The Hadamard test is a widely used routine in quantum computing to compute the real and imaginary part of expectation values of unitary operators. However, all papers I have come across in the literature that make use of the Hadamard test fail to cite its original reference (cf., e.g., this paper by Mitarai and Fujii or this pedagogical Los Alamos review). The Wikipedia page on the Hadamard test only refers to this paper by Aharonov, Jones, and Landau, which includes the passage "the well known Hadamard test" without referring the original reference once more.

What is the paper where the Hadamard test is originally introduced?

  • $\begingroup$ If you look up "Hadamard test" on scholar.google.com, you'll find that this is also a term used in mathematics. (The eponymous Hadamard was a mathematician). I'm not sure if the quantum algorithm use of this term and the math use of this term are related. In any case, this may also complicate finding the first usage of the term. $\endgroup$ Nov 17, 2022 at 20:54

1 Answer 1


I believe I have found the original reference to the Hadamard test: It is "Quantum Algorithms Revisited" by R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca.


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