Classical Shadow Tomography with Mutually Unbiased Bases

Abstract: Classical shadow tomography, harnessing randomized informationally complete (IC) measurements, provides an effective avenue for predicting many properties of unknown quantum states with sample-efficient precision. Projections onto $2^n+1$ mutually unbiased bases (MUBs) are widely recognized as minimal and optimal IC measurements for full-state tomography. We study how to use MUBs circuits as the ensemble in classical shadow tomography. For the general observables, the variance to predict their expectation value is shown to be exponential to the number of qubits $n$. However, for a special class termed as appropriate MUBs-average (AMA) observables, the variance decreases to $poly(n)$. Additionally, we find that through biased sampling of MUBs circuits, the variance for non-AMA observables can again be reduced to $poly(n)$ with the MUBs-sparse condition. The performance and complexity of using the MUBs and Clifford circuits as the ensemble in the classical shadow tomography are compared in the end.