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PreprintCC BY 4.0arXiv

Conditioning-Free Non-Uniform Quantum Fourier and Chebyshev Transforms

  • Chaowen Guan
  • Akshit Katiyar
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Abstract

We present an efficient quantum algorithm for the non-uniform Chebyshev transform. It is defined as the projection of a function onto Chebyshev polynomials sampled at given nodes that are uniform in x∈[−1,1], and hence non-uniform in the angle θ=arccos x, a setting that QFT-based quantum Chebyshev transforms cannot handle. Our construction is based on an improvement of an existing Non-uniform Quantum Fourier Transform (NUQFT) whereby we remove the conditioning from non-uniform node sampling. Hence, error bounds are independent of the geometry-dependent parameter κ of prior work. We use the fact that Chebyshev transform matrix is the average of two Type-II non-uniform DFTs, which we implement with a single controlled NUQFT circuit. We provide explicit oracle constructions, including the row-access oracle previously left as an assumption. The resulting ε-accurate block encoding has O(1) normalization and uses O(L) qubits and Õ(L2) gates, where L=log N+log(1/ε). We give an end-to-end implementation with complexity analysis, including the success probability and output-state error.

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Cite

BibTeXRIS
BibTeX
@misc{guan2026conditioningfree,
  title = {{Conditioning-Free Non-Uniform Quantum Fourier and Chebyshev Transforms}},
  author = {Chaowen Guan and Akshit Katiyar},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.40277},
  eprint = {2609.40277},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40277},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Conditioning-Free Non-Uniform Quantum Fourier and Chebyshev Transforms
AU  - Chaowen Guan
AU  - Akshit Katiyar
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.40277
UR  - https://arxiv.org/abs/2609.40277
AB  - We present an efficient quantum algorithm for the non-uniform Chebyshev transform. It is defined as the projection of a function onto Chebyshev polynomials sampled at given nodes that are uniform in x∈[−1,1], and hence non-uniform in the angle θ=arccos x, a setting that QFT-based quantum Chebyshev transforms cannot handle. Our construction is based on an improvement of an existing Non-uniform Quantum Fourier Transform (NUQFT) whereby we remove the conditioning from non-uniform node sampling. Hence, error bounds are independent of the geometry-dependent parameter κ of prior work. We use the fact that Chebyshev transform matrix is the average of two Type-II non-uniform DFTs, which we implement with a single controlled NUQFT circuit. We provide explicit oracle constructions, including the row-access oracle previously left as an assumption. The resulting ε-accurate block encoding has O(1) normalization and uses O(L) qubits and Õ(L^2) gates, where L=log N+log(1/ε). We give an end-to-end implementation with complexity analysis, including the success probability and output-state error.
ER  -