Demystifying Quantum Gate Fidelity for Electronics Engineers

Abstract

The implementation of quantum gates by means of microwave cryo-RFICs controlling qubits is a promising path toward scalable quantum processors. Quantum gate fidelity quantifies how well an actual quantum gate produces a quantum state close to the desired ideal one. Regrettably, the literature usually reports on quantum gate fidelity in a highly theoretical way, making it hard for RFIC designers to understand. This paper explains quantum gate fidelity by moving from Shannon’s concept of fidelity and proposing a detailed mathematical proof of a valuable integral formulation of quantum gate fidelity. Shannon’s information theory and the simple mathematics adopted for the proof are both expected to be in the background of electronics engineers. By using Shannon’s fidelity, this paper rationalizes the integral formulation of quantum gate fidelity. Because of the simple mathematics adopted, this paper also demystifies to electronics engineers how this integral formulation can be reduced to a more practical algebraic product matrix. This paper makes evident the practical utility of this matrix formulation by applying it to the specific examples of one- and two-qubit quantum gates. Moreover, this paper also compares mixed states, entanglement fidelity, and the error rate’s upper bound.

1. Introduction

A scalable and fault-tolerant quantum processor is the enabling device for the second quantum revolution, which, exploiting the counter-intuitive laws of quantum mechanics, promises to revolutionize, in the near-medium future, how information is processed and transmitted, with relevant impacts in several practical fields, such as cryptography, drug synthesis, and artificial intelligence [1].

Microelectronic technologies seem to assure scalability. In particular, cryogenic RFICs (radio frequency integrated circuits) are a promising solution for scaled quantum microprocessors [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The application of a microwave pulse with an appropriate amplitude, frequency, and time duration to a two-level quantum physical system (quantum bit or qubit for short) indeed enables the implementation of one-qubit quantum gates; likewise, two-qubit quantum gates can be implemented by applying suitable microwave pulses to a pair of qubits [22]. The useful frequency range spans from 100 MHz to 50 GHz for electron spin qubits and from 500 MHz to 15 GHz for superconductive qubits [23]. RFICs are interesting because they are microminiaturized microwave sources that can be placed close to the qubits, minimizing the number of microwave cables connecting the quantum microprocessor to the external room temperature environment. In this way, the size of the whole system is reduced and the adiabaticity of the cryostat is improved.

On the one hand, microelectronic technologies seem to assure scalability. However, on the other hand, the intrinsic brittle coherence of quantum objects constitutes the main challenge to achieving fault tolerance because it tends to increase the quantum computation error. The quantum error rate is the correct figure of merit for evaluating the fault-tolerance degree of a quantum processor [24]. An important theorem of quantum information theory, the threshold theorem, states that there exists a threshold error rate under which a quantum processor, even if faulty, is able to correctly run a quantum algorithm by using an efficient number of qubits and quantum gates [25]. The theorem guarantees that a quantum processor is feasible but only on the condition that the quantum error rate is kept low enough. Because of the aforementioned brittleness of the quantum coherence, this condition can be attained only by adopting quantum error correction (QEC) codes [25].

The unavoidable non-idealities in the actual implementation of quantum gates also prevent quantum gates from functioning ideally. Fidelity is the figure of merit that quantifies the difference between the output states generated by actual and ideal quantum gates [26]. The key point is that the QEC codes can be successfully applied only if quantum gate fidelities are higher than a given threshold [27]. A high enough quantum gate fidelity is therefore the prerequisite for achieving quantum fault tolerance.

The non-idealities in the electronic circuitry in an RFIC may limit the quantum gate fidelity [28,29]. For instance, the phase noise, affecting the microwave signal generated by the oscillator/phase locked loop in the RFIC, and the inaccuracies in the amplitude of the controlling microwave signal due to the output pre-drive amplifier in the RFIC can reduce the fidelity. The RFIC should be placed close to the qubits inside the cryostat chamber, whose thermal budget nevertheless limits the amount of power the RFIC is allowed to dissipate. Designing RFICs under power dissipation constraints may lead to device size, bias voltage, and low power circuit solutions that sacrifice ideal behavior for power [4,20]. For instance, power dissipation is traded for sensitivity to parametric fluctuations, the spurious free dynamic range (SFDR), and linearity.

Since quantum gate fidelity enables the application of the QEC codes, it is an important specification for electronics engineers designing the qubit RFIC controllers. Typical quantum gate fidelity requirements pose a significant challenge to the RFIC designers [10,30,31], who should therefore be aware of what quantum gate fidelity is, in order to properly account for it alongside other design specifications. In addition, being aware of quantum gate fidelity also facilitates discussion and information exchange between RFIC designers and physicists. Collaboration between these two professional figures is of paramount importance for successfully addressing the fabrication of quantum microprocessors.

Nevertheless, the literature typically addresses quantum gate fidelity by definition, without providing a rationalized explanation that aids understanding. Moreover, it employs advanced abstract mathematical formalism that does not help electronics engineers grasp the concept. This paper intends to contribute to overcoming these obstacles.

To achieve this, Section 2 of this paper introduces quantum gate fidelity by leveraging the concept of fidelity as found in information theory, which is generally familiar to electronics engineers. Moreover, Section 3 introduces an analogy between a quantum gate and a transmission channel, thus linking the input and output states of the quantum gate to, respectively, the transmitted and received signals on the channel. Section 4 reports a step-by-step, detailed mathematical proof of a relatively well-known formula for quantum gate fidelity [32]. This formula is particularly relevant for RFIC designers because it reduces the calculation of the fidelity to matrix analysis, a domain familiar to electronics engineers. Regrettably, in [32], as it often occurs in other papers on fidelity, see, for instance, refs. [33,34,35], the mathematics is often presented using an abstract formalism characteristic of theoretical physics. This makes it difficult for electronics engineers to interpret the formulation of fidelity, thereby discouraging its use, despite fidelity being a relevant design specification. In contrast, the mathematical proof proposed in Section 4 adopts a more accessible mathematical formalism, aligning with the mathematical background of electronics engineers. It is the authors’ opinion that addressing mathematical issues in this way helps to demystify concepts that may remain otherwise obscure. With the aim of further clarifying the relevance of the formula above for RFIC designers, Section 5 discusses brief case studies that exemplify how the RFIC non-idealities impact quantum gate fidelity, and Section 6 addresses the relationship between quantum gate fidelity and the energy efficiency of the quantum microprocessor. Section 7 discusses quantum gate fidelity in the more involved case of mixed states. Eventually, Section 8 closes this paper by summarizing the obtained results and drawing some conclusions.

2. Fidelity According to Shannon

The concept of fidelity is rooted in the communication theory conceived by Claude Shannon (1916–2001) in 1948 at Bell Laboratories [36]. Shannon introduced the term fidelity to indicate the degree of exactness in recovering, at the receiver side R_X of a transmission channel, the signal generated by a source at the transmitter side T_X. In particular, his historical paper introduces the probability $P\left(m_{IN},m_{OUT}\right)$, which represents, as follows, the likelihood that $m_{OUT}$ and $m_{IN}$ are, respectively, the input and output signals:

$$P\left(m_{IN},m_{OUT}\right)=P\left(m_{IN}\right)P\left(m_{OUT}|m_{IN}\right)$$

(1)

where $P\left(m_{IN}\right)$ is the probability that the input message source generates the signal $m_{IN}$ and $P\left(m_{OUT}|m_{IN}\right)$ is the conditional probability that the received signal is $m_{OUT}$ given that the sent message is $m_{IN}$. The conditional probability in Equation (1) implicitly attributes a stochastic nature to the transmission channel. Indeed, it introduces the fact that, for a given input signal, different output signals may be observed, each with its own conditional probability. All possible messages $m_{IN}$ and $m_{OUT}$ form the message space $S_m$. The probability $P\left(m_{IN},m_{OUT}\right)$ fully statistically describes the whole transmission system (message source plus transmission channel) in terms of the pairs of messages $m_{IN}$ and $m_{OUT}$. If the probability $P\left(m_{IN},m_{OUT}\right)$ is known for the whole space $S_m$, the fidelity is also known in a qualitative sense. Nevertheless, if you desire a more operative and quantitative definition of fidelity, you need to introduce a function $F\left\{·\right\}$ that yields a number from $P\left(m_{IN},m_{OUT}\right)$. Shannon interpreted this number as the distance between the ideally expected and the actually received output signals. He dubbed this distance fidelity because it quantifies the degree of exactness in receiving the transmitted message. In his paper, Shannon mathematically introduced fidelity as the weighted average distance calculated over the message space, where the probability $P\left(m_{IN},m_{OUT}\right)$ serves as the weighting factor [36]:

$$F\left\{P\left(m_{IN},m_{OUT}\right)\right\}=\sum _{S_m}P\left(m_{IN},m_{OUT}\right)\delta \left(m_{IN},m_{OUT}\right)$$

