Efficient Pauli-decomposition and multistage state-refinement for tensor network based differential equation solver

• URL: http://arxiv.org/abs/2607.08166v1
• Date: Thu, 09 Jul 2026 07:10:01 GMT
• Title: Efficient Pauli-decomposition and multistage state-refinement for tensor network based differential equation solver
• Abstract summary: tensor network (TN) methods can be highly productive for PDEs giving rise to Sturm--Liouville problems.<n>Most methods for generating MPOs of large operators without explicit tensor-product structure require prohibitively large memory.<n>We present techniques to mitigate both challenges for certain sparse, structured differential operators.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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