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PreprintCC BY 4.0arXiv

Low-Depth Random Unitaries without Ancillae

  • Zhenyu Du
  • Siyuan Cheng
  • Xiongfeng Ma
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Abstract

Random unitaries are fundamental to quantum information and many-body physics, with widespread applications ranging from quantum learning and metrology to device benchmarking. A central pursuit is to minimize the space and circuit depth required to generate them. However, existing methods for generating low-depth random unitaries rely heavily on an extensive number of ancillary qubits, imposing severe spatial overhead. In this work, we prove that random unitaries can be generated in optimal depth without ancillae. For multiplicative-error approximate k-designs on n qubits, our circuits achieve a depth of Õ (k) (log n)1/δ on δ-dimensional architectures and Õ(k) log log n with all-to-all connectivity. Furthermore, by introducing a general exactification lemma, we lift our construction to optimal-depth exact k-designs, yielding an exponential resource reduction over state-of-the-art exact constructions. Our results minimize the space-time costs for a wide range of quantum protocols.

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Cite

BibTeXRIS
BibTeX
@misc{du2026lowdepth,
  title = {{Low-Depth Random Unitaries without Ancillae}},
  author = {Zhenyu Du and Siyuan Cheng and Xiongfeng Ma},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06528},
  eprint = {2609.06528},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06528},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Low-Depth Random Unitaries without Ancillae
AU  - Zhenyu Du
AU  - Siyuan Cheng
AU  - Xiongfeng Ma
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06528
UR  - https://arxiv.org/abs/2609.06528
AB  - Random unitaries are fundamental to quantum information and many-body physics, with widespread applications ranging from quantum learning and metrology to device benchmarking. A central pursuit is to minimize the space and circuit depth required to generate them. However, existing methods for generating low-depth random unitaries rely heavily on an extensive number of ancillary qubits, imposing severe spatial overhead. In this work, we prove that random unitaries can be generated in optimal depth without ancillae. For multiplicative-error approximate k-designs on n qubits, our circuits achieve a depth of Õ (k) (log n)^1/δ on δ-dimensional architectures and Õ(k) log log n with all-to-all connectivity. Furthermore, by introducing a general exactification lemma, we lift our construction to optimal-depth exact k-designs, yielding an exponential resource reduction over state-of-the-art exact constructions. Our results minimize the space-time costs for a wide range of quantum protocols.
ER  -