What is fidelity in quantum computing?

Question

As I studied quantum computing, I saw term 'Fidelity' in many papers that related to quantum algorithm. So, I really wonder about following two things.

What is the real meaning of 'Fidelity' (As I searched, it is reliablity of thing)?

Answer 1

Fidelity can refer to either state fidelity or gate/unitary fidelity depending on the use case. In both cases, it is a number that answers the question "how similar is A to B?" on a scale from 0 (as different as possible) to 1 (the same thing). As user1271772 wrote, you can define the state fidelity between two states $\rho$ and $\sigma$ as

$$F = \Big|\text{tr}\big(\sqrt{\sqrt{\rho}\sigma \sqrt{\rho}}\big)\Big|^2,$$

which simplifies to

$$F_{pure} = \big|\langle \psi_{\rho} | \psi_{\sigma} \rangle \big|^2$$

for pure states (wikipedia).

You can define the gate fidelity between two $d \times d$ unitaries $U$ and $V$ as

$$F = \frac{1}{d^2} \Big|\text{tr}( U^\dagger V )\Big|^2.$$

This second fidelity is used to measure the quality of a quantum logic gate, and is used in e.g. quantum optimal control as part of the objective function. You can think of the gate fidelity as a sort of average of the state fidelities over the output states of $U$ and $V$ acting on the standard basis, although it also takes into account the phases of these output states to make sure they are consistent with each other.

As a measure of reliability, you can compute the fidelity between an expected logical operation (such as an $X$ gate) and the actual operation that was performed. For example, if you want to perform a logical $X$ operation

$$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$$

but instead your quantum computer performs the operation

$$X' = \begin{pmatrix} 0.141 & 0.99 \\ 0.99 & -0.141 \end{pmatrix},$$

your $X$ gate fidelity would be $\frac{1}{d^2}|\text{tr}(X'^\dagger X)|^2 = 0.9801$.

Answer 2

"What is the real meaning of 'Fidelity' (As I searched, it is reliablity of thing)?"

Since "fidelity" means "reliability of a thing" in everyday (not necessarily quantum) English, you have to search "fidelity quantum" to get the answer you are seeking.

Repeating what you'll find in the second paragraph of that article, a quantum state $\rho$ has the following fidelity with respect to another quantum state $\sigma$:

$$ F = \left(\textrm{tr}\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}\right)^2.\tag{1} $$