(2)

where $\delta \left(m_{IN},m_{OUT}\right)$ quantifies, based on a given criterion, the distance between the received $m_{OUT}$ and the transmitted $m_{IN}$ signals. This makes sense because for an ideal transmission channel, the ideal received output signal $m_{OUT}^I=m_{IN}$, where the superscript “I” stands for ideal. The quantity $\delta \left(m_{IN},m_{OUT}\right)$ therefore measures the distance between the ideal and actual output messages. Since in Equations (1) and (2), $m_{OUT}$ is the actual output signal, for reasons of unambiguousness, it is good practice to replace the symbol $m_{OUT}$ with $m_{OUT}^R$ in these equations. With these representations, the substitution of Equation (1) into Equation (2) yields:

$$F\left\{P\left(m_{OUT}^I,m_{OUT}^R\right)\right\}=\sum _{S_m}P\left(m_{IN}\right)P\left(m_{OUT}^R|m_{IN}\right)\delta \left(m_{OUT}^I,m_{OUT}^R\right)$$

(3)

where, in the mathematical expressions of the distance $\delta$ and fidelity $F$, it has been emphasized that $m_{OUT}^I=m_{IN}$. In this way, it is made plain that the distance $\delta$ and fidelity $F$ address the difference between the ideal $m_{OUT}^I$ and actual $m_{OUT}^R$ output messages.

A deterministic (not stochastic) transmission channel implies $P\left(m_{OUT}^R|m_{IN}\right)=1$, reducing the mathematical expression of the fidelity to the following:

$$F\left\{P\left(m_{OUT}^I,m_{OUT}^R\right)\right\}=\sum _{S_m}P\left(m_{IN}\right)\delta \left(m_{OUT}^I,m_{OUT}^R\right)$$

(4)

It is worth noticing that, being deterministic, the channel does not imply an ideal channel. The distance $\delta$ in Equation (4) accounts for that. In particular, since $m_{OUT}^I=m_{IN}$, Equation (4) shows that different $m_{IN}$ signals contribute differently to the fidelity.

In cases where the input source generates input messages with a uniform probability, the probability $P\left(m_{IN}\right)$ is a constant, so you can factor it out from the summation:

$$F\left\{P\left(m_{OUT}^I,m_{OUT}^R\right)\right\}=P\left(m_{IN}\right)\sum _{S_m}\delta \left(m_{OUT}^I,m_{OUT}^R\right)$$

(5)

Apart from a constant, the fidelity can therefore be rewritten as follows:

$$F\left\{P\left(m_{OUT}^I,m_{OUT}^R\right)\right\}=\sum _{S_m}\delta \left(m_{OUT}^I,m_{OUT}^R\right)$$

(6)

If the space $S_m$ is continuous instead of discrete, as assumed above, an integral replaces the summation:

$$F\left\{P\left(m_{OUT}^I,m_{OUT}^R\right)\right\}={\int }_{S_m}\delta \left(m_{OUT}^I,m_{OUT}^R\right)d{S}_m$$

(7)

with $d{S}_m$ being the infinitesimal volume in $S_m$. Equations (6) and (7) state that the fidelity is the average distance between $m_{OUT}^I$ and $m_{OUT}^R$ on the message space $S_m$.

It is worth concluding this section by emphasizing that Equations (6) and (7) embed two hypotheses: the transmission channel is deterministic, and all input signals are generated with the same probability.

3. The Fidelity of a Quantum Gate

Quantum gate fidelity, as well as the fidelity for transmitted and received signals in Section 2, quantifies the difference between ideal and actual quantum gates as the average distance between ideal and actual quantum output states. A fidelity equal to 100% means that the quantum gates behave ideally. In practice, it is useful to introduce the error rate of a quantum gate as the complement to 100% of the quantum fidelity [37], so that an ideal quantum gate exhibits an error rate equal to zero.

In quantum information theory, a quantum operation, such as a quantum gate, is an operator $A$ [26,38]. Following the considerations in Section 2, this paper assimilates a real quantum gate to a real channel, as depicted in Figure 1. The output quantum state $|{\psi }_{OUT}^R⟩=A|{\psi }_{IN}⟩$ coincides with the ideal one $|{\psi }_{OUT}^I⟩$ only if the operator is ideal, exactly as the output signal $m_{OUT}$ is identical to the input signal $m_{IN}$ only if the channel behaves ideally. If the effect of the quantum gate in Figure 1 on the input state $|{\psi }_{IN}⟩$ does not introduce phase decoherence, the output state $|{\psi }_{OUT}^R⟩$ is a pure state as long as the input state is pure. Appendix A shortly reviews the definition of pure and mixed states, along with some further considerations.

As observed for a deterministic transmission channel, a coherent quantum gate does not imply that it behaves ideally. When both input and output states are pure, the distance $\delta \left(m_{OUT}^I,m_{OUT}^R\right)$ is the state overlap probability $\delta \left(|{\psi }_{OUT}^I⟩,|{\psi }_{OUT}^R⟩\right)={\left|⟨{\psi }_{OUT}^I|{\psi }_{OUT}^R⟩\right|}^2$, which is the probability that the actual output state $|{\psi }_{OUT}^R⟩$ coincides with the desired ideal $|{\psi }_{OUT}^I⟩$ [38]. It is worth emphasizing here that, in quantum mechanics, the quantity $⟨{\psi }_{OUT}^I|{\psi }_{OUT}^R⟩$ is dubbed the inner product, and it is a complex-valued scalar. By assuming that all input states are equally probable, the fidelity $F$ of the quantum gate can be therefore introduced from Equation (7) as follows:

$$F={\int }_{BS}{\left|⟨{\psi }_{OUT}^I|{\psi }_{OUT}^R⟩\right|}^{2}dV$$

(8)

where $dV$ is the infinitesimal volume of the Bloch sphere (BS), which is a geometrical figure useful for visualizing a quantum state [26,38]. For a n-level quantum state (qudit), the Bloch sphere is a hypersphere in a (2n−1)-dimensional space $S^{2n-1}$ and it can be described as a nested Bloch sphere [39]. A qubit is a qudit for n = 2. In this case, the Bloch hypersphere reduces to the usual three-dimensional sphere, because 2n−1 = 3. It is worth pointing out that, as Equation (7) is an average value on the message space $S_m$, Equation (8) describes the fidelity $F$ as the probability, averaged on the entire Bloch hypersphere, that $|{\psi }_{OUT}^I⟩$ and $|{\psi }_{OUT}^R⟩$ are identical.

If $A_I$ and $A_R$ are the ideal and actual forms, respectively, of the operator $A$, you can express $|{\psi }_{OUT}^I⟩=A_I|{\psi }_{IN}⟩$ and $|{\psi }_{OUT}^R⟩=A_R|{\psi }_{IN}⟩$. Thus:

$$F={\int }_{BS}{\left|⟨{\psi }_{IN}|A_I^{†}{A}_R|{\psi }_{IN}⟩\right|}^{2}dV$$

(9)

where $A_I^{†}$ is the conjugate transpose (adjoint) of $A_I$. If the input state $|{\psi }_{IN}⟩$ is a vector of an n-dimensional complex Hilbert space:

$${|\psi }_{IN}⟩=\sum _{i=1}^n{\alpha }_i{|\psi }_i⟩$$

(10)

where ${|\psi }_i⟩$ for i=1…n, is an orthonormal basis for the Hilbert space, Equation (9) can be reformulated as follows:

$$F={\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|A_I^{†}{A}_R|{\psi }_{IN}⟩\right|}^{2}dV$$

(11)

In Equation (11), $|{\psi }_{IN}⟩$ describes a qudit and the integral is calculated based on the Bloch hypersphere in the multi-dimensional space $S^{2n-1}$.

