Quantum Probability Oracles & Multidimensional Amplitude Estimation

Author Joran van Apeldoorn

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Joran van Apeldoorn
  • Institute for Information Law, University of Amsterdam, The Netherlands
  • QuSoft, Centrum Wiskunde & Informatica, Amsterdam, The Netherlands


I would like to thank Ronald de Wolf, András Gilyén, Ashley Montanaro, and Srinivasan Arunachalam for insightful discussions and comments. I would also like to thank the TQC2021 reviewers for their useful comments.

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Joran van Apeldoorn. Quantum Probability Oracles & Multidimensional Amplitude Estimation. In 16th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 197, pp. 9:1-9:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We give a multidimensional version of amplitude estimation. Let p be an n-dimensional probability distribution which can be sampled from using a quantum circuit U_p. We show that all coordinates of p can be estimated up to error ε per coordinate using Õ(1/(ε)) applications of U_p and its inverse. This generalizes the normal amplitude estimation algorithm, which solves the problem for n = 2. Our results also imply a Õ(n/ε) query algorithm for 𝓁₁-norm (the total variation distance) estimation and a Õ(√n/ε) query algorithm for 𝓁₂-norm. We also show that these results are optimal up to logarithmic factors.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Quantum computation theory
  • quantum algorithms
  • amplitude estimation
  • monte carlo


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