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

Yu Wang

TL;DR

This Letter introduces a quantum approach based on qudit classical shadow tomography, significantly reducing computational complexity from O(d^{2}) down to O[poly(logd)] in typical cases and at least to O[dpoly(logd)] in the worst case.

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 \(ρ\) and arbitrary BN-observable \( O \), we show that \(\operatorname{tr}(ρO)\) 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.

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

TL;DR

This Letter introduces a quantum approach based on qudit classical shadow tomography, significantly reducing computational complexity from O(d^{2}) down to O[poly(logd)] in typical cases and at least to O[dpoly(logd)] in the worst case.

Abstract

The computation of \(\operatorname{tr}(AB)\) is essential in quantum science and artificial intelligence, yet classical methods for -dimensional matrices and 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 and Hermitian matrices 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 -qubit stabilizer state and arbitrary BN-observable , we show that \(\operatorname{tr}(ρO)\) 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 . These advances open new avenues for efficient high-dimensional data analysis and modeling.
Paper Structure (16 sections, 5 theorems, 94 equations, 3 figures)

This paper contains 16 sections, 5 theorems, 94 equations, 3 figures.

Table of Contents

  1. Classical shadow tomography framework with random Clifford measurements
  2. Efficient random projections in DDB-ST versus Clifford-ST
  3. Classical shadow tomography with random projective measurements or a single POVM.
  4. Clifford circuits versus DDB circuits.
  5. Preprocessing complexity from the POVM perspective.
  6. Preprocessing cost of sampling DDB projectors.
  7. Proof of Theorem 1: calculation of the reconstruction channel
  8. Proof of Proposition 1: constant-time post-processing and storage analysis in a single measurement for DDB-ST
  9. Proof of theoreom 2: performance guarantee
  10. Upper bound for the worst case
  11. Average performance analysis
  12. Proof of lemma 1: approximate average state case
  13. More exact variance formula
  14. Efficient estimation of stabilizer states with DDB-ST
  15. Proof of Theorem 3
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\rho = \sum_{j,k=0}^{d-1} \rho_{jk} |j\rangle\langle k|$, and define $P_k = |k\rangle\langle k|$ as the projectors onto the computational basis. The corresponding quantum channel $\mathcal{M}$ takes the form: Its inverse reconstruction channel $\mathcal{M}^{-1}$ is given by:

Figures (3)

  • Figure 1: Overview of the proposed framework. (a) Pipeline from inputs $\{(\rho,O)\}$ to the estimation of $\mathrm{tr}(\rho O)$. (b) Comparison of complexities. For dense $d{\times}d$ matrices, classical storage and direct evaluation cost $O(d^2)$. Throughout, the observable $O$ is assumed to be specified by its matrix elements in the computational basis.
  • Figure 2: Proportion of approximately DDB-average states.
  • Figure 3: Classical vs quantum estimation pipeline: enabling advantage with DDB-ST. DDB-ST offers exponential or near-quadratic speedup in the estimation stage, bridging the gap between quantum state outputs and downstream applications in AI, optimization, and scientific computing rao1973linearhorn2012matrix. Although a quantum state $\rho$ has limited lifetime, it could be compactly stored as polynomial-size classical data via well-designed random measurements by classical shadow tomography.

Theorems & Definitions (9)

  • Theorem 1: Reconstruction channel for DDB-ST
  • Theorem 2: Performance Guarantee
  • Definition 1: Approximately DDB-Average State
  • Lemma 1
  • Theorem 3: Informal
  • Remark : Storage complexity
  • Remark : Representation of observables
  • Lemma 2
  • proof