A multistage state-refinement heuristic that accelerates ITE convergence, reducing convergence time by up to two orders of magnitude is proposed.
Abstract
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 $N\times N$ can be represented as a matrix product operator (MPO) using only $n=\log_2(N)$ 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 $\mathcal{O}(2^{n+1})$ to $\mathcal{O}(2n)$. 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 $10^6$ 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.
The corner transfer matrix renormalization group (CTMRG) is one of the standard contraction methods for infinite projected entangled-pair states (iPEPS), but its computational cost is dominated by repeated truncated singular value decompositions (SVDs). We introduce subspace-iteration CTMRG (SI-CTMRG), a QR-based projector construction that replaces each large-matrix SVD with an SVD of a much smaller matrix. The resulting algorithm shifts the dominant cost from decompositions to tensor contractions, making it highly suited to GPU acceleration and yielding speedups of up to two orders of magnitude over standard CTMRG. We demonstrate the efficiency and accuracy of the method for the triangular-lattice Heisenberg antiferromagnet, reaching state-of-the-art iPEPS results on a single H100 GPU in approximately 10 hours of computation.
The Koopman-von Neumann framework has been proposed to design quantum algorithms for non-linear dynamics. It maps a non-linear ordinary differential equation to a linear partial differential equation (PDE) governing a probability amplitude. Previous works represents this amplitude in the Hermite-function basis, equivalently as a bosonic state, and truncates the total Hermite degree to obtain a representation over $\Theta(m\log N)$ qubits, where $N$ is the number of variables and $m$ the truncation order. We extend this approach to a broader class of linear PDEs whose differential operators have a structured polynomial form. We prove convergence of the truncation for both time-dependent dynamics and gapped ground-state problems under explicit regularity and stability assumptions. We then introduce a qubit encoding that supports efficient block encodings of the truncated operators. Finally, we apply the framework to Bayesian inverse problems with Gaussian priors and observation noise, reducing posterior-state preparation to the preparation of a structured Hamiltonian's ground state.
Tensor networks are powerful formats for compressing large-scale data. However, their application to general data processing has been limited by the difficulty of performing nonlinear operations. Here, we introduce iterative tensor network transformations (ITNTs), a general algorithmic framework for the element-wise evaluation of elementary and nonlinear filtering functions on data encoded as tensor trains (TTs), a class of tensor networks. Our approach operates entirely in the compressed domain, enabling efficient computation on exponentially large datasets while maintaining a controlled computational cost. We demonstrate its power in two key areas: (I) evaluating highly nonlinear elementary and filtering functions on a 3D reactive flow field, enabling high-fidelity reaction rate computation and region filtering, and (II) finding extrema in complex optimization problems, such as solving Max-SAT instances on spaces up to $2^{70}$ configurations. These results establish ITNT as a foundational tool that provides tensor network methods with the capability for general-purpose data science and large-scale optimization.
Xiao Wang, Tomohiro Hashizume, Pia Siegl et al.· 2 citations
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.
Abhijatmedhi Chotrattanapituk, Michael J. Landry, Chuliang Fu et al.· 0 citations
The Variational Quantum Eigensolver (VQE) and its adaptive variants, such as ADAPT-VQE, are central to the study of strongly correlated quantum systems. However, the classical simulation of the ansatz growth process remains constrained by the exponential scaling of operator space and the associated computational cost of unitary evolution. We introduce the Greedy Operator Commutativity Partitioning (GOCP) framework, an analytical methodology designed to optimize both operator selection and state evolution. By reformulating complex unitary rotations as a chained sequence of fifth-order O(5) Taylor series expansions, GOCP bypasses the need for explicit matrix exponentiation, reducing the computational task to a sequence of sparse matrix-vector operations. We evaluate the performance of this framework across diverse molecular systems, including BeH2 and strongly correlated H2O geometries, utilizing both Jordan-Wigner and Bravyi-Kitaev mappings. Our results demonstrate that the GOCP framework maintains exceptional numerical fidelity-exceeding 1 - 10^-6 in state fidelity-while achieving sub-chemical accuracy in ground-state energy calculations. By enabling the simulation of operator manifolds exceeding 2.68 x 10^8 elements with high efficiency, this approach provides a scalable and rigorous pathway for exploring deep variational circuits in complex quantum many-body systems.