Computing inner products of the form $\mathrm{tr}(AB)$, where $A$ is a $d$-dimensional density matrix [with $\mathrm{tr}(A)=1$, $A\ge 0$] and $B$ is a bounded-norm observable [Hermitian with $\mathrm{tr}(B^2)\le O[\mathrm{poly}(\log d)]$ and $\mathrm{tr}(B)$ 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 Letter, 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 $O[d\mathrm{poly}(\log d)]$ 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 $\mathrm{tr}(\rho O)$ for any known stabilizer state $\rho$ and arbitrary bounded-norm observable $O$, using polynomial computational resources. Crucially, it ensures constant-time classical postprocessing 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.

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• Quantum algorithms & computation
• Quantum information theory
• Qudits