4. Quantum Gate Fidelity in Matrix Form

Calculating the fidelity from Equation (11) is not a trivial task, as it involves a multi-dimensional integral. The aim of this section is to formulate Equation (11) in a more practical matrix form, to make it easy to calculate the quantum fidelity once the operators $A_I$ and $A_R$ are provided in the matrix form. It is worth emphasizing that, once a basis for the Hilbert space is provided, an operator is mathematically represented by a matrix [40]; in this way, an operator enjoys all the properties of the matrix. In particular, the application of the Toeplitz decomposition [41] to the $A_{TOT}=A_I^{†}{A}_R$ total operator makes it possible to decompose $A_{TOT}$ into the sum of a Hermitian $H$ and an anti-Hermitian $A$ operator. Equation (11) can thus be rewritten as follows:

$$F={\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|A_{TOT}|{\psi }_{IN}⟩\right|}^{2}dV={\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|H+A|{\psi }_{IN}⟩\right|}^{2}dV$$

(12)

Because of linearity, Equation (12) can be reformulated as follows:

$$F={\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|H|{\psi }_{IN}⟩+⟨{\psi }_{IN}|A|{\psi }_{IN}⟩\right|}^{2}dV$$

(13)

By assuming that the state $|{\psi }_{IN}⟩$ is normalized, the quantities $⟨{\psi }_{IN}|H|{\psi }_{IN}⟩$ and $⟨{\psi }_{IN}|A|{\psi }_{IN}⟩$ are the expectation values of the operators $H$ and $A$, respectively [42,43]. Since the expectation value of an operator is the average values of its eigenvalues [42,43], you can write:

$$\begin{array}{c}⟨{\psi }_{IN}|H|{\psi }_{IN}⟩={\displaystyle \sum _i}{\left|{\alpha }_i\right|}^2{h}_i\\ ⟨{\psi }_{IN}|A|{\psi }_{IN}⟩={\displaystyle \sum _i}{\left|{\alpha }_i\right|}^2{a}_i\end{array}$$

(14)

where ${\alpha }_i$ is the components of the state vector $|{\psi }_{IN}⟩$ and $h_i$ ($a_i$) are the eigenvalues of the operator $H$ ($A$). By emphasizing that the eigenvalues of a Hermitian operator are real numbers while the eigenvalues of an anti-Hermitian operator are pure imaginary numbers [42], you can state that $h_i$ are real numbers while $a_i$ are imaginary numbers. In addition, since ${\left|{\alpha }_i\right|}^2$ are real numbers, you can conclude, from Equation (14), that $⟨{\psi }_{IN}|H|{\psi }_{IN}⟩$ is real while $⟨{\psi }_{IN}|A|{\psi }_{IN}⟩$ is purely imaginary. The argument of the integral in Equation (13) is therefore of the general form ${\left|x+jy\right|}^2$, with $x$ and $y$ real numbers. By emphasizing that ${\left|x+jy\right|}^2={\left|x\right|}^2+{\left|y\right|}^2$, Equation (13) can be split into two contributions:

$$F={\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|H|{\psi }_{IN}⟩\right|}^{2}dV+{\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|A|{\psi }_{IN}⟩\right|}^{2}dV$$

(15)

It is now also worth emphasizing that the Hermitian and anti-Hermitian operators are normal operators [44]. They therefore enjoy the spectral theorem from the matrix analysis [41], which states that every normal operator is unitarily similar to a diagonal operator, which is an operator represented by a diagonal matrix [44]. In particular, the matrix entries on the main diagonal are exactly the eigenvalues $d_{Hi}$ for the diagonal operator $D_H$, which is similar to the operator $H$, and $d_{Ai}$ for the diagonal operator $D_A$, which is similar to the operator $A$. Since similar operators exhibit the same eigenvalues, $h_i=d_{Hi}$ and $a_i=d_{Ai}$. Thus:

$$\begin{array}{c}⟨{\psi }_{IN}|D_H|{\psi }_{IN}⟩={\displaystyle \sum _i}{\left|{\alpha }_i\right|}^2{d}_{Hi}={\displaystyle \sum _i}{\left|{\alpha }_i\right|}^2{h}_i\\ ⟨{\psi }_{IN}|D_A|{\psi }_{IN}⟩={\displaystyle \sum _i}{\left|{\alpha }_i\right|}^2{d}_{Ai}={\displaystyle \sum _i}{\left|{\alpha }_i\right|}^2{a}_i\end{array}$$

(16)

Of course, with $h_i$ ($a_i$) being real (imaginary) numbers, $d_{Hi}$ ($d_{Ai}$) are also real (imaginary) numbers. The comparison of Equation (14) with Equation (16) leads to the following reformulation of Equation (15):

$$F={\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|D_H|{\psi }_{IN}⟩\right|}^{2}dV+{\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|D_A|{\psi }_{IN}⟩\right|}^{2}dV$$

(17)

Since $d_{Ai}$ is wholly imaginary, $d_{Ai}/j$ is a real number and, since $\left|j\right|=1$, Equation (17) can be reformulated by replacing $D_A$, whose entries are $d_{Ai}$, with $D_A/j$, whose entries are $d_{Ai}/j$:

$$F={\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|D_H|{\psi }_{IN}⟩\right|}^{2}dV+{\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|D_A/j|{\psi }_{IN}⟩\right|}^{2}dV$$

(18)

In this way, the fidelity $F$ has been expressed by means of the two real diagonal matrices $D_H$ and $D_A/j$. For the sake of brevity, it therefore makes sense to address the following integral $\mathfrak{H}$:

$$\mathfrak{H}={\int }_{S^{2n-1}}{\left|⟨{\psi }_{IN}|D|{\psi }_{IN}⟩\right|}^{2}dV$$

(19)

where $D$ is a real diagonal matrix of real-valued eigenvalues $d_i$. The expectation value $⟨{\psi }_{IN}|D|{\psi }_{IN}⟩$ is thus also real-valued, because of the usual expression:

$$⟨{\psi }_{IN}|D|{\psi }_{IN}⟩=\sum _i{\left|{\alpha }_i\right|}^2{d}_i$$

(20)

Since the squared module of a real number is the squared number ${\left|⟨{\psi }_{IN}|D|{\psi }_{IN}⟩\right|}^2={⟨{\psi }_{IN}|D|{\psi }_{IN}⟩}^2$, Equation (20) yields:

$${\left|⟨{\psi }_{IN}|D|{\psi }_{IN}⟩\right|}^2={\left(\sum _i{\left|{\alpha }_i\right|}^2{d}_i\right)}^2=\sum _i{\left|{\alpha }_i\right|}^4{d}_i^2+2\sum _{i,j\ne i}{\left|{\alpha }_i\right|}^2{\left|{\alpha }_j\right|}^2{d}_i{d}_j$$

(21)

Equation (21) is the generalization to $n$ terms of the Leibniz trinomial for n = 2 ${\left(a+b+c\right)}^2=a^2+b^2+c^2+2ab+2ac+2bc$.

