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

Yu Wang

doi:10.1103/92ky-ln8f

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arXiv · 2408.16244

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

Yu Wang

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Abstract

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.

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Yu Wang. 2024-08-29. Quantum Advantage via Efficient Post-processing on Qudit Classical Shadow tomography. https://doi.org/10.1103/92ky-ln8f

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