Solving differential equations is one of the most computationally expensive problems in classical computing, occupying the vast majority of high-performance computing resources devoted towards practical applications in various fields of science and engineering. Despite recent progress made in the field of quantum computing and quantum algorithms, its end-to-end application towards practical realization still remains unattainable. In this article, we tackle one of the primary obstacles towards this ultimate objective, specifically the encoding of matrices derived via finite difference method solving Poisson partial differential equations in simple boundary-value problems. To that end, we propose a novel methodology called block-diagonalization, which provides a common decomposition form for our matrices, and similarly a common procedure for block-encoding these matrices inside a unitary operator of a quantum circuit. The depth of these circuits is double-logarithmic in the matrix size, which is an exponential improvement over existing quantum methods and a superexponential improvement over existing classical methods. These improvements come at the price of a constant multiplicative overhead on the number of qubits and the number of gates. Combined with quantum linear solver algorithms, we can utilize these quantum circuits to produce a quantum state representation of the solution to the Poisson partial differential equations and their boundary-value problems.
Solving differential equations is one of the most computationally expensive problems in classical computing, occupying the vast majority of high-performance computing resources devoted towards practical applications in various fields of science and engineering. Despite recent progress made in the field of quantum computing and quantum algorithms, its end-to-end application towards practical realization still remains unattainable. In this article, we tackle one of the primary obstacles towards this ultimate objective, specifically the encoding of matrices derived via finite difference method solving Poisson partial differential equations in simple boundary-value problems. To that end, we propose a novel methodology called block-diagonalization, which provides a common decomposition form for our matrices, and similarly a common procedure for block-encoding these matrices inside a unitary operator of a quantum circuit. The depth of these circuits is double-logarithmic in the matrix size, which is an exponential improvement over existing quantum methods and a superexponential improvement over existing classical methods. These improvements come at the price of a constant multiplicative overhead on the number of qubits and the number of gates. Combined with quantum linear solver algorithms, we can utilize these quantum circuits to produce a quantum state representation of the solution to the Poisson partial differential equations and their boundary-value problems.
One of the most promising applications of quantum computers is solving partial differential equations (PDEs). By using the Schrödingerization technique—which converts nonconservative PDEs into Schrödinger equations—the problem can be reduced to Hamiltonian simulations. The particular class of Hamiltonians we consider is shown to be sufficient for simulating almost any linear PDE. In particular, these Hamiltonians consist of discretizations of polynomial products and sums of position and momentum operators. This paper addresses an important gap by efficiently loading these Hamiltonians into the quantum computer through block-encoding. The construction is explicit and efficient in terms of one- and two-qubit operations, forming a fundamental building block for constructing the unitary evolution operator for that class of Hamiltonians. The proposed algorithm demonstrates a squared logarithmic scaling with respect to the spatial partitioning size, offering a polynomial speedup over classical finite-difference methods in the context of spatial partitioning for solving PDEs. Furthermore, the algorithm is extended to the multidimensional case, achieving an exponential acceleration with respect to the number of dimensions, alleviating the curse of dimensionality problem. This work provides an essential foundation for developing explicit and efficient quantum circuits for PDEs, Hamiltonian simulations, and ground state and thermal state preparation.
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.
Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency ω multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a O(poly(n)) gate complexity, where n is the number of qubits, in the worst-case scenario used for banded matrices, and O(n) when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.
Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.
Many computational fluid dynamics (CFD) algorithms solve partial differential equations by discretization, resulting in large and sparse systems of linear equations. While iterative methods are widely used to solve these systems classically, most existing quantum linear system solvers target the solution through matrix inversion rather than approximating it using an iterative procedure. In this work, we develop a quantum implementation of the Jacobi method based on quantum singular value transformation (QSVT). By reformulating the Jacobi iteration as a polynomial transformation of a block-encoded operator, the algorithm requires only a constant ancilla overhead with respect to the number of iterations while maintaining a circuit depth that scales linearly with the iteration number. We demonstrate the algorithm for one- and two-dimensional Poisson problems, including the pressure Poisson equation arising in Chorin's projection method for the lid-driven cavity flow. The proposed algorithm provides a promising building block for future quantum implementations of multigrid methods and preconditioning techniques, bringing quantum algorithms closer to established CFD solution strategies.
The data input model is a fundamental component of every quantum algorithm, as its efficiency is crucial for achieving potential speed-ups over classical methods. For quantum linear algebra tasks that utilize quantum eigenvalue or singular value transformations, block encoding is the established technique for accessing matrix data. A key application of this is solving partial differential equations, where the Laplacian operator and its finite difference discretization serve as foundational examples. In this article, we present an efficient and explicit block encoding method for the Laplacian with periodic boundary conditions that enhances existing approaches in key aspects. We detail the construction of the quantum algorithm and illustrate how it leverages the unique structure of finite difference discretizations. Furthermore, we analytically derive the scaling of the sub-normalization factor and of the success probability of the block encoding with respect to the problem dimension, the grid width of the finite difference grid and the regularity of the exact solution, and we give resource estimates.