The substitution of Equation (21) into Equation (19) yields:

$$\mathfrak{H}={\int }_{S^{2n-1}}\sum _i{\left|{\alpha }_i\right|}^4{d}_i^{2}dV+{\int }_{S^{2n-1}}2\sum _{i,j\ne i}{\left|{\alpha }_i\right|}^2{\left|{\alpha }_j\right|}^2{d}_i{d}_{j}dV$$

(22)

from which:

$$\mathfrak{H}=\sum _i\left({\int }_{S^{2n-1}}{\left|{\alpha }_i\right|}^{4}dV\right)d_i^2+\sum _i\left({\int }_{S^{2n-1}}{\left|{\alpha }_i\right|}^2{\left|{\alpha }_j\right|}^{2}dV\right)2d_i{d}_j$$

(23)

The eigenvectors can be factored out from the integral because the integral is calculated on the (2n − 1)-dimensional Bloch hypersphere by varying the vector state $\left|{\psi }_{IN}\right.⟩$, i.e., by varying its components. Appendix B details the mathematical proof of the following pair of integrals:

$$\begin{array}{l}{\int }_{S^{2n-1}}{\left|{\alpha }_i\right|}^{4}dV={\displaystyle \frac{2}{n(n+1)}}\\ {\int }_{S^{2n-1}}{\left|{\alpha }_i\right|}^2{\left|{\alpha }_j\right|}^{2}dV={\displaystyle \frac{1}{n(n+1)}}\end{array}$$

(24)

The substitution of Equation (24) into Equation (23) yields:

$$\mathfrak{H}={\displaystyle \frac{2}{n(n+1)}}\sum _i{d}_i^2+{\displaystyle \frac{1}{n(n+1)}}\sum _{i}2d_i{d}_j$$

(25)

By remarking that the trace $Tr\left\{D\right\}$ of the matrix $D$ is also a real-valued number, you can write:

$${\left|Tr\left\{D\right\}\right|}^2={Tr}^2\left\{D\right\}={\left(\sum _i{d}_i\right)}^2=\sum _i{d}_i^2+2\sum _{i,j\ne i}{d}_i{d}_j=Tr\left\{D^2\right\}+2\sum _{i,j\ne i}{d}_i{d}_j$$

(26)

where it has been remarked that the first summation on the right-hand side of Equation (26) is the trace $Tr\left\{D^2\right\}$ of $D^2$.

Eventually, the substitution of Equation (26) into Equation (25) yields:

$$\mathfrak{H}={\displaystyle \frac{2}{n(n+1)}}\sum _i{d}_i^2+{\displaystyle \frac{1}{n(n+1)}}\left({\left|Tr\left\{D\right\}\right|}^2-Tr\left\{D^2\right\}\right)$$

(27)

Again, the summation in Equation (27) is $Tr\left\{D^2\right\}$, so that the Equation can be rearranged by factoring out $Tr\left\{D^2\right\}$:

$$\mathfrak{H}=Tr\left\{D^2\right\}\left[{\displaystyle \frac{2}{n(n+1)}}-{\displaystyle \frac{1}{n(n+1)}}\right]+{\displaystyle \frac{1}{n(n+1)}}{\left|Tr\left\{D\right\}\right|}^2={\displaystyle \frac{Tr\left\{D^2\right\}+{\left|Tr\left\{D\right\}\right|}^2}{n(n+1)}}$$

(28)

The application of Equation (28) to the real diagonal matrices $D_H$ and $D_A/j$ in Equation (18) leads to:

$$\begin{array}{cc}F\hfill & ={\displaystyle \frac{Tr\left\{D_H^2\right\}+{\left|Tr\left\{D_H\right\}\right|}^2}{n(n+1)}}+{\displaystyle \frac{Tr\left\{{\left(D_A/j\right)}^2\right\}+{\left|Tr\left\{D_A/j\right\}\right|}^2}{n(n+1)}}\hfill \\ & ={\displaystyle \frac{Tr\left\{D_H^2\right\}+{\left|Tr\left\{D_H\right\}\right|}^2}{n(n+1)}}+{\displaystyle \frac{-Tr\left\{D_A^2\right\}+{\left|Tr\left\{D_A\right\}\right|}^2}{n(n+1)}}\hfill \end{array}$$

(29)

Since $D_H$ ($D_A$) is a diagonal matrix, whose entries are pure real (imaginary), $D_H=D_H^{†}$ ($D_A=-D_A^{†}$) and thus:

$$\begin{array}{l}{D}_H^2=D_H{D}_H^{†} \\ D_A^2=-D_A{D}_A^{†}\end{array}$$

(30)

Since $H$ and $D_H$ ($D_A$) are unitarily similar, by definition, $D_H$ ($D_A$) is obtained from $H$ ($A$) by means of the following similarity transformation:

$$\begin{array}{l}{D}_H=U_H^{-1}H{U}_H \\ D_A=U_A^{-1}A{U}_A\end{array}$$

(31)

where $U_H$ ($U_A$) is the diagonalizing unitary matrix. The substitution of Equation (31) into Equation (30) yields:

$$\begin{array}{l}{D}_H^2=U_H^{-1}H{U}_H{\left(U_H^{-1}H{U}_H\right)}^{†}=U_H^{-1}H{U}_H{\left[U_H^{-1}\left(H{U}_H\right)\right]}^{†} \\ D_A^2=-U_A^{-1}A{U}_A{\left(U_A^{-1}A{U}_A\right)}^{†}=-U_A^{-1}A{U}_A{\left[U_A^{-1}\left(A{U}_A\right)\right]}^{†}\end{array}$$

(32)

Since, for a unitary matrix, $U^{-1}=U^{†}$, and, for a couple of matrices $A$ and $B$, ${\left(AB\right)}^{†}=B^{†}{A}^{†}$, Equation (32) can be reformulated as follows:

$$\begin{array}{l}{D}_H^2=U_H^{-1}H{U}_H {\left(H{U}_H\right)}^{†}{U}_H^{-1†}=U_H^{-1}H{U}_H{U}_H^{†}{H}^{†}{U}_H=U_H^{-1}H{H}^{†}{U}_H\\ D_A^2=-U_A^{-1}A{U}_A {\left(A{U}_A\right)}^{†}{U}_A^{-1†}=-U_A^{-1}A{U}_A{U}_A^{†}{A}^{†}{U}_A=-U_A^{-1}A{A}^{†}{U}_A\end{array}$$

(33)

By emphasizing that, for a pair of matrices $A$ and $B$, $Tr\left\{AB\right\}=Tr\left\{BA\right\}$, from Equation (33), you get:

$$\begin{array}{l}Tr\left\{D_H^2\right\}=Tr\left\{\left(U_H^{-1}H\right)\left(H^{†}{U}_H\right)\right\}=Tr\left\{\left(H^{†}{U}_H\right)\left(U_H^{-1}H\right)\right\}=Tr\left\{H^{†}H\right\}\\ Tr\left\{D_A^2\right\}=-Tr\left\{\left(U_A^{-1}A\right)\left(A^{†}{U}_A\right)\right\}-Tr\left\{\left(A^{†}{U}_A\right)\left(U_A^{-1}A\right)\right\}=-Tr\left\{A^{†}A\right\}\end{array}$$

(34)

and from Equation (31):

$$\begin{array}{l}Tr\left\{D_H\right\}=Tr\left\{\left(U_H^{-1}H\right)U_H\right\}=Tr\left\{U_H\left(U_H^{-1}H\right)\right\}=Tr\left\{U_H{U}_H^{-1}H\right\}=Tr\left\{H\right\}\\ Tr\left\{D_A\right\}=Tr\left\{A\right\}\end{array}$$

(35)

The substitution of Equations (34) and (35) into Equation (29) yields:

$$F={\displaystyle \frac{Tr\left\{H^{†}H\right\}+Tr\left\{A^{†}A\right\}+{\left|Tr\left\{H\right\}\right|}^2+{\left|Tr\left\{A\right\}\right|}^2}{n(n+1)}}$$

(36)

By emphasizing that $A_{TOT}=H+A$, you can write:

$$Tr\left\{A_{TOT}{A}_{TOT}^{†}\right\}=Tr\left\{\left(H+A\right){\left(H+A\right)}^{†}\right\}=Tr\left\{\left(H+A\right)\left(H-A\right)\right\}$$

(37)

where it has been emphasized that, by definition, $H^{†}=H$ and $A^{†}=-A$. Due to the above-emphasized property $Tr\left\{AB\right\}=Tr\left\{BA\right\}$, it holds that $Tr\left\{HA\right\}=Tr\left\{AH\right\}$. Thus, Equation (37) can be shortened as follows:

$$Tr\left\{A_{TOT}{A}_{TOT}^{†}\right\}=Tr\left\{HH\right\}-Tr\left\{HA\right\}+Tr\left\{AH\right\}-Tr\left\{AA\right\}=Tr\left\{H{H}^{†}\right\}+Tr\left\{A{A}^{†}\right\}$$

(38)

From Equation (31), you can observe that similar matrices own the same trace:

$$\begin{array}{l}Tr\left\{D_H\right\}=Tr\left\{\left(U_H^{-1}H\right)U_H\right\}=Tr\left\{U_H{U}_H^{-1}H\right\}=Tr\left\{H\right\} \\ Tr\left\{D_A\right\}=Tr\left\{A\right\}\end{array}$$

