We develop a finite element method (FEM) for partial differential equation (PDE) solver based on the entanglement-basis representation introduced in our companion work. By lifting non-linear finite-element equations into an augmented coefficient space, the governing PDE together with boundary, initial, and inter-element constraints can be expressed through a unified quadratic residual minimization. Although this augmented space grows exponentially with the number of elements, its tensor-product structure allows it to be represented efficiently using tensor networks. Using the matrix product state (MPS) as a concrete example, we show that density matrix renormalization group (DMRG) sweeps enable element-by-element optimization without explicitly constructing the full augmented space. For bounded bond dimension, the resulting computational cost scales polynomially with the number of finite elements. We extend the framework to time-dependent problems through implicit temporal discretization and demonstrate convergence under both mesh and polynomial refinement using diffusion equations.
We present a quantum-inspired tensor-network framework for solving advection–diffusion–reaction (ADR) partial differential equations. Discretized solution fields are encoded as matrix product states (MPS), while differential operators are represented as matrix product operators (MPOs). Time integration is performed entirely in tensor-network form using explicit Euler updates with controlled truncation. The method is evaluated on one- and two-dimensional ADR problems and compared with high-accuracy Runge–Kutta reference solutions. Numerical results show that the proposed representation remains compact, stable, and accurate across a range of dynamical regimes. The solver captures both local solution profiles and global observables while maintaining small bond dimensions throughout the simulation. These results highlight the potential of tensor networks as efficient structure-preserving tools for PDE simulation in multiple spatial dimensions.
We present a quantum-inspired tensor-network framework for solving advection-diffusion-reaction (ADR) partial differential equations. Discretized solution fields are encoded as matrix product states (MPS), while differential operators are represented as matrix product operators (MPOs). Time integration is performed entirely in tensor-network form using explicit Euler updates with controlled truncation. The method is evaluated on one- and two-dimensional ADR problems and compared with high-accuracy Runge-Kutta reference solutions. Numerical results show that the proposed representation remains compact, stable, and accurate across a range of dynamical regimes. The solver captures both local solution profiles and global observables while maintaining small bond dimensions throughout the simulation. These results highlight the potential of tensor networks as efficient structure-preserving tools for PDE simulation in multiple spatial dimensions.
Classical numerical techniques for solving partial differential equations (PDEs) become computationally expensive as the dimension of the discretized differential operator increases. For PDEs giving rise to Sturm--Liouville problems, tensor network (TN) methods can be highly productive: an operator of dimension
N×N can be represented as a matrix product operator (MPO) using only
n=log2(N) qubits, enabling computation of eigenvalues and eigenvectors via imaginary time evolution (ITE). However, this remains computationally challenging. First, most methods for generating MPOs of large operators without explicit tensor-product structure require prohibitively large memory. Second, the number of Trotterization steps for convergence in conventional ITE increases rapidly with
n. We present techniques to mitigate both challenges for certain sparse, structured differential operators. To address the first, we construct the MPO by expanding the operator in the Pauli-string basis, enabled by an analytical expression for the Pauli basis coefficients that reduces the memory requirement from
O(2n+1) to
O(2n). To address the second, we propose a multistage state-refinement heuristic that accelerates ITE convergence, reducing convergence time by up to two orders of magnitude. Using this TN framework, we compute the first 32 eigenstates of a Laplacian of dimension exceeding
106 with fidelity above
0.95 using a 20-qubit MPO. We further validate the method on the 2D anharmonic oscillator and investigate disordered systems, where increasing random potential strength degrades accuracy and limits the approach.
Accurately solving high-dimensional partial differential equations (PDEs) remains a central challenge in computational mathematics. Traditional numerical methods, while effective in low-dimensional settings or on coarse grids, often struggle to deliver the precision required in practical applications. Recent machine learning-based approaches offer flexibility but frequently fall short in terms of accuracy and reliability, particularly in industrial contexts. In this work, we explore a quantum-inspired method based on quantized tensor trains (QTT), enabling efficient and accurate solutions to PDEs in a variety of challenging scenarios. Through several representative examples, we show that the QTT approach can achieve logarithmic scaling in memory and computational cost for linear PDEs when the relevant QTT ranks remain moderate. We also develop QTT space-time formulations that treat time as an additional dimension, allowing the full temporal evolution to be represented and solved globally rather than through sequential time stepping. For the nonlinear Burgers equation, we study both time-stepping and a frozen-coefficient space-time Picard scheme in QTT form, and report empirical convergence behavior on smooth one-dimensional viscous test problems. Additionally, we present a proof-of-concept data-driven workflow within the quantum-inspired framework, in which sampled source data are interpolated into QTT form and then incorporated directly into the structured PDE solver.
Operator equations (OEs) underpin quantitative modeling across science and engineering. Finite-element (FE) methods discretize continuous OEs into finite-dimensional algebraic systems, whereas tensor networks (TNs) provide flexible variational representations of correlated discrete systems. Here, we develop a framework that connects FE with TN for analytic OEs. The power of this method comes from its ability to convert highly non-linear partial differential equations into linear matrix equations. In particular, we show that FE discretization induces a hierarchy of multilinear interaction tensors, through which differential, integral, nonlinear, memory, and delay equations can be expressed within a common algebraic structure. The resulting systems are reformulated as weighted-residual optimization problems over TN degrees of freedom. Matrix-product-state calculations for one-dimensional linear and nonlinear diffusion reproduce conventional solutions with controlled error while preserving continuity and Neumann boundary conditions. The framework provides a common variational language for analytic OEs and establishes a direct connection between FE numerical formalism and TN variational algorithms, offering a general foundation for TN-based and quantum-inspired approaches to solving OEs.
We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions.