lagrange-encoded VQA. This approach shifts the paradigm by incorporating entanglement directly within the encoding block while maintaining a shallow variational ansatz. Inspired by the Hadamard test structure, the design enables an efficient representation of the complete set of Lagrange polynomial basis functions using a limited number of gates and facilitates the use of the Hadamard test differentiation method, thereby reducing the computational load. VQA performance is analysed through the accuracy of the result in combination with the gate count and circuit count metrics, which serve as proxies for runtime and noise accumulation on NISQ devices. The proposed VQA is benchmarked on multiple problems: (i) a first-order linear ODEs and (ii) the second-order ODEs damped mass-spring system, both comparing Lagrange- and Chebyshev-encoded VQAs, (iii) the Poisson equation, comparing the Lagrange-encoded VQA with the discretized approach of Sato et al., and (iv) a first-order ODE with nonlinear coefficients addressed with the Lagrange-encoded VQA. Across these cases, the Lagrange-encoded VQA consistently achieves comparable or improved accuracy while significantly reducing circuit depth and computational overload, making it a promising candidate for NISQ devices.
This paper proposes a variational quantum algorithm to address the large-scale and low computational efficiency issues of high-dimensional partial differential equations (PDEs) in climate science. In this method, parameterized quantum circuits encode the discrete solutions of PDEs into quantum states. Then, through repeated quantum measurements, we can obtain the expectation values of the operators. By iteratively adjusting the circuit parameters, a hybrid quantum-classical optimization loop can reduce the error in the solution. The Schrödinger equation and the Poisson equation are benchmark partial differential equations for numerical experiments, and they are four-dimensional and five-dimensional, respectively. According to the above results, the convergence speed is relatively fast; after 100 iterations, the median relative errors of all test cases are less than 1.6% and 3.1%, respectively, with the final target loss ranging between 0.014 and 0.045. The above comparison indicates that in high-dimensional and ill-posed problems, quantum solvers can achieve the same or better accuracy and robustness compared to traditional methods. Resource analysis indicates that quantum algorithms use less memory and fewer sub-exponential growth gates and runtime in larger systems. According to the above findings, variational quantum solvers can be used for complex scientific computing problems. In addition, they hope that future climate models will be more accurate and efficient.
Structural analysis in high-performance computing (HPC) faces challenges related to computational complexity, energy efficiency, and solution accuracy. This research explores Quantum-Inspired Algorithms (QIAs) as an innovative approach to enhance computational efficiency and accuracy in large-scale structural simulations. The proposed methodology integrates a Quantum-Inspired Evolutionary Algorithm (QIEA) with a Hybrid Quantum-Inspired Neural Network (HQINN) for improved structural performance prediction. The study evaluates QIAs on three benchmark structural problems: Bridge Load Distribution Analysis – Achieves a computational speed-up of 45% compared to classical solvers while maintaining an error rate of <0.5%. The Quantum-Inspired Variational Monte Carlo (QIVMC) method is applied to solve complex eigenvalue problems, achieving an 8× acceleration in solving large-scale stiffness matrices compared to traditional iterative solvers. Experimental validation on a high-performance computing cluster using 1,024 cores demonstrates a 55% improvement in processing speed and a 37% reduction in energy consumption. Results confirm that Quantum-Inspired Algorithms significantly outperform traditional numerical methods in structural analysis, paving the way for their adoption in next-generation engineering simulations. Future work will focus on hybrid quantum-classical frameworks and their real-world applications in civil, aerospace, and automotive engineering.
This paper presents a novel approach to simulating complex fluid dynamics, specifically turbulent flows, utilizing quantum-inspired algorithms. Traditional numerical methods, such as Navier-Stokes solvers, often struggle with the computational demands of accurately representing turbulence, leading to significant processing times and limitations in practical applications. We introduce a simulation strategy based on quantum amplitude estimation (QAE), a technique inspired by quantum computing principles, to accelerate the solution process. The core idea is to approximate the probability amplitude associated with different fluid states, thereby reducing the number of iterations required to reach a stable solution. We demonstrate, through theoretical analysis and a simplified numerical framework, that QAE offers a promising path toward more efficient and accurate simulations of turbulent flows, potentially bridging the gap between classical computational fluid dynamics (CFD) and the computational benefits anticipated from full-scale quantum simulation. This work lays the groundwork for future research exploring the full potential of quantum-inspired algorithms in tackling complex fluid dynamics problems.