(39)

Equation (39) shows that the trace of $H$ ($A$) is real (imaginary), because the entries of $D_H$ ($D_A$) are real (imaginary). Consequently, you can write:

$${\left|Tr\left\{A_{TOT}\right\}\right|}^2={\left|Tr\left\{H+A\right\}\right|}^2={\left|Tr\left\{H\right\}+Tr\left\{A\right\}\right|}^2={\left|Tr\left\{H\right\}\right|}^2+{\left|Tr\left\{A\right\}\right|}^2$$

(40)

The substitution of Equations (38) and (40) into Equation (36) yields:

$$F={\displaystyle \frac{Tr\left\{A_{TOT}{A}_{TOT}^{†}\right\}+{\left|Tr\left\{A_{TOT}\right\}\right|}^2}{n(n+1)}}$$

(41)

It is now worth emphasizing that the operator describing a quantum gate is unitary and thus represented by a unitary matrix. Since $A_I$ and $A_R$ are unitary operators, being the operators of a quantum gate, the operator $A_{TOT}$, which is their product, is also unitary and thus $A_{TOT}^{-1}=A_{TOT}^{†}$. Equation (41) therefore takes the following form:

$$F={\displaystyle \frac{Tr\left\{I\right\}+{\left|Tr\left\{A_{TOT}\right\}\right|}^2}{n(n+1)}}={\displaystyle \frac{n+{\left|Tr\left\{{A_I^{†}A}_R\right\}\right|}^2}{n(n+1)}}$$

(42)

Equation (42) provides the mathematical expression for the fidelity of a quantum gate exploiting an n-level quantum system, as it has been calculated for a general qudit. In the case of a one-qubit gate (e.g., X Pauli gate), n = 2, since there are two energy levels, while for a two-qubit gate (e.g., CNOT gate), n = 4 because there are four energy levels [22].

5. Brief Case Studies

To better grasp the relevance of Equation (42) as an RFIC designer, let us consider the brief case studies of the one-qubit Pauli X and two-qubit CNOT quantum gates implemented in the spin of electrons confined in quantum wells [22]. The quantum gate is implemented by irradiating the qubit, via a micro-antenna, with a microwave pulse at the resonance Larmor frequency and for a suitable time duration $\mathsf{\Delta }t$; this microwave irradiation induces the following rotation angle $\theta$ of the state vector:

$$\mathit{\theta }={\displaystyle \frac{\mathit{g}{\mathit{\mu }}_{\mathit{B}}}{2\mathit{\hslash }}}{\mathit{B}}_1\mathsf{\Delta }\mathit{t}$$

(43)

where $\mathit{g}$ is the electron spin g-factor, ${\mathit{\mu }}_{\mathit{B}}$ is the Bohr magneton, $\mathit{\hslash }$ is the reduced Planck constant, and ${\mathit{B}}_{\mathbf{1}}$ is the magnitude of the magnetic component of the microwave electromagnetic pulse applied to the qubit. The magnitude ${\mathit{B}}_{\mathbf{1}}$ depends on the amplitude of the microwave signal ${\mathit{s}}_{\mathit{R}\mathit{F}}$, generated by the RFIC, which excites the micro-antenna used to irradiate the qubit.

5.1. Pauli X Quantum Gate

The ideal matrix ${\mathit{A}}_{\mathit{X}\mathit{I}}$ of the Pauli X quantum gate operator is obtained from the matrix ${\mathit{A}}_{\mathit{R}\mathit{X}}$ of the ${\mathit{R}}_{\mathit{X}}$ operator [22]:

$${\mathit{A}}_{\mathit{R}\mathit{X}}=\left[\begin{array}{cc}\mathit{c}\mathit{o}\mathit{s}{\displaystyle \frac{\mathit{\theta }}{2}}& -\mathit{j}\mathit{s}\mathit{i}\mathit{n}{\displaystyle \frac{\mathit{\theta }}{2}}\\ -\mathit{j}\mathit{s}\mathit{i}\mathit{n}{\displaystyle \frac{\mathit{\theta }}{2}}& \mathit{c}\mathit{o}\mathit{s}{\displaystyle \frac{\mathit{\theta }}{2}}\end{array}\right]$$

(44)

by forcing the rotation angle of the qubit state vector $\mathit{\theta }=\mathit{\pi }$ and by factorizing and neglecting the global phase term $-\mathit{j}$:

$${\mathit{A}}_{\mathit{X}\mathit{I}}=\left[\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}\end{array}\right]$$

(45)

Let us suppose that the actual rotation angle is affected by an error $\mathit{\epsilon }$, so that $\mathit{\theta }=\mathit{\pi }+\mathit{\epsilon }$, because the electronics circuitry, designed to control the qubit, causes errors in the amplitude of ${\mathit{s}}_{\mathit{R}\mathit{F}}$ and/or in the duration $\mathsf{\Delta }\mathit{t}$. From Equation (44), the actual matrix ${\mathit{A}}_{\mathit{X}\mathit{R}}$ of the Pauli X quantum gate operator is:

$${\mathit{A}}_{\mathit{X}\mathit{R}}=\left[\begin{array}{cc}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\pi }}{\mathbf{2}}}+{\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& -\mathit{j}\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\pi }}{\mathbf{2}}}+{\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\\ -\mathit{j}\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\pi }}{\mathbf{2}}}+{\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\pi }}{\mathbf{2}}}+{\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]=-\left[\begin{array}{cc}\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\\ \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]$$

(46)

Since the entries of the matrix ${\mathit{A}}_{\mathit{X}\mathit{I}}$ are real values, ${\mathit{A}}_{\mathit{X}\mathit{I}}={\mathit{A}}_{\mathit{X}\mathit{I}}^{†}$, and thus from Equations (45) and (46), you obtain:

$${\mathit{A}}_{\mathit{X}\mathit{I}}^{†}{\mathit{A}}_{\mathit{X}\mathit{R}}=-\left[\begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}\end{array}\right]\left[\begin{array}{cc}\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\\ \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]=-\left[\begin{array}{cc}\mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\\ \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]$$

(47)

from which:

$${\left|\mathit{T}\mathit{r}\left\{{\mathit{A}}_{\mathit{X}\mathit{I}}^{†}{\mathit{A}}_{\mathit{X}\mathit{R}}\right\}\right|}^{\mathbf{2}}={\left|-\mathbf{2}\mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\right|}^{\mathbf{2}}=\mathbf{4}{\mathit{c}\mathit{o}\mathit{s}}^{\mathbf{2}}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)$$

(48)

The substitution of Equation (48) into Equation (42) yields:

$$\mathit{F}={\displaystyle \frac{\mathbf{2}+\mathbf{4}{\mathit{c}\mathit{o}\mathit{s}}^{\mathbf{2}}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)}{\mathbf{2}(\mathbf{2}+\mathbf{1})}}={\displaystyle \frac{\mathbf{1}+\mathbf{2}{\mathit{c}\mathit{o}\mathit{s}}^{\mathbf{2}}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)}{\mathbf{3}}}$$

(49)

where it has been emphasized that n = 2 for a one-qubit quantum gate. As expected, Equation (49) yields $\mathit{F}=\mathbf{1}$ when $\mathit{\epsilon }=\mathbf{0}$.

