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arXiv:2605.16749

Boundary-Aware QFT Block-Encoding of Fractional Laplacians

Younes Javanmard, Sina Kazemian

quant-phSubmitted May 16, 2026• Announced May 19, 2026• 0 citations
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Abstract

We study the quantum Fourier transform (QFT) block-encoding of the semi-discrete fractional Laplacian on bounded domains with open, zero-extension boundary conditions. In the notation of the main construction, the target operator is the finite Toeplitz truncation \(A^{(N)}_{α,h}\) obtained from the full-lattice semi-discrete operator with symbol \(|ξ|^α\). A finite QFT register, however, diagonalizes circulant matrices rather than Toeplitz truncations. The native QFT circuit therefore implements a periodic surrogate \(\widetilde A^{(N)}_{α,h}\), not the open-boundary operator. We identify this mismatch through an exact Toeplitz-to-circulant aliasing identity. To recover the open-boundary action, we zero-pad the state into a larger \(M\)-point QFT register, apply the same Fourier-symbol block-encoding, and compress back to the physical subspace. The resulting compressed block satisfies \(P_{N\to M}^{\dagger}\widetilde A^{(M)}_{α,h}P_{N\to M} = A^{(N)}_{α,h}+E^{(M)}\), where \(E^{(M)}\) is controlled by the tail of the semi-discrete convolution kernel. Thus, the QFT layer implements the fractional symbol, while zero-padding supplies the open-boundary geometry. The construction is an operator-compilation primitive for boundary-aware quantum simulation rather than a complete PDE solver.

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We study the quantum Fourier transform (QFT) block-encoding of the semi-discrete fractional Laplacian on bounded domains with open, zero-extension boundary conditions. In the notation of the main construction, the target operator is the finite Toeplitz truncation (A^{(N)}{α,h}) obtained from the full-lattice semi-discrete operator with symbol (|ξ|^α). A finite QFT register, however, diagonalizes circulant matrices rather than Toeplitz truncations. The native QFT circuit therefore implements a periodic surrogate (\widetilde A^{(N)}{α,h}), not the open-boundary operator. We identify this mismatch through an exact Toeplitz-to-circulant aliasing identity.
To recover the open-boundary action, we zero-pad the state into a larger (M)-point QFT register, apply the same Fourier-symbol block-encoding, and compress back to the physical subspace. The resulting compressed block satisfies (P_{N\to M}^{\dagger}\widetilde A^{(M)}{α,h}P{N\to M} = A^{(N)}_{α,h}+E^{(M)}), where (E^{(M)}) is controlled by the tail of the semi-discrete convolution kernel. Thus, the QFT layer implements the fractional symbol, while zero-padding supplies the open-boundary geometry. The construction is an operator-compilation primitive for boundary-aware quantum simulation rather than a complete PDE solver.
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.
We present quantum algorithms to approximate the exponential of banded Toeplitz matrices. These matrices have central importance in many PDE related problems, but their quantum implementation is hindered by their possibly large norms. Constructing directly the exponential we can circumvent this limitation. By relating the lower/upper shift operators to circulant and skewcirculant generators, which are diagonalised by the QFT, we construct (i) an LCU-based block encoding of banded Toeplitz matrices, (ii) an efficient, controllably truncated Pauli-string decomposition of the circulant eigenphases with a closed-form error bound, and (iii) a specialized QFT-Trotter product formula. As an application we build a block encoding of the propagator of the discretised heat equation with periodic, Dirichlet and Neumann boundary conditions, using a frequency cutoff.
Quantum computing holds the promise of solving computational mechanics problems in polylogarithmic time, meaning computational time scales as O((log⁡N)c)\mathscr{O}((\log N)^c)O((logN)c), where NNN is the problem size and ccc a constant. We propose a quantum spectral method with polylogarithmic complexity for solving non-periodic boundary value problems with arbitrary Dirichlet boundary conditions. Our method extends the recently proposed approach by Liu et al. (2025), in which periodic problems are discretised using truncated Fourier series. In such spectral methods, the discretisation of boundary value problems with constant coefficients leads to a set of algebraic equations in the Fourier space. We implement the respective diagonal solution operator by first approximating it with a polynomial and then quantum encoding the polynomial. The mapping between the physical and Fourier spaces is accomplished using the quantum Fourier transform (QFT). To impose zero Dirichlet boundary conditions, we double the domain size and reflect all physical fields antisymmetrically. The respective reflection matrix defines the quantum sine transform (QST) by pre- and post-multiplying with the QFT. For non-zero Dirichlet boundary conditions, the solution is decomposed into a boundary-conforming and a homogeneous part. The homogenous part is determined by solving a problem with a suitably modified forcing vector. We illustrate the basic approach with a Dirichlet-Poisson problem and demonstrate its generality by applying it to a fractional stochastic PDE for modelling spatial random fields. We discuss the circuit implementation of the proposed approach and provide numerical evidence confirming its polylogarithmic complexity.
This paper presents a quantum algorithm for solving the fractional Poisson equation (−Δ)su=f(-Δ)^s u = f(−Δ)su=f with s∈(0,1)s \in (0,1)s∈(0,1) on bounded domains. The proposed approach combines rational approximation techniques with quantum linear system solvers to achieve exponential quantum advantage. The rational approximation represents the inverse fractional Laplacian as a weighted sum of standard resolvents, transforming the original nonlocal problem into a collection of shifted integer-order partial differential equations. These equations are consolidated into a single large linear system through a modified right-hand side construction that simplifies the quantum implementation. To enable practical implementation, we develop explicit quantum circuits via the Schrödingerization technique, which converts the non-unitary dynamics of the linear system into a higher-dimensional Schrödinger-type equation, allowing the use of standard Hamiltonian simulation. The circuit construction leverages the decomposition of shift operators to realize the discrete Laplacian and employs controlled operations to implement the select oracle. Under finite difference discretization, we provide detailed algorithmic procedures utilizing block-encoding techniques for the coefficient matrices. A comprehensive complexity analysis demonstrates that the quantum algorithm achieves a dependence on the inverse mesh size h−1h^{-1}h−1 that is independent of the spatial dimension ddd, in stark contrast to classical methods which suffer from exponential growth in high dimensions. This establishes an exponential quantum advantage for high-dimensional fractional problems, effectively overcoming the curse of dimensionality that limits classical approaches.
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.
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.