Quantum Advantage via Efficient Post-processing on Qudit Classical Shadow tomography

The computation of $\operatorname{tr}(AB)$ is essential in quantum science and artificial intelligence, yet classical methods for $d$-dimensional matrices $A$ and $B$ require $O(d^2)$ complexity, which becomes infeasible for exponentially large systems. We introduce a quantum approach based on qudit shadow tomography that reduces both computational and storage complexities to $O(\text{poly}(\log d))$ in specific cases. The proposed method applies to quantum density matrices $A$ and Hermitian matrices $B$ with given $\operatorname{tr}(B)$ and $\operatorname{tr}(B^2)$ bounded by a constant (referred to as BN-observables). It guarantees at least a quadratic speedup ($O(d^2) \to O(d)$) in the worst case and achieves exponential speedup for approximately average cases. For any $n$-qubit stabilizer state $\rho$ and arbitrary BN-observable $O$, we show that $\operatorname{tr}(\rho O)$ can be efficiently estimated with $\text{poly}(n)$ computations. Moreover, our approach significantly reduces the post-processing complexity in shadow tomography using random Clifford measurements, and it is applicable to arbitrary dimensions $d$. These advances open new avenues for efficient high-dimensional data analysis and modeling.