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?
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.
Also, you can check the so called process fidelity in Qiskit.
qiskit.org/documentation/stubs/qiskit.quantum_info.process_fidelity.html
Given two unitary operations $U$ and $V$, the process fidelity calculates $F(U,V) = \text{tr}(UV^\dagger)/d^2$, where $d$ is the space dimension.
Could you explain what's the difference between process fidelity and gate fidelity? Under what scenario would you take the process fidelity into consideration?
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 [1]:
$$
F_{pro}(\mathcal{E}, U) = \frac{1}{d^2}\sum_k |\text{tr}(U^\dagger V_k)|^2, \tag{4}
$$
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 [2], 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 [3]:
$$
F_{avg}(\mathcal{E}) \equiv F_{avg}(\mathcal{E}, I) = \frac{dF_e(\mathcal{E}) + 1}{d+1}. \tag{5}
$$
This metric can be further generalized into the gate entanglement fidelity $F_e(\mathcal{E}, U)$ in ref [4]. 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 [4]. 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 $||U|| \equiv \sqrt{\text{tr}U^\dagger U}$ [5].
3. Summary
Experimentally, these two metrics can both be utilized [6]. 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).
References:
[1] Fortunato, Evan M., et al. "Design of strongly modulating pulses to implement precise effective Hamiltonians for quantum information processing." The Journal of chemical physics 116.17 (2002): 7599-7606.
[2] Nielsen, Michael A. "A simple formula for the average gate fidelity of a quantum dynamical operation." Physics Letters A 303.4 (2002): 249-252.
[3] Horodecki, Michał, Paweł Horodecki, and Ryszard Horodecki. "General teleportation channel, singlet fraction, and quasidistillation." Physical Review A 60.3 (1999): 1888.
[4] Fortunato, Evan M., et al. "Implementation of universal control on a decoherence-free qubit." New Journal of Physics 4.1 (2002): 5.
[5] Wu, Re-Bing, et al. "Data-driven gradient algorithm for high-precision quantum control." Physical Review A 97.4 (2018): 042122.
[6] Chow, J. M., et al. "Randomized benchmarking and process tomography for gate errors in a solid-state qubit." Physical Review Letters 102.9 (2009): 090502.