# Demystifying Quantum Gate Fidelity for Electronics Engineers
## 5 Brief Case Studies
### N-Qubit Quantum Gate  
Equations (60) and (61) suggest that the entanglement fidelity \(F_{e,NQUBIT}\) of an n-qubit quantum gate can be approximated as follows for small values of \(\varepsilon\):  
\[\mathbf{F_{e,NQUBIT}}\cong\mathbf{1}-\left(\frac{\varepsilon}{\mathbf{n}}\right)^{2} \tag{62}\]  
The substitution of Equation (57) into Equation (62) yields:  
\[\frac{\left|\mathbf{Tr}\{A_{I}^{\dagger}A_{R}\}\right|^{2}}{\mathbf{n}^{2}}=\mathbf{1}- \left(\frac{\varepsilon}{\mathbf{n}}\right)^{2} \tag{63}\]and thus:  
\[\left|\mathbf{Tr}\Big{\{}A_{I}^{\dagger}A_{R}\Big{\}}\right|^{2}=n^{2}-\varepsilon^{2} \tag{64}\]  
By adding \(n\) on both sides, you obtain:  
\[\left|\mathbf{Tr}\Big{\{}A_{I}^{\dagger}A_{R}\Big{\}}\right|^{2}+n=n+n^{2}- \varepsilon^{2} \tag{65}\]  
from which:  
\[\frac{\left|\mathbf{Tr}\big{\{}A_{I}^{\dagger}A_{R}\big{\}}\right|^{2}+n}{n+n^{2} }=\mathbf{1}-\frac{\varepsilon^{2}}{n+n^{2}} \tag{66}\]  
By referring to Equation (42), Equation (66) can be written in a shorter form:  
\[\mathbf{F}=\mathbf{1}-\frac{\varepsilon^{2}}{n+n^{2}} \tag{67}\]  
from which:  
\[\varepsilon=\sqrt{n(n+1)(\mathbf{1}-\mathbf{F})} \tag{68}\]  
Equation (68) is identical to the mathematical expression of the upper bound error claimed in [24] for an n-qubit quantum gate. It should be pointed out that the authors in [24] propose the Pauli distance as tool for a more accurate investigation of quantum gates.  
For the sake of convenience, Table 1 summarizes the main obtained mathematical expressions for fidelity and entanglement fidelity.  
Table 2 reports some experimental values of quantum fidelity for one-qubit and two-qubit quantum gates. It shows that achieving higher fidelities is simpler for one-qubit than for two-qubit quantum gates. On the other hand, Equation (67) implies that, for a given error, the fidelity increases with increasing \(n\). This comparison suggests that actual quantum gates also suffer from environmental decoherence [25]. Equation (42), of which Equation (67) was demonstrated to be a consequence, can indeed account for coherent sources of error only, as it holds for pure states [25].  
\begin{table}
\begin{tabular}{c c c c} \hline
**Quantum Gate** & \(\mathbf{F}\) & \(\mathbf{F_{\varepsilon}}\) & \(\mathbf{F_{\varepsilon}(\ll\)**1)** \\ \hline One-qubit (Pauli X) & \(\frac{\mathbf{1}+2 cos^{2}\left(\frac{\varepsilon}{2}\right)}{3}\) & \(\mathbf{1}-\mathbf{sin^{2}\left(\frac{\varepsilon}{2}\right)}\) & \(\mathbf{1}-\frac{\varepsilon^{2}}{4}\) \\ \hline Two-qubit (CNOT) & \(\frac{\mathbf{1}+\left[1+cos\left(\frac{\varepsilon}{2}\right)\right]^{2}}{5}\) & \(\frac{\mathbf{1}-\frac{1}{2}sin^{2}\left(\frac{\varepsilon}{2}\right)+cos\left( \frac{\varepsilon}{2}\right)}{2}\) & \(\mathbf{1}-\frac{\varepsilon^{2}}{16}\) \\ \hline N-qubit & & & \(\mathbf{1}-\left(\frac{\varepsilon}{n}\right)^{2}\) \\ \hline \end{tabular}
\end{table}
Table 1: Mathematical expressions for fidelity and entanglement fidelity.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Number of Qubits** & **Qubit Technology** & **Fidelity [\%]** & **Reference** \\ \hline
1 & Superconducting & 99.9 & [46] \\ \hline
1 & Superconducting & 99.8 & [47] \\ \hline
1 & Superconducting & \textgreater{}99.9 & [48] \\ \hline
1 & Electron Spin & 99.9 & [49] \\ \hline
2 & Superconducting & 99.4 & [47] \\ \hline
2 & Superconducting & 99.4 & [46] \\ \hline
2 & Trapped ions & 99.3 & [47] \\ \hline
2 & Neutral atom & 99.5 & [37] \\ \hline \end{tabular}
\end{table}
Table 2: Experimental quantum fidelity values for one- and two-qubit quantum gates.