Quantum Advantage via Efficient Post-processing on Qudit Shadow tomography

Abstract: Computing inner products of the form ( \operatorname{tr}(AB) ), where ( A ) is a ( d )-dimensional density matrix (with ( \operatorname{tr}(A) = 1 ), ( A \geq 0 )) and ( B ) is a bounded-norm observable (Hermitian with ( \operatorname{tr}(B^2) \le O(\mathrm{poly}(\log d)) ) and ( \operatorname{tr}(B) ) known), is fundamental across quantum science and artificial intelligence. Classically, both computing and storing such inner products require $O(d^2)$ resources, which rapidly becomes prohibitive as $d$ grows exponentially. In this work, we introduce a quantum approach based on qudit classical shadow tomography, significantly reducing computational complexity from $O(d^2)$ down to $O(\mathrm{poly}(\log d))$ in typical cases and at least to $O(d~ \text{poly}(\log d))$ in the worst case. Specifically, for (n)-qubit systems (with $n$ being the number of qubit and (d = 2^n)), our method guarantees efficient estimation of (\operatorname{tr}(\rho O)) for any known stabilizer state (\rho) and arbitrary bounded-norm observable (O), using polynomial computational resources. Crucially, it ensures constant-time classical post-processing per measurement and supports qubit and qudit platforms. Moreover, classical storage complexity of $A$ reduces from $O(d^2)$ to $O(m \log d)$, where the sample complexity $m$ is typically exponentially smaller than $d^2$. Our results establish a practical and modular quantum subroutine, enabling scalable quantum advantages in tasks involving high-dimensional data analysis and processing.