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Einstein Metrics and Quantum Control on Symmetric Tensor Product Quotients

  • Honghuai Fang
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Abstract

Let Mn=(PU(2)nrtimes Sn)backslashPU(2n) be the space of projective n-qubit unitaries modulo local unitaries and permutations of the qubits. We prove that for all sufficiently large n the manifold Mn carries exactly two PU(2n)-invariant Einstein metrics up to scaling. Both Einstein metrics have uniformly bounded diameter after normalizing two-qubit costs, although approximating a Haar-random unitary requires exponentially many two-qubit gates. In contrast, we construct non-Einstein metrics with nonnegative Ricci curvature and exponentially growing typical control distances, and determine the sharp curvature threshold for this family.

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Cite

BibTeXRIS
BibTeX
@misc{fang2026einstein,
  title = {{Einstein Metrics and Quantum Control on Symmetric Tensor Product Quotients}},
  author = {Honghuai Fang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38340},
  eprint = {2610.38340},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38340},
}
RIS
TY  - UNPB
TI  - Einstein Metrics and Quantum Control on Symmetric Tensor Product Quotients
AU  - Honghuai Fang
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38340
UR  - https://arxiv.org/abs/2610.38340
AB  - Let M_n=(PU(2)^nrtimes S_n)backslashPU(2^n) be the space of projective n-qubit unitaries modulo local unitaries and permutations of the qubits. We prove that for all sufficiently large n the manifold M_n carries exactly two PU(2^n)-invariant Einstein metrics up to scaling. Both Einstein metrics have uniformly bounded diameter after normalizing two-qubit costs, although approximating a Haar-random unitary requires exponentially many two-qubit gates. In contrast, we construct non-Einstein metrics with nonnegative Ricci curvature and exponentially growing typical control distances, and determine the sharp curvature threshold for this family.
ER  -