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PreprintCC BY 4.0arXiv

Three-dimensional QCA that are not stably circuits

  • Bowen Yang
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Abstract

We prove that the three-dimensional chiral Zp Clifford quantum cellular automata of Haah and of Sun--Yang--Wang--Tantivasadakarn--Chen are, for every odd prime p, not finite-depth quantum circuits, even after tensoring with arbitrary ancillary degrees of freedom and composing with translations. Previously such statements were known only inside the Clifford category. As a consequence, the group of three-dimensional QCA modulo stabilized circuits and translations is not finitely generated. The key ingredient is a new invariant of invertible subalgebras, which we conjecture to be the square of the total quantum dimension of the hosted anyon theory, modulo rational squares.

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Cite

BibTeXRIS
BibTeX
@misc{yang2026threedimensional,
  title = {{Three-dimensional QCA that are not stably circuits}},
  author = {Bowen Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40235},
  eprint = {2610.40235},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40235},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Three-dimensional QCA that are not stably circuits
AU  - Bowen Yang
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40235
UR  - https://arxiv.org/abs/2610.40235
AB  - We prove that the three-dimensional chiral Z_p Clifford quantum cellular automata of Haah and of Sun--Yang--Wang--Tantivasadakarn--Chen are, for every odd prime p, not finite-depth quantum circuits, even after tensoring with arbitrary ancillary degrees of freedom and composing with translations. Previously such statements were known only inside the Clifford category. As a consequence, the group of three-dimensional QCA modulo stabilized circuits and translations is not finitely generated. The key ingredient is a new invariant of invertible subalgebras, which we conjecture to be the square of the total quantum dimension of the hosted anyon theory, modulo rational squares.
ER  -