5.2. CNOT Quantum Gate

As for the CNOT quantum gate operator, its ideal matrix ${\mathit{A}}_{\mathit{C}\mathit{N}\mathit{O}\mathit{T}\mathit{I}}$ is obtained from the matrix ${\mathit{A}}_{\mathit{\omega }\mathit{R}1,\mathit{I}}$ [22]:

$${\mathit{A}}_{\mathit{\omega }\mathit{R}\mathbf{1},\mathit{I}}=\left[\begin{array}{cccc}-\mathit{j}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& -\mathit{j}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)& -\mathit{j}\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)\\ \mathbf{0}& \mathbf{0}& -\mathit{j}\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)& \mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)\end{array}\right]$$

(50)

by forcing the rotation angle of the qubit state vector $\mathit{\theta }=\mathit{\pi }$ and by factorizing and neglecting the global phase term $-\mathit{j}$:

$${\mathit{A}}_{\mathit{C}\mathit{N}\mathit{O}\mathit{T}\mathit{I}}=\left[\begin{array}{cc}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}\end{array}& \begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\\ \begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}& \begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}\end{array}\end{array}\right]$$

(51)

As in the previous case, let the actual rotation angle be affected by an error $\mathit{\epsilon }$, so that $\theta =\pi +\epsilon$. Equation (50) yields the actual matrix $A_{CNOTR}$ of the CNOT quantum gate operator:

$$\begin{array}{cc}{\mathit{A}}_{\mathit{C}\mathit{N}\mathit{O}\mathit{T}\mathit{R}}\hfill & =\left[\begin{array}{cccc}-\mathit{j}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& -\mathit{j}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\pi }}{\mathbf{2}}}+{\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& -\mathit{j}\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\pi }}{\mathbf{2}}}+{\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\\ \mathbf{0}& \mathbf{0}& -\mathit{j}\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\pi }}{\mathbf{2}}}+{\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\pi }}{\mathbf{2}}}+{\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]\hfill \\ & =\left[\begin{array}{cccc}-\mathit{j}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& -\mathit{j}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& -\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& -\mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)\\ \mathbf{0}& \mathbf{0}& -\mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)& -\mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]=-\left[\begin{array}{cccc}\mathit{j}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathit{j}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)\\ \mathbf{0}& \mathbf{0}& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]\hfill \end{array}$$

(52)

Since the entries of the matrix $A_{CNOTI}$ are real values, $A_{CNOTI}=A_{CNOTI}^{†}$. Thus, from Equations (51) and (52), you obtain:

$$\begin{array}{c}{\mathit{A}}_{\mathit{C}\mathit{N}\mathit{O}\mathit{T}\mathit{I}}^{†}{\mathit{A}}_{\mathit{C}\mathit{N}\mathit{O}\mathit{T}\mathit{R}}=-\left[\begin{array}{cc}\begin{array}{cc}\mathbf{1}& \mathbf{0}\\ \mathbf{0}& \mathbf{1}\end{array}& \begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\\ \begin{array}{cc}\mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}& \begin{array}{cc}\mathbf{0}& \mathbf{1}\\ \mathbf{1}& \mathbf{0}\end{array}\end{array}\right]\left[\begin{array}{cccc}\mathit{j}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathit{j}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)\\ \mathbf{0}& \mathbf{0}& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]=\hfill \\ =-\left[\begin{array}{cccc}\mathit{j}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathit{j}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)\\ \mathbf{0}& \mathbf{0}& \mathit{s}\mathit{i}\mathit{n}\left({\displaystyle \frac{\mathit{\theta }}{\mathbf{2}}}\right)& \mathit{j}\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\end{array}\right]\hfill \end{array}$$

(53)

from which:

$${\left|\mathit{T}\mathit{r}\left\{{\mathit{A}}_{\mathit{C}\mathit{N}\mathit{O}\mathit{T}\mathit{I}}^{†}{\mathit{A}}_{\mathit{C}\mathit{N}\mathit{O}\mathit{T}\mathit{R}}\right\}\right|}^{\mathbf{2}}={\left|-\mathbf{2}\mathit{j}\left[\mathbf{1}+\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\right]\right|}^{\mathbf{2}}=\mathbf{4}{\left[\mathbf{1}+\mathit{c}\mathit{o}\mathit{s}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)\right]}^{\mathbf{2}}$$

(54)

The substitution of Equation (54) into Equation (42) yields:

$$\mathit{F}={\displaystyle \frac{\mathbf{4}+\mathbf{4}{\left[\mathbf{1}+\mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)\right]}^{\mathbf{2}}}{\mathbf{4}(\mathbf{4}+\mathbf{1})}}={\displaystyle \frac{\mathbf{1}+{\left[\mathbf{1}+\mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)\right]}^{\mathbf{2}}}{\mathbf{5}}}$$

(55)

where it has been emphasized that n = 4 for a two-qubit quantum gate. As expected, Equation (55) yields $F=1$ when $\epsilon =0$.

5.3. Entanglement Fidelity

Finally, it is worth noting that the authors in [45] expressed the fidelity in Equation (42) in terms of the entanglement fidelity $F_e$:

$$\mathit{F}={\displaystyle \frac{\mathit{n}{\mathit{F}}_{\mathit{e}}+\mathbf{1}}{\mathit{n}+\mathbf{1}}}$$

(56)

where $F_e$ quantifies the exactness in teleporting information coded in a quantum state over a quantum transmission channel formed by entangled states (e.g., Bell states). Equation (56) further reinforces the analogy proposed in Figure 1 between a quantum gate and a transmission channel. The substitution of Equation (42) into Equation (56) yields:

$${\mathit{F}}_{\mathit{e}}={\displaystyle \frac{{\left|\mathit{T}\mathit{r}\left\{{{\mathit{A}}_{\mathit{I}}^{†}\mathit{A}}_{\mathit{R}}\right\}\right|}^{\mathbf{2}}}{{\mathit{n}}^{\mathbf{2}}}}$$

(57)

which is the mathematical expression used in [28] to investigate the impact of the non-ideal behavior of the electronics on quantum fidelity. In the above case studies with the Pauli X and CNOT quantum gates, the substitution of Equation (48) into Equation (57) yields the entanglement fidelity $F_{e,X}$ for the Pauli X quantum gate:

$${\mathit{F}}_{\mathit{e},\mathit{X}}={\displaystyle \frac{\mathbf{4}{\mathit{c}\mathit{o}\mathit{s}}^{\mathbf{2}}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)}{{\mathbf{2}}^{\mathbf{2}}}}={\mathit{c}\mathit{o}\mathit{s}}^{\mathbf{2}}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)=\mathbf{1}-{\mathit{s}\mathit{i}\mathit{n}}^{\mathbf{2}}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)$$

(58)

and the subsititution of Equation (54) into Equation (57) yields the entanglement fidelity $F_{e,CNOT}$ for the CNOT quantum gate:

$$\begin{gathered} {\mathit{F}}_{\mathit{e},\mathit{C}\mathit{N}\mathit{O}\mathit{T}}={\displaystyle \frac{\mathbf{4}{\left[\mathbf{1}+\mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)\right]}^{\mathbf{2}}}{{\mathbf{4}}^{\mathbf{2}}}}={\displaystyle \frac{\mathbf{2}-{\mathit{s}\mathit{i}\mathit{n}}^{\mathbf{2}}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)+\mathbf{2}\mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)}{\mathbf{4}}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{\mathbf{1}-{\frac{\mathbf{1}}{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}}^{\mathbf{2}}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)+\mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)}{\mathbf{2}}} \end{gathered}$$

(59)

For small values of $\epsilon$, $sin\left({\displaystyle \raisebox{1ex}{$\epsilon $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$ can be approximated with ${\displaystyle \raisebox{1ex}{$\epsilon $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ and $cos\left({\displaystyle \raisebox{1ex}{$\epsilon $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$ with $\mathbf{1}$, so that, from Equations (58) and (59), respectively, you obtain:

$${\mathit{F}}_{\mathit{e},\mathit{X}}\cong \mathbf{1}-{\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)}^{\mathbf{2}}=\mathbf{1}-{\displaystyle \frac{{\mathit{\epsilon }}^{\mathbf{2}}}{\mathbf{4}}}$$

(60)

$${\mathit{F}}_{\mathit{e},\mathit{C}\mathit{N}\mathit{O}\mathit{T}}\cong {\displaystyle \frac{\mathbf{1}-\frac{\mathbf{1}}{\mathbf{2}}{\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)}^{\mathbf{2}}+\mathbf{1}}{\mathbf{2}}}=\mathbf{1}-{\displaystyle \frac{{\mathit{\epsilon }}^{\mathbf{2}}}{\mathbf{16}}}$$

(61)

Equation (60) agrees with the mathematical form claimed in [24,28] for one-qubit quantum gates facing phase errors in the controlling microwave signal.

