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Here is the definition I found:
The difference between an ideal gate (a logical gate) and what the corresponding physical gate that a quantum hardware offers is called gate infidelity.

What is the difference here? Is it the difference in computational accuracy?

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The gate fidelity formula is here qiskit.org/documentation/stubs/qiskit.quantum_info.average_gate_fidelity.html
You can think the stuff in the integral as the square of the state fidelity between $U|\psi\rangle$ and $\mathcal{E}(|\psi\rangle\langle\psi|)$, with $U$ the ideal gate, and $\mathcal{E}$ the noisy version, i.e. a quantum channel.

Then the gate fidleity is just the Haar average of that.

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Could you explain what's the difference between process fidelity and gate fidelity? Under what scenario would you take the process fidelity into consideration?

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Sure. Thanks for bringing this nice question! Basically, we have two kinds of fidelity here.

1. The average gate fidelity:
$$F_{avg}(\mathcal{E}, U) = \int \langle \psi|U^\dagger \mathcal{E}(|\psi\rangle\langle\psi|)U|\psi\rangle ~ d\psi, \tag{1}$$
which calculates the average state fidelity $F(U|\psi\rangle\langle\psi|U^\dagger, \mathcal{E}(|\psi\rangle\langle\psi|))$ among many input states $|\psi\rangle$ (Haar just means uniform in most cases). Note $U$ is the ideal unitary quantum gate and $\mathcal{E}$ is the noisy/experimental quantum operation in the language of quantum channels (i.e., trace-preserving completely positive maps as the generalization of quantum gates). A quantum channel can be expressed in the operator-sum representation:
$$\mathcal{E}(\rho) = \sum_k V_k\rho V_k^\dagger, \tag{2}$$
where {$V_k$} are called Kraus operators and satisfies the completeness condition $\sum_k V_k^\dagger V_k = I$. Recall the state fidelity $F$
$$F(\rho, \sigma) = \big(\text{tr} |\sqrt{\rho} \sqrt{\sigma}| \big)^2 = \bigg(\text{tr}\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}} \bigg)^2, \tag{3}$$
where the absolute value of an operator is defined as $|U| \equiv \sqrt{U^\dagger U}$. When we use Eq. (1) to experimentally determine the quality of our quantum control technique (how close is the actual quantum process $\mathcal{E}$ and the ideal process $U$), there is an input state dependence which is not handy when optimizing in the pulse level.

Note: If you want to know more about Haar measure, check the following Pennylane post.

2. The process fidelity
Let's first write down a more general expression for the process fidelity Eq. (10) from ref :
$$F_{pro}(\mathcal{E}, U) = \frac{1}{d^2}\sum_k |\text{tr}(U^\dagger V_k)|^2, \tag{4}$$
where {$V_k$} are the Kraus operators for the actual quantum process $\mathcal{E}$. This metric is originally known as the entanglement fidelity $F_e(\mathcal{E})$ of process $\mathcal{E}$ in the literature , which measures how well entanglement with other systems is preserved by the action of $\mathcal{E}$. A beautiful formula connects $F_{avg}$ and $F_e$ is originally given in ref :
$$F_{avg}(\mathcal{E}) \equiv F_{avg}(\mathcal{E}, I) = \frac{dF_e(\mathcal{E}) + 1}{d+1}. \tag{5}$$
This metric can be further generalized into the gate entanglement fidelity $F_e(\mathcal{E}, U)$ in ref . The authors further reduce the expression by setting the partial state into a fully mixed state $\rho = I/d$ and finally reach Eq. (4). For details, check IV B in ref . This expression is state independent and can be useful when optimizing the pulse level quantum control, although people working in this area often use $||U- V||^2$ as the metric (the operator norm defined as $||U|| \equiv \sqrt{\text{tr}U^\dagger U}$ .

3. Summary
Experimentally, these two metrics can both be utilized . However, the process fidelity usually requires Quantum Process Tomography (QPT) to obtain the characteristic $\chi$ matrix then compare $F_{pro} \propto \text{tr} \big[ \chi_{\text{ideal}^{-1}} \chi \big]$.

Overall, the key idea behind these two metrics is the same. When two unitary operations $U$ and $V$ are identical, we would expect $U^\dagger V=UV^\dagger=I_{d\times d}$. $\dagger$ can be understood as running the quantum circuit backward. Nowadays, the standard tool to charactrize gate errors in a quantum computer is Randomization benchmarking (RB).