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arXiv:2605.04861

Quantum algorithm for solving differential equations using SLAC derivatives

Rakshit M. Gharat, Gopikrishnan Muraleedharan, Dominic W. Berry, Gavin K. Brennen

quant-phSubmitted May 6, 2026• Announced May 7, 2026• 0 citations• Revised May 25, 2026 (v2)
v1 · May 6, 2026v2 · May 25, 2026
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Abstract

In numerical approaches to solving differential equations on a lattice, a representation of the derivative operator that correctly matches the continuum behaviour of functions of momentum up to the band limit must be non-local. We present the construction of efficient linear-combination-of-unitaries ($\mathrm{LCU}$)-based block-encodings for the first-order derivative and Laplacian operators in the non-local \(N=2^n\)-dimensional SLAC representation. We use state-preparation techniques designed for smoothly decaying functions to prepare the dense $\mathrm{LCU}$ amplitudes with high success probability and low gate cost. Furthermore, we demonstrate how Shannon wavelet transforms can be applied to these block-encodings to obtain multiscale representations of the SLAC derivative operators. We then show how to apply a diagonal preconditioner that reduces the condition number of these matrices in the multiscale wavelet basis to a small constant. This enables the solution of partial differential equations (PDEs) with SLAC-discretised derivative operators on a finite lattice using the quantum linear solving algorithm ($\mathrm{QLSA}$). For a $d$-dimensional PDE, after projection away from the nullspace, the resulting quantum linear-system algorithm has overall gate complexity ${O}(dn^3α^{(k)}\log(1/\varepsilon))$, where $α^{(k)}$ is the subnormalisation factor of the order-$k$ SLAC block-encoding and $\varepsilon$ denotes the algorithmic approximation error.

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37 Pages 9 Figures (Version 2: added new figures and expanded discussion)

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In numerical approaches to solving differential equations on a lattice, a representation of the derivative operator that correctly matches the continuum behaviour of functions of momentum up to the band limit must be non-local. We present the construction of efficient linear-combination-of-unitaries (LCU\mathrm{LCU}LCU)-based block-encodings for the first-order derivative and Laplacian operators in the non-local (N=2^n)-dimensional SLAC representation. We use state-preparation techniques designed for smoothly decaying functions to prepare the dense LCU\mathrm{LCU}LCU amplitudes with high success probability and low gate cost. Furthermore, we demonstrate how Shannon wavelet transforms can be applied to these block-encodings to obtain multiscale representations of the SLAC derivative operators. We then show how to apply a diagonal preconditioner that reduces the condition number of these matrices in the multiscale wavelet basis to a small constant. This enables the solution of partial differential equations (PDEs) with SLAC-discretised derivative operators on a finite lattice using the quantum linear solving algorithm (QLSA\mathrm{QLSA}QLSA). For a ddd-dimensional PDE, after projection away from the nullspace, the resulting quantum linear-system algorithm has overall gate complexity O(dn3α(k)log⁡(1/ε)){O}(dn^3α^{(k)}\log(1/\varepsilon))O(dn3α(k)log(1/ε)), where α(k)α^{(k)}α(k) is the subnormalisation factor of the order-kkk SLAC block-encoding and ε\varepsilonε denotes the algorithmic approximation error.
Partial differential equations (PDEs) are ubiquitous in science and engineering. Prior quantum algorithms for solving the system of linear algebraic equations obtained from discretizing a PDE have a computational complexity that scales at least linearly with the condition number κκκ of the matrices involved in the computation. For many practical applications, κκκ scales polynomially with the size NNN of the matrices, rendering a polynomial complexity in NNN for these algorithms. Here we present a quantum algorithm with a complexity that is polylogarithmic in NNN but is independent of κκκ for a large class of PDEs. Our algorithm generates a quantum state from which features of the solution can be extracted. Central to our methodology is using a wavelet basis as an auxiliary system of coordinates in which the condition number of associated matrices becomes independent of NNN by a simple diagonal preconditioner. We present numerical simulations showing the effect of the wavelet preconditioner for several differential equations. Our work could provide a practical way to boost the performance of quantum simulation algorithms where standard methods are used for discretization.
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.
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