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Optimal qudit overlapping tomography and optimal measurement order

Shuowei Ma, Qianfan Wang, Lvzhou Li, Fei Shi

TL;DR

The paper addresses efficient tomography for high-dimensional quantum systems by introducing optimal qudit overlapping tomography based on generalized Gell-Mann matrices. It reveals a precise link between minimal qudit measurement schemes and covering arrays, providing two explicit CA-based constructions (Zero-Sum and Bush) that achieve optimality in key regimes. For $n$-qutrits, it proves an upper bound $\\phi_2(n,3)\\le 8+56\\left\\lceil \\log_{8} n \\rceil$ with an explicit scheme, and presents an algorithmic approach to minimize measurement switching costs, achieving about a $50\%$ reduction on representative cases. These results offer a practical pathway to efficiently characterize qudit systems, with significant implications for quantum communication and computation and potential extensions to mixed-dimension systems.

Abstract

Quantum state tomography is essential for characterizing quantum systems, but it becomes infeasible for large systems due to exponential resource scaling. Overlapping tomography addresses this challenge by reconstructing all $k$-body marginals using few measurement settings, enabling the efficient extraction of key information for many quantum tasks. While optimal schemes are known for qubits, the extension to higher-dimensional qudit systems remains largely unexplored. Here, we investigate optimal qudit overlapping tomography, constructing local measurement settings from generalized Gell-Mann matrices. By establishing a correspondence with combinatorial covering arrays, we present two explicit constructions of optimal measurement schemes. For $n$-qutrit systems, we prove that pairwise tomography requires at most $8 + 56\left\lceil \log_{8} n \right\rceil$ measurement settings, and provide an explicit scheme achieving this bound. Furthermore, we develop an efficient algorithm to determine the optimal order of these measurement settings, minimizing the experimental overhead associated with switching configurations. Compared to the worst-case ordering, our optimized schedule reduces switching costs by approximately 50\%. These results provide a practical pathway for efficient characterization of qudit systems, facilitating their application in quantum communication and computation.

Optimal qudit overlapping tomography and optimal measurement order

TL;DR

The paper addresses efficient tomography for high-dimensional quantum systems by introducing optimal qudit overlapping tomography based on generalized Gell-Mann matrices. It reveals a precise link between minimal qudit measurement schemes and covering arrays, providing two explicit CA-based constructions (Zero-Sum and Bush) that achieve optimality in key regimes. For -qutrits, it proves an upper bound with an explicit scheme, and presents an algorithmic approach to minimize measurement switching costs, achieving about a reduction on representative cases. These results offer a practical pathway to efficiently characterize qudit systems, with significant implications for quantum communication and computation and potential extensions to mixed-dimension systems.

Abstract

Quantum state tomography is essential for characterizing quantum systems, but it becomes infeasible for large systems due to exponential resource scaling. Overlapping tomography addresses this challenge by reconstructing all -body marginals using few measurement settings, enabling the efficient extraction of key information for many quantum tasks. While optimal schemes are known for qubits, the extension to higher-dimensional qudit systems remains largely unexplored. Here, we investigate optimal qudit overlapping tomography, constructing local measurement settings from generalized Gell-Mann matrices. By establishing a correspondence with combinatorial covering arrays, we present two explicit constructions of optimal measurement schemes. For -qutrit systems, we prove that pairwise tomography requires at most measurement settings, and provide an explicit scheme achieving this bound. Furthermore, we develop an efficient algorithm to determine the optimal order of these measurement settings, minimizing the experimental overhead associated with switching configurations. Compared to the worst-case ordering, our optimized schedule reduces switching costs by approximately 50\%. These results provide a practical pathway for efficient characterization of qudit systems, facilitating their application in quantum communication and computation.
Paper Structure (8 sections, 4 theorems, 15 equations, 1 figure, 3 tables, 4 algorithms)

This paper contains 8 sections, 4 theorems, 15 equations, 1 figure, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Optimal qudit overlapping tomography and covering array
  3. Constructions of optimal qudit overlapping tomography
  4. upper bounds on optimal qudit overlapping tomography
  5. Optimal measurement order
  6. Conclusions
  7. $\mathop{\rm CA}(64;2,8,8)$
  8. Optimal measurement order algorithm

Key Result

Theorem 1

$\phi_{k}(k+1,d) = (d^2 - 1)^k$.

Figures (1)

  • Figure 1: Comparison between the optimal measurement order and the worst measurement order. (a). The red line represents the minimum cost achieved by the optimal measurement order, while the blue line represents the maximum cost corresponding to the worst measurement order, as the number of qudits increases from $4$ to $27$. The yellow region indicates the portion of cost that is reduced through optimization. (b) The green line represents the optimization rate, which is calculated as $\frac{\mathrm{Maximum\ cost} - \mathrm{Minimum\ cost}}{\mathrm{Maximum\ cost}} \times 100\%$.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Definition 3