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.

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doi.org/10.4230/LIPIcs.TQC.2021.9
Leibniz International Proceedings in Informatics
Quantum Software Consortium
16th Conference on the Theory of Quantum Computation, Communication and Cryptography, TQC 2021
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

van Apeldoorn, J.T.S. (2021). Quantum probability oracles & multidimensional amplitude estimation. In Conference on the Theory of Quantum Computation, Communication and Cryptography (pp. 9:1–9:11). doi:10.4230/LIPIcs.TQC.2021.9