5.4. 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 $\epsilon$:

$${\mathit{F}}_{\mathit{e},\mathit{N}\mathit{Q}\mathit{U}\mathit{B}\mathit{I}\mathit{T}}\cong \mathbf{1}-{\left({\displaystyle \frac{\mathit{\epsilon }}{\mathit{n}}}\right)}^{\mathbf{2}}$$

(62)

The substitution of Equation (57) into Equation (62) yields:

$${\displaystyle \frac{{\left|\mathit{T}\mathit{r}\left\{{{\mathit{A}}_{\mathit{I}}^{†}\mathit{A}}_{\mathit{R}}\right\}\right|}^{\mathbf{2}}}{{\mathit{n}}^{\mathbf{2}}}}=\mathbf{1}-{\left({\displaystyle \frac{\mathit{\epsilon }}{\mathit{n}}}\right)}^{\mathbf{2}}$$

(63)

and thus:

$${\left|\mathit{T}\mathit{r}\left\{{{\mathit{A}}_{\mathit{I}}^{†}\mathit{A}}_{\mathit{R}}\right\}\right|}^{\mathbf{2}}={\mathit{n}}^{\mathbf{2}}-{\mathit{\epsilon }}^{\mathbf{2}}$$

(64)

By adding $n$ on both sides, you obtain:

$${\left|\mathit{T}\mathit{r}\left\{{{\mathit{A}}_{\mathit{I}}^{†}\mathit{A}}_{\mathit{R}}\right\}\right|}^{\mathbf{2}}+\mathit{n}={\mathit{n}+\mathit{n}}^{\mathbf{2}}-{\mathit{\epsilon }}^{\mathbf{2}}$$

(65)

from which:

$${\displaystyle \frac{{\left|\mathit{T}\mathit{r}\left\{{{\mathit{A}}_{\mathit{I}}^{†}\mathit{A}}_{\mathit{R}}\right\}\right|}^{\mathbf{2}}+\mathit{n}}{{\mathit{n}+\mathit{n}}^{\mathbf{2}}}}=\mathbf{1}-{\displaystyle \frac{{\mathit{\epsilon }}^{\mathbf{2}}}{{\mathit{n}+\mathit{n}}^{\mathbf{2}}}}$$

(66)

By referring to Equation (42), Equation (66) can be written in a shorter form:

$$\mathit{F}=\mathbf{1}-{\displaystyle \frac{{\mathit{\epsilon }}^{\mathbf{2}}}{{\mathit{n}+\mathit{n}}^{\mathbf{2}}}}$$

(67)

from which:

$$\mathit{\epsilon }=\sqrt{\mathit{n}(\mathit{n}+1)(\mathbf{1}-\mathit{F})}$$

(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. Mathematical expressions for fidelity and entanglement fidelity.

Quantum Gate	$\mathit{F}$	${\mathit{F}}_{\mathit{e}}$	${\mathit{F}}_{\mathit{e}}(\mathit{\epsilon }\ll \mathbf{1})$
One-qubit (Pauli X)	${\displaystyle \frac{\mathbf{1}+\mathbf{2}{\mathit{c}\mathit{o}\mathit{s}}^{\mathbf{2}}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)}{\mathbf{3}}}$	$\mathbf{1}-{\mathit{s}\mathit{i}\mathit{n}}^{\mathbf{2}}\left({\displaystyle \frac{\mathit{\epsilon }}{\mathbf{2}}}\right)$	$\mathbf{1}-{\displaystyle \frac{{\mathit{\epsilon }}^{\mathbf{2}}}{\mathbf{4}}}$
Two-qubit (CNOT)	${\displaystyle \frac{\mathbf{1}+{\left[\mathbf{1}+\mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)\right]}^{\mathbf{2}}}{\mathbf{5}}}$	${\displaystyle \frac{\mathbf{1}-{\frac{\mathbf{1}}{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}}^{\mathbf{2}}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)+\mathit{c}\mathit{o}\mathit{s}\left(\frac{\mathit{\epsilon }}{\mathbf{2}}\right)}{\mathbf{2}}}$	$\mathbf{1}-{\displaystyle \frac{{\mathit{\epsilon }}^{\mathbf{2}}}{\mathbf{16}}}$
N-qubit			$\mathbf{1}-{\left({\displaystyle \frac{\mathit{\epsilon }}{\mathit{n}}}\right)}^{\mathbf{2}}$

summarizes the main obtained mathematical expressions for fidelity and entanglement fidelity.

Table 2. Experimental quantum fidelity values for one- and two-qubit quantum gates.

Number of Qubits	Qubit Technology	Fidelity [%]	Reference
1	Superconducting	99.9	[46]
1	Superconducting	99.8	[47]
1	Superconducting	>99.9	[48]
1	Electron Spin	99.9	[49]
2	Superconducting	99.4	[47]
2	Superconducting	99.4	[46]
2	Trapped ions	99.3	[47]
2	Neutral atom	99.5	[37]

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 $\mathit{n}$. 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].

6. Energy Efficiency

As outlined in Section 1, in order to address problems of practical utility, quantum microprocessors need to protect the qubits against errors by adopting QEC codes. Nevertheless, these codes require fidelity that is better than a threshold value $F_{threshold}$ [37,50]. A QEC code implies merging $N_{qubit}$ physical qubits into one logic qubit to achieve a desired logic error rate ${\epsilon }_{logic}$, which decreases with increasing $N_{qubit}$ [51]:

$${\mathit{\epsilon }}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}}={\mathit{K}}_{\mathbf{1}}{\left(\sqrt{{\displaystyle \frac{\mathbf{1}-\mathit{F}}{{\mathbf{1}-\mathit{F}}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}}}}\right)}^{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}\left(\sqrt{{\displaystyle \frac{{\mathit{N}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}+\mathbf{1}}{\mathbf{2}}}}+\mathbf{1}\right)}$$

(69)

with ${\mathit{K}}_1$ being a suitable constant. By emphasizing the definition of the error rate of a quantum gate given in Section 3, Equation (69) can be reformulated as follows:

$${\mathit{\epsilon }}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}}={\mathit{K}}_{\mathbf{1}}{\left(\sqrt{{\displaystyle \frac{\mathbf{1}-\mathit{F}}{{\mathit{\epsilon }}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}}}}\right)}^{{\displaystyle \frac{\mathbf{1}}{\mathbf{2}}}\left(\sqrt{{\displaystyle \frac{{\mathit{N}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}+\mathbf{1}}{\mathbf{2}}}}+\mathbf{1}\right)}$$

(70)

where ${\mathit{\epsilon }}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}$ is the error rate threshold required by the QEC code. From Equation (70), you can obtain:

$${\mathit{N}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}=\mathbf{2}{\left[\mathbf{1}-\mathbf{2}{\displaystyle \frac{\mathit{l}\mathit{o}\mathit{g}\left({\mathit{\epsilon }}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}}\right)-\mathit{l}\mathit{o}\mathit{g}\left({\mathit{K}}_{\mathbf{1}}\right)}{\mathit{l}\mathit{o}\mathit{g}\left(\frac{\mathbf{1}-\mathit{F}}{{\mathit{\epsilon }}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}}\right)}}\right]}^{\mathbf{2}}-\mathbf{1}\cong \mathbf{8}{\left[{\displaystyle \frac{\mathit{l}\mathit{o}\mathit{g}\left({\mathit{\epsilon }}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}}\right)}{\mathit{l}\mathit{o}\mathit{g}\left(\frac{\mathbf{1}-\mathit{F}}{{\mathit{\epsilon }}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}}\right)}}\right]}^{\mathbf{2}}$$

(71)

where the approximation is based on ${\mathit{\epsilon }}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}}$ being very small, in the order of ${\mathbf{10}}^{-\mathbf{10}}$ [51]. By defining ${\mathit{P}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}$ the power dissipated to control each physical qubit, Equation (71) provides the total dissipated power ${\mathit{P}}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}\mathit{a}\mathit{l}}$ for the logical qubit:

$${\mathit{P}}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}\mathit{a}\mathit{l}}={\mathit{P}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}{\left[{\displaystyle \frac{\mathit{l}\mathit{o}\mathit{g}\left({\mathit{\epsilon }}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}}\right)}{\mathit{l}\mathit{o}\mathit{g}\left(\frac{\mathbf{1}-\mathit{F}}{{\mathit{\epsilon }}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}}\right)}}\right]}^{\mathbf{2}}$$

