# Demystifying Quantum Gate Fidelity for Electronics Engineers
## 8 Conclusions  
This paper provided a rationalization of the concept of quantum gate fidelity, starting from [PERSON]'s fidelity and drawing an analogy between a quantum gate and a transmission channel. This approach introduces quantum gate fidelity to electronics engineers in a more accessible manner, leveraging their general background. Bringing quantum fidelity back to information theory concepts facilitates understanding its integral form in Equation (11). In contrast, other authors, such as in [32], introduce this integral form abstractly and associate it with fidelity without any justification. This paper also proposed a step-by-step mathematical proof of Equation (42) using mathematical tools familiar to electronics engineers. Apart from some fundamentals of quantum mechanics and matrix analysis, for which reference textbooks [40; 41; 42; 43; 44] are suggested if needed, the proof primarily requires algebraic steps and hyperspace integrations (see Appendices B and C). Though longer than the one in [32], this proposed proof is significant, because Equation (42) is of relevance for RFIC designers. It indeed simplifies the quantum gate fidelity calculation from the abstract multi-dimensional integral in Equation (11) to the product between the matrices \(A_{I}\) and \(A_{R}\), describing, respectively, the wanted ideal and actual behavior of the quantum gate. In particular, the proof begins by applying the Toeplitz decomposition, focusing on the single integral form in Equation (19), from which Equation (28) follows. In contrast, in [32], the Toeplitz decomposition appears only in a later phase, and multi-dimensional integrals are just stated. A mathematical expression close to Equation (28) is reported in [32], relying on particularly simple matrices and abstract formulas, but lacking proof or useful references. To further clarify the use of Equation (42), Section 5 addressed case studies of Pauli X and CNOT quantum gates encoded in an electron spin, specifically analyzing the impact of phase error on fidelity. The implications of these results are then discussed in the light of entanglement fidelity and the error upper bound, reinforcing the central role of Equation (42). The main implications of the results above are a facilitated understanding of quantum gate fidelity to electronics engineers and improved collaboration between engineers and physicists, which is important for the practice of applied research. Section 6 also contributed to reinforcing the engineering relevance of Equation (42) by elucidating its impact on the energy efficiency of a quantum microprocessor. Even if the experimental investigation of fidelity is not the subject of this paper, for the sake of completeness, it is worth closing these conclusions by remarking that the experimental verification of Equation (42) needs independent measurements of the fidelity and the actual matrix \(A_{R}\) of the quantum gate operator. For this purpose, several experimental techniques, each with its own advantages and downsides, can be adopted, such as the randomized benchmarking (RB) protocol [54], quantum process tomography (QPT) [26], or gate set tomography (GST) [49]. In this case, the fidelity is a process fidelity, calculated as the trace of the product matrix between the ideal process matrix and the Hermitian and unitary matrix that best fits the tomographic experimental data [60]. The experimental characterization of quantum gate fidelity is an important activity because it allows us to optimize microwave pulse parameters, such as shape, amplitude, and rise/fall times, for the best possible fidelity [49]. With this perspective, artificial intelligence (AI), when applied to experimental techniques for the characterization of quantum gate fidelity, may be a useful tool for optimizing the microwave control of the qubits [61; 62].