Hadamard test (quantum computation)

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Hadamard test measure real.png

In quantum computation the Hadamard test is a method used to create a random variable whose expected value is the expected real part of the observed value of a state with respect to some unitary operator.[1]

Let be a state which can be efficiently generated, and let be a unitary gate. The Hadamard test produces a random variable whose image is in and whose expected value is exactly . A variant of the test produces a random variable whose expected value is .[1]

To perform the Hadamard test we first calculate the state . We then apply the unitary operator on conditioned on the first qubit to obtain the state . We then apply the Hadamard gate to the first qubit, yielding .

Measuring the first qubit, the result is with probability , in which case we output . The result is with probability , in which case we output . The expected value of the output will then be the difference between the two probabilities, which is

To obtain a random variable whose expectation is follow exactly the same procedure but start with .

The Hadamard test has many applications in quantum algorithms such as the Aharonov-Jones-Landau algorithm.

References[edit]

  1. ^ a b Dorit Aharonov Vaughan Jones, Zeph Landau (2009). "A Polynomial Quantum Algorithm for Approximating the Jones Polynomial". Algorithmica. 55 (3): 395–421. arXiv:quant-ph/0511096. doi:10.1007/s00453-008-9168-0.