(72)

where ${\mathit{P}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}$ depends on the adopted qubit-driving method [52]. The power ${\mathit{P}}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}\mathit{a}\mathit{l}}$ appears, magnified by the ratio between the room ${\mathit{T}}_{\mathit{r}\mathit{o}\mathit{o}\mathit{m}}$ and the qubit cryogenic ${\mathit{T}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}$ temperatures, in the power dissipated by the cryostat ${\mathit{P}}_{\mathit{c}\mathit{r}\mathit{y}\mathit{o}}$ [53]:

$${\mathit{P}}_{\mathit{c}\mathit{r}\mathit{y}\mathit{o}}={\displaystyle \frac{{\mathit{T}}_{\mathit{r}\mathit{o}\mathit{o}\mathit{m}}}{{\mathit{T}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}}}{\mathit{P}}_{\mathit{q}\mathit{u}\mathit{b}\mathit{i}\mathit{t}}{\left[{\displaystyle \frac{\mathit{l}\mathit{o}\mathit{g}\left({\mathit{\epsilon }}_{\mathit{l}\mathit{o}\mathit{g}\mathit{i}\mathit{c}}\right)}{\mathit{l}\mathit{o}\mathit{g}\left(\frac{\mathbf{1}-\mathit{F}}{{\mathit{\epsilon }}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}}\right)}}\right]}^{\mathbf{2}}$$

(73)

Equation (73) demonstrates the importance of achieving a quantum gate fidelity that is as high as possible, as this makes the ratio $(1-F)/{\epsilon }_{threshold}$ lower and reduces, in this way, $P_{cryo}$. Equation (73) also shows that hot qubits may help in reducing $P_{QCPU}$, because they reduce the temperature ratio $T_{room}/T_{qubit}$. Interestingly, one- and two-qubit quantum gates encoded in semiconductor electron spins operating above 1 K have been recently demonstrated to produce fidelity, respectively, of 99.6% and 98.9% [54].

7. Mixed States

Equation (42) has been proven under the assumptions that the input state is pure and uniformly distributed and that quantum gates do not introduce phase decoherence, ensuring that the output state is also pure. In practice, the uniform distribution of the input states may not be considered a serious limitation, because the characterization of a quantum gate is typically intended to explore how the quantum gate behaves for all possible input states, as it occurs in the case of the quantum process tomography [55]. Several papers on average fidelity indeed assume a uniform distribution, such as, for instance, [34,56]. On the other hand, the case of mixed states is more involved and it deserves a more comprehensive discussion for the sake of comparison with the literature.

In cases where the input state ${|\psi }_{IN}⟩$ of the density matrix is pure and the output state is mixed ${\rho }_{OUT}$, Wilde introduces the expected fidelity as an average distance $〈\delta 〉$ between pure states [38]:

$$〈\delta 〉=\sum _X{p}_X{\left|⟨{\psi }_{IN}|{\psi }_X⟩\right|}^2$$

(74)

where $p_X$ is the probability used to describe the mixed state as a mixture of pure states in terms of a linear combination of the density matrix of pure states ${|\psi }_X⟩$ (see Appendix A):

$${\rho }_{OUT}=\sum _X{p}_X{|\psi }_X⟩⟨\left.{\psi }_X\right|$$

(75)

Since, as observed in Appendix A, a pure state is an ensemble of identical pure states, in cases where the output state is pure, Equation (75) reduces to ${\left|⟨{\psi }_{IN}|{\psi }_{OUT}⟩\right|}^2$, being ${|\psi }_X⟩={|\psi }_{OUT}⟩$ and $\sum _X{p}_X=1$ for the probability normalization.

Wilde shows that a short algebraic manipulation leads to the following compact mathematical expression of Equation (75):

$$〈\delta 〉=⟨{\psi }_{IN}|{\rho }_{OUT}|{\psi }_{IN}⟩$$

(76)

Equivalently, Equation (76) can be expressed by means of the trace $Tr$ [57]:

$$〈\delta 〉=Tr\left\{{|\psi }_{IN}⟩⟨\left.{\psi }_{IN}\right|{\rho }_{OUT}\right\}=Tr\left\{{\rho }_{IN}{\rho }_{OUT}\right\}$$

(77)

Nielsen reports, as a root fidelity, a mathematical expression very close to Equation (76) [26]:

$$F=\sqrt{⟨{\psi }_{IN}|{\rho }_{OUT}|{\psi }_{IN}⟩}$$

(78)

In the more involved case where, in addition to the output state, the input state is also mixed, Wilde invokes Uhlmann’s fidelity [38], which, after [58] and [59], is defined as the maximum probability that the purifications of the two mixed states are coincident. Appendix A reports short notes about the concept of purification. Uhlmann’s theorem allows the following compact mathematical expression for the fidelity [38]:

$$F={‖\sqrt{{\rho }_{IN}}\sqrt{{\rho }_{OUT}}‖}_1$$

(79)

where ${‖·‖}_1$ is the Schatten 1-norm, defined as $Tr\left\{\sqrt{M^{†}M}\right\}$ for a given operator $M$. Note that $\sqrt{{\rho }_{IN}}\sqrt{{\rho }_{OUT}}$ is indeed an operator, with the density matrix ${\rho }_{IN}$ and ${\rho }_{OUT}$ being operators. Moving from Equation (79), Wilde reports that it is possible to write Uhlmann’s fidelity in the following trace form:

$$F={\left\{Tr\sqrt{\sqrt{{\rho }_{IN}}{\rho }_{OUT}\sqrt{{\rho }_{IN}}}\right\}}^2$$

(80)

Equation (80) reduces to Equation (76) when the input state is pure, because, for a pure state, it holds that ${\rho }^2=\rho$ and $Tr\left\{{\rho }^2\right\}=1$ [43]. The property ${\rho }^2=\rho$ indeed allows for writing:

$$F={\left\{Tr\sqrt{{\rho }_{IN}{\rho }_{OUT}{\rho }_{IN}}\right\}}^2={\left\{Tr\sqrt{{|\psi }_{IN}⟩⟨\left.{\psi }_{IN}\right|{\rho }_{OUT}{|\psi }_{IN}⟩⟨\left.{\psi }_{IN}\right|}\right\}}^2$$

(81)

and since $⟨\left.{\psi }_{IN}\right|{\rho }_{OUT}{|\psi }_{IN}⟩$ is scalar, being an expectation value [43], Equation (81) can be rearranged as follows:

$$F={\left\{\sqrt{⟨\left.{\psi }_{IN}\right|{\rho }_{OUT}{|\psi }_{IN}⟩}Tr\sqrt{{|\psi }_{IN}⟩⟨\left.{\psi }_{IN}\right|}\right\}}^2=⟨\left.{\psi }_{IN}\right|{\rho }_{OUT}{|\psi }_{IN}⟩{Tr}^2\left\{\sqrt{{\rho }_{IN}}\right\}$$

(82)

Having emphasized that $Tr\left\{cM\right\}=cTr\left\{M\right\}$ for a given scalar $c$ and operator $M$, invoking again the properties ${\rho }^2=\rho$ and $Tr\left\{{\rho }^2\right\}=1$ yields:

$$F=⟨\left.{\psi }_{IN}\right|{\rho }_{OUT}{|\psi }_{IN}⟩{Tr}^2\left\{{\rho }_{IN}\right\}=⟨\left.{\psi }_{IN}\right|{\rho }_{OUT}{|\psi }_{IN}⟩$$

(83)

which is equal to Equation (76). Equation (80) reduces to the distance ${\left|⟨{\psi }_{IN}|{\psi }_{OUT}⟩\right|}^2$, used in Equation (8), when both the input and output states are pure. Having indeed proved that an input pure state leads from Equation (80) to Equation (76), it is just enough to state that Equation (76) is equivalent to Equation (74) and that the pure output state reduces Equation (74) to ${\left|⟨{\psi }_{IN}|{\psi }_{OUT}⟩\right|}^2$, because $\sum _X{p}_X=1$.

8. Conclusions

This paper provided a rationalization of the concept of quantum gate fidelity, starting from Shannon’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 Appendix B and Appendix 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].

QEC codes, briefly cited in Section 6, and AI applied to experimental characterization can therefore be considered promising strategies to improve the fidelity of quantum gates in the presence of practical limitations imposed by thermal noise and the accuracy of electronics circuitry.