Periodic Clifford Ensembles Attain Adaptive Minimax Trace Norm Rates for Polynomial Spectral Decay Tomography

The authors derive uniform fourth-moment bounds on block projectors that hold for arbitrary entangled test states, enabling covariance concentration and positivity arguments to produce statewise oracle inequalities. These bounds depend only on each state's actual spectral tail. For admissible logarithmic block sizes the construction uses only logarithmic elementary gate depth while remaining nonadaptive and single-copy.

The same estimators automatically attain the minimax rate on every fixed-rank class, delivering adaptation that earlier nonadaptive schemes lacked. Lower bounds are obtained via a shared-rotation information argument on spectral-tail packings, showing the upper bounds are tight up to constants.

Polynomial-time reconstruction algorithms and Lean formalizations are supplied, together with numerical comparisons. The work therefore supplies both theoretical optimality and practical implementability for near-term quantum devices.

Next steps include hardware demonstrations on 5-10 qubit systems and extension to continuous-variable and noisy intermediate-scale settings.