Quantum Channel Polynomial Processing

Abstract

We introduce a quantum algorithmic framework based on probabilistic mixtures of unitary channels that, similar to the framework of quantum singular value transformations, enables the application of arbitrary polynomials of hermitian operators onto arbitrary initial states. We show that our framework supports a flexible tradeoff between sample- and query complexity ranging from optimal query complexity, meaning logarithmic in the error, and exponentially scaling sample complexity to sub-polynomial query complexity in the error and polynomial sample complexity. Combined with the considerably lower quantum circuit complexity, compared to quantum singular value transformations with a linear combination of unitaries block encoding, we argue that our framework can be seamlessly scaled from NISQ to fault-tolerant quantum computing.

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BibTeX RIS

BibTeX

@misc{liu2026quantum,
  title = {{Quantum Channel Polynomial Processing}},
  author = {Tianhan Liu and Fedor Simkovic IV and Martin Leib},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.06557},
  eprint = {2607.06557},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.06557},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Quantum Channel Polynomial Processing
AU  - Tianhan Liu
AU  - Fedor Simkovic IV
AU  - Martin Leib
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2607.06557
UR  - https://arxiv.org/abs/2607.06557
AB  - We introduce a quantum algorithmic framework based on probabilistic mixtures of unitary channels that, similar to the framework of quantum singular value transformations, enables the application of arbitrary polynomials of hermitian operators onto arbitrary initial states. We show that our framework supports a flexible tradeoff between sample- and query complexity ranging from optimal query complexity, meaning logarithmic in the error, and exponentially scaling sample complexity to sub-polynomial query complexity in the error and polynomial sample complexity. Combined with the considerably lower quantum circuit complexity, compared to quantum singular value transformations with a linear combination of unitaries block encoding, we argue that our framework can be seamlessly scaled from NISQ to fault-tolerant quantum computing.
ER  -

Abstract

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