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PreprintCC BY 4.0arXiv

Quantum algorithms for the exponentiation of Toeplitz matrices and applications in partial differential equations

  • Xabier Gutiérrez
  • Nicola Mariella
  • Javier González-Conde
  • Sergiy Zhuk
  • Mikel Sanz
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{gutierrez2026quantum,
  title = {{Quantum algorithms for the exponentiation of Toeplitz matrices and applications in partial differential equations}},
  author = {Xabier Gutiérrez and Nicola Mariella and Javier González-Conde and Sergiy Zhuk and Mikel Sanz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.31372},
  eprint = {2609.31372},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31372},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quantum algorithms for the exponentiation of Toeplitz matrices and applications in partial differential equations
AU  - Xabier Gutiérrez
AU  - Nicola Mariella
AU  - Javier González-Conde
AU  - Sergiy Zhuk
AU  - Mikel Sanz
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.31372
UR  - https://arxiv.org/abs/2609.31372
AB  - 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.
ER  -