Exact Clifford Optimality for Two-Sided Locally Randomized Classical Shadows

@misc{nie2026exact,
  title = {{Exact Clifford Optimality for Two-Sided Locally Randomized Classical Shadows}},
  author = {Xiaotian Nie and Tao Zhang and Yadong Wu and Linghui Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04621},
  eprint = {2610.04621},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04621},
}

TY  - UNPB
TI  - Exact Clifford Optimality for Two-Sided Locally Randomized Classical Shadows
AU  - Xiaotian Nie
AU  - Tao Zhang
AU  - Yadong Wu
AU  - Linghui Chen
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.04621
UR  - https://arxiv.org/abs/2610.04621
AB  - Entangling measurements can reduce the cost of learning many-body correlations. We consider classical shadows in which a fixed unitary on a k-qubit region is sandwiched between two layers of random single-qubit Clifford gates, and ask which middle unitary best estimates Pauli strings with full support on the region. Combining symplectic support counting with a constraint from operator-size parity, we prove that within this architecture the minimum squared shadow norm over all Clifford middle unitaries is C_k^opt=4· 9^k/(3· 5^k+3^k−6) for odd k and C_k^opt=4· 9^k/(3· 5^k−3^k+2) for even k. A dual-axis rotation circuit U_* attains this minimum for every k. Asymptotically C_k^opt≃(4/3)·(9/5)^k: the exponential base of Wu et al.'s contractive unitary is optimal, and the prefactor improves from $2$ to 4/3. For 1≤ k≤7, exact certificates show that U_* is optimal over the entire unitary group U(2^k), and we conjecture that this holds for every k.
ER  -