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.
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.
Discrete Laplacian operators arise ubiquitously in scientific computing and frequently appear in quantum algorithms for tasks such as linear algebra, Hamiltonian simulation, and partial differential equations. Block encoding provides the standard method for accessing matrix data within quantum circuits. Efficient implementations of such algorithms require efficient block encodings of the discretized operator. While several general-purpose techniques exist for block encoding arbitrary matrices, they usually require deep quantum circuits. Moreover, existing efficient constructions that exploit Laplacian structure are limited in scope, typically assuming fixed boundary conditions or uniform grid resolutions. In this work, we present a unified framework for efficiently block encoding finite-difference discretizations of the Laplacian that supports Dirichlet, periodic, and Neumann boundary conditions in arbitrary spatial dimensions. Our construction allows different boundary conditions and grid sizes to be specified independently along each coordinate axis, enabling mixed-boundary and anisotropic discretizations within a single modular circuit architecture. We provide analytical gate-complexity estimates and perform circuit-level benchmarks after transpilation to an IBM hardware gate set. Across one-, two-, and three-dimensional examples, the resulting circuits exhibit substantially lower gate counts and higher success probabilities when compared to certain existing approaches.
Block encoding is the standard technique for accessing matrix data in quantum linear-algebra algorithms. Its implementation directly affects its subnormalization, which in turn controls the algorithm's success probability, simulation time, and downstream costs. Explicit construction of block encodings with provably optimal subnormalization exists for only a handful of operators, with bespoke calculations used in its design. In this work, we develop a framework to block encode translation-invariant finite-difference operators on a periodic grid. These operators are the finite-difference discretizations of the constant-coefficient partial differential equations that sit at the core of scientific computing. We show that the moment order of these stencils can be used to simultaneously determine the continuum operator approximated, the vanishing order of the Fourier symbol, and the cost of the block encoding. From there, we derive a closed-form optimality criterion as a function of the stencil coefficients, which certifies whether the construction attains the optimal subnormalization for an entire operator family, uniformly in grid size, and quantifies the gap when it does not. The framework subsumes optimal constructions for the Laplacian operator in the literature and can be used to certify new instances at higher even orders, including the biharmonic operator. Furthermore, we derive success-probability floors parameterized by spectral properties of the operator's symbol and find explicit constants for the block encoding of the advection-diffusion family for which no prior explicit spatial block encoding exists.
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.
The biharmonic equation is a prototypical fourth-order partial differential equation whose high-dimensional discretization suffers from rapidly growing degrees of freedom and severe ill-conditioning. We develop QSVT--VTAA quantum linear-system algorithms by constructing explicit block-encodings tailored to periodic, simply supported, and Dirichlet--Neumann boundary conditions. For periodic and simply supported problems, Fourier and sine-transform diagonalizations yield augmented Poisson systems with the condition-number scaling of a second-order operator. For Dirichlet--Neumann problems, we introduce a second-order boundary-corrected finite-difference discretization, establish mesh-independent stability, and construct an explicit block-encoding of the resulting nonsymmetric matrix. We also formulate a coupled-Laplace system with additional boundary unknowns and characterize its complexity in terms of the condition number of the complete augmented matrix. The analysis covers discretization error, block-encoding normalization, gate complexity, and solution extraction under an amplitude-input and quantum-state-output model. Numerical experiments validate the proposed discretizations and the corresponding linear solves.
Abstract We propose an explicit quantum framework for numerically simulating general linear partial differential equations (PDEs), extending previous work (Guseynov et al 2025 Phys. Rev. Res. 7 033100) to incorporate (a) Robin boundary conditions—which include Neumann and Dirichlet conditions as special cases–(b) inhomogeneous terms, and (c) variable coefficients in space and time. Our approach begins with a general finite-difference discretization and applies the Schrödingerisation technique to transform the resulting system into one that admits unitary quantum evolution, enabling quantum simulation. For the Schrödinger equation corresponding to the discretized PDE, we construct an efficient block-encoding of the Hamiltonian H that scales polylogarithmically with the number of grid points N . This encoding is compatible with quantum signal processing and allows for the implementation of the evolution operator e − i H t . The explicit circuit construction in our method permits complexity to be measured in fundamental gate units–namely, CNOT gates and single-qubit rotations–bypassing the inefficiencies of oracle queries. Consequently, the overall algorithm scales polynomially with N and linearly with the spatial dimension d . Under certain input/output assumptions our method achieves a polynomial speedup in N and an exponential advantage in d for a wide class of PDEs, thereby mitigating the classical curse of dimensionality. The validity and efficiency of the proposed approach are further substantiated by numerical simulations. By explicitly defining the quantum operations and quantifying their resource requirements, our approach offers a practical alternative for numerically solving PDEs, distinct from others that rely on oracle queries and purely asymptotic scaling methods.
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.