Abstract In this article, we introduce an original hybrid quantum–classical algorithm based on a variational quantum algorithm for solving systems of differential equations. The algorithm relies on a spectral decomposition of the trial functions that are encoded directly in the quantum states generated by different parametrized circuits, and transforms the task of solving the differential equations into an optimization problem. We first describe the principle of the algorithm from a theoretical point of view. We provide a detailed pseudo-code of the algorithm, on which we elaborate preliminary elements for a complexity analysis to highlight some of its scaling properties. We apply our algorithm to a set of examples, running on emulators and real hardware showcasing its applicability across diverse sets of differential equations. We discuss the advantages of our method and potential avenues for further exploration and refinement 1 1 The algorithm described in this paper is subject to a patent submission (EP24306601). Use of this method requires a license, which can be obtained from ColibriTD. .
Abstract In this article, we introduce an original hybrid quantum–classical algorithm based on a variational quantum algorithm for solving systems of differential equations. The algorithm relies on a spectral decomposition of the trial functions that are encoded directly in the quantum states generated by different parametrized circuits, and transforms the task of solving the differential equations into an optimization problem. We first describe the principle of the algorithm from a theoretical point of view. We provide a detailed pseudo-code of the algorithm, on which we elaborate preliminary elements for a complexity analysis to highlight some of its scaling properties. We apply our algorithm to a set of examples, running on emulators and real hardware showcasing its applicability across diverse sets of differential equations. We discuss the advantages of our method and potential avenues for further exploration and refinement 1 1 The algorithm described in this paper is subject to a patent submission (EP24306601). Use of this method requires a license, which can be obtained from ColibriTD. .
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.
We propose several approaches for solving regression problems and differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented using automatic differentiation of quantum circuits. While previously quantum kernel methods primarily targeted classification tasks, here we consider their applicability to regression tasks, based on available data and differential constraints. We use two strategies to approach these problems. First, we devise a mixed model regression with a trial solution represented by kernel-based functions, which is trained to minimize a loss for specific differential constraints or datasets. Second, we use support vector regression that accounts for the structure of differential equations. The developed methods are capable of solving both linear and nonlinear systems. Contrary to prevailing hybrid variational approaches for parametrized quantum circuits, we perform training of the weights of the model classically. Under certain conditions this corresponds to a convex optimization problem, which can be solved with provable convergence to global optimum of the model. The proposed approaches also favor hardware implementations, as optimization only uses evaluated Gram matrices, but require a quadratic number of function evaluations. We highlight trade-offs when comparing our methods to those based on variational quantum circuits such as the recently proposed differentiable quantum circuits approach. The proposed methods offer potential quantum enhancement through the rich kernel representations using the power of quantum feature maps, and start the quest towards provably trainable quantum DE solvers.
Quantum computers have been proposed as a solution for efficiently solving non-linear differential equations (DEs), a fundamental task across diverse technological and scientific domains. However, a crucial milestone in this regard is to design protocols that are hardware-aware, making efficient use of limited available quantum resources. We focus here on promising variational methods derived from scientific machine learning: differentiable quantum circuits (DQC), addressing specifically their cost in number of circuit evaluations. Reducing the number of quantum circuit evaluations is particularly valuable in hybrid quantum/classical protocols, where the time required to interface and run quantum hardware at each cycle can impact the total wall-time much more than relatively inexpensive classical post-processing overhead. Here, we propose and test two sample-efficient protocols for solving non-linear DEs, achieving exponential savings in quantum circuit evaluations. These protocols are based on redesigning the extraction of information from DQC in a ``measure-first" approach, by introducing engineered cost operators similar to the randomized-measurement toolbox (i.e. classical shadows). In benchmark simulations on one and two-dimensional DEs, we report up to ∼ 100 fold reductions in circuit evaluations. Our protocols thus hold the promise to unlock larger and more challenging non-linear differential equation demonstrations with existing quantum hardware.
Quantum computing has the potential to solve certain compute-intensive problems faster than classical computing by leveraging the quantum mechanical properties of superposition and entanglement. This capability can be particularly useful for solving Partial Differential Equations (PDEs), which are challenging to solve even for High-Performance Computing (HPC) systems, especially for multidimensional PDEs. This led researchers to investigate the usage of Quantum-Centric High-Performance Computing (QC-HPC) to solve multidimensional PDEs for various applications. However, the current quantum computing-based PDE-solvers, especially those based on Variational Quantum Algorithms (VQAs) suffer from limitations, such as low accuracy, long execution times, and limited scalability. In this work, we propose an innovative algorithm to solve multidimensional PDEs with two variants. The first variant uses Finite Difference Method (FDM), Classical-to-Quantum (C2Q) encoding, and numerical instantiation, whereas the second variant utilizes FDM, C2Q encoding, and Column-by-Column Decomposition (CCD). We evaluated the proposed algorithm using the Poisson equation as a case study and validated it through experiments conducted on noise-free and noisy simulators, as well as hardware emulators and real quantum hardware from IBM. Our results show higher accuracy, improved scalability, and faster execution times in comparison to variational-based PDE-solvers, demonstrating the advantage of our approach for solving multidimensional PDEs.
Different hybrid quantum-classical algorithms have recently been developed as a near-term way to solve linear systems of equations on quantum devices. However, the focus has so far been mostly on the methods, rather than the problems that they need to tackle. In fact, these algorithms have been run on real hardware only for problems in quantum physics, such as Hamiltonians of a few qubits systems. These problems are particularly favorable for quantum hardware, since their matrices are the sum of just a few unitary terms and since only shallow quantum circuits are required to estimate the cost function. However, for many interesting problems in linear algebra, it appears far less trivial to find an efficient decomposition and to trade it off with the depth of the cost quantum circuits. A first simple yet interesting instance to consider are tridiagonal systems of equations. These arise, for instance, in the discretization of one-dimensional finite element analyses. This work presents a method to solve a class of tridiagonal systems of equations with the variational quantum linear solver (VQLS), a recently proposed variational hybrid algorithm for solving linear systems. In particular, we present a new decomposition for this class of matrices based on both Pauli strings and multi--qubit gates, resulting in less terms than those obtained by just using Pauli gates. Based on this decomposition, we discuss the tradeoff between the number of terms and the near-term implementability of the quantum circuits. Furthermore, we present the first simulated and real-hardware results obtained by solving tridiagonal linear systems with VQLS, using the decomposition proposed.
Quantum computers have the potential to solve some complex problems much faster than classical equivalents. Significant research efforts are aimed at determining which algorithms give which advantages for different use-cases. An application where quantum computers can possibly bring great advantages is solving (partial) differential equations, which has a broad set of applications, for instance in wave-propagation models. Few differential equations admit an analytical solution. For most heuristic methods are required to approximate a solution. Well-known heuristic techniques include the finite element and finite difference method, where the considered space is partitioned and systems of linear equations resulting from this partitioning need to be solved. Quantum computing puts forward new methods to solve these systems of linear equations and hence these differential equations. First, the HHL algorithm gives an efficient way to solve a linear system of equations. The HHL algorithm comes with drawbacks, but in this specific use-case some of these objections might be circumvented. Quantum computers furthermore offer the variational approach: an optimization-based path to solving differential equations. With this approach, the devices might even ‘learn’ the noise patterns emerging in present day quantum computers and compensate for it. In this work we revisit quantum methods for solving partial differential equations and consider how well they work in solving differential equations in practice. We also discuss possible caveats and bottlenecks of the approaches.