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

TL;DR

This paper introduces a quantum method using qudit shadow tomography that drastically reduces computational complexity for estimating traces of large matrices, enabling efficient analysis of high-dimensional quantum data.

Contribution

It presents a novel quantum approach that lowers complexity from quadratic to polylogarithmic in dimension for trace estimation, with broad applicability.

Findings

Achieves quadratic and exponential speedups in trace computations.

Efficiently estimates ext{tr}( ho O) for stabilizer states and BN-observables.

Reduces post-processing complexity in shadow tomography.

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.