Hadamard test (quantum computation)
In quantum computation, the Hadamard test is a method used to create a random variable whose expected value is the expected real part , where is a quantum state and is a unitary gate acting on the space of .  The Hadamard test produces a random variable whose image is in and whose expected value is exactly . It is possible to modify the circuit to produce a random variable whose expected value is .
Description of the circuit
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. Via a very simple modification it can be used to compute inner product between two states and : instead of starting from a state it suffice to start from the ground state , and perform two controlled operations on the ancilla qubit. Controlled on the ancilla register being , we apply the unitary that produces in the second register, and controlled on the ancilla register being in the state , we create in the second register. The expected value of the measurements of the ancilla qubits leads to an estimate of . The number of samples needed to estimate the expected value with absolute error is , because of a Chernoff bound. This value can be improved to using amplitude estimation techniques.
- 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.
- "quantumalgorithms.org - Hadamard test". Open Publishing. Retrieved 27 February 2022.
- "quantumalgorithms.org - Modified hadamard test". Open Publishing. Retrieved 27 February 2022.