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On Estimating the Quantum 𝓁_α Distance

Authors Yupan Liu , Qisheng Wang

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Quantum query complexity
  • Theory of computation → Quantum information theory
  • Theory of computation → Quantum complexity theory
Keywords
  • quantum algorithms
  • quantum state testing
  • trace distance
  • Schatten norm

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Abstract

We study the computational complexity of estimating the quantum 𝓁_α distance T_α(ρ₀,ρ₁), defined via the Schatten α-norm ‖A‖_α := tr(|A|^α)^{1/α}, given poly(n)-size state-preparation circuits of n-qubit quantum states ρ₀ and ρ₁. This quantity serves as a lower bound on the trace distance for α > 1. For any constant α > 1, we develop an efficient rank-independent quantum estimator for T_α(ρ₀,ρ₁) with time complexity poly(n), achieving an exponential speedup over the prior best results of exp(n) due to Wang, Guan, Liu, Zhang, and Ying (IEEE Trans. Inf. Theory 2024). Our improvement leverages efficiently computable uniform polynomial approximations of signed positive power functions within quantum singular value transformation, thereby eliminating the dependence on the rank of the states. 
Our quantum algorithm reveals a dichotomy in the computational complexity of the Quantum State Distinguishability Problem with Schatten α-norm (QSD_α), which involves deciding whether T_α(ρ₀,ρ₁) is at least 2/5 or at most 1/5. This dichotomy arises between the cases of constant α > 1 and α = 1:  
- For any 1+Ω(1) ≤ α ≤ O(1), QSD_α is BQP-complete. 
- For any 1 ≤ α ≤ 1+1/n, QSD_α is QSZK-complete, implying that no efficient quantum estimator for T_α(ρ₀,ρ₁) exists unless BQP = QSZK.  The hardness results follow from reductions based on new rank-dependent inequalities for the quantum 𝓁_α distance with 1 ≤ α ≤ ∞, which are of independent interest.

Cite As Get BibTex

Yupan Liu and Qisheng Wang. On Estimating the Quantum 𝓁_α Distance. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 106:1-106:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.106

Author Details

Yupan Liu
  • Graduate School of Mathematics, Nagoya University, Japan
Qisheng Wang
  • School of Informatics, University of Edinburgh, UK

Funding

  • Liu, Yupan: Supported in part by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) Quantum Leap Flagship Program (Q-LEAP) under Grant JPMXS0120319794, in part by Japan Science and Technology Agency (JST) Support for Pioneering Research Initiated by the Next Generation (SPRING) under Grant JPMJSP2125 and "THERS Make New Standards Program for the Next Generation Researchers", and in part by the Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) under Grant 24H00071.
  • Wang, Qisheng: Supported by the Engineering and Physical Sciences Research Council under Grant EP/X026167/1.

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