Circuit-Based Compact Model of Electron Spin Qubit

Abstract

Today, an electron spin qubit on silicon appears to be a very promising physical platform for the fabrication of future quantum microprocessors. Thousands of these qubits should be packed together into one single silicon die in order to break the quantum supremacy barrier. Microelectronics engineers are currently leveraging on the current CMOS technology to design the manipulation and read-out electronics as cryogenic integrated circuits. Several of these circuits are RFICs, as VCO, LNA, and mixers. Therefore, the availability of a qubit CAD model plays a central role in the proper design of these cryogenic RFICs. The present paper reports on a circuit-based compact model of an electron spin qubit for CAD applications. The proposed model is described and tested, and the limitations faced are highlighted and discussed.

1. Introduction

Relevant mathematical problems in several technical and scientific fields, such as computational chemistry or cryptography, require an amount of computing resources super-polynomial with the size of the problem, even when addressed with the best traditional algorithms. Following Noble Prize winner Richard Feynman’s suggestion of adopting algorithms based on quantum mechanics laws [1], several quantum algorithms have been proposed, such as Shor’s algorithm for the factorization of integer numbers [2,3] and Grove’s algorithm for database searching [4]. The efficiency of quantum algorithms in ab-initio computational chemistry has demonstrated the correctness of Feynman’s vision [5,6,7]. Nevertheless, running a quantum algorithm on a classic microprocessor has the challenge of the super-polynomial solving time. To be efficient, a quantum algorithm needs to run on a hardware making quantum mechanics laws available. Thus, the qubit, the elementary chip of information in a quantum algorithm, needs its physical counterpart. Nowadays, several microelectronics and computer science industries are betting on the transmon as physical implementation of the qubit [8,9,10,11]. Another approach explored exploits the spin of a single electron confined in a Quantum Dot (QD) and exposed to a magnetic field [12,13]. Weak spin-orbit coupling and a non-magnetic nucleus for the large part (95%) of the natural silicon isotopes [14], potential of directly leveraging the actual CMOS microelectronics technology, the small footprint, longer coherence times, and higher working temperatures make the electron spin qubit a very promising candidate for the fabrication of large-scale quantum micro-processors [15]. This type of micro-processor should be packed with thousands of qubits, in order to offer advantages over the classical type. Unlike the case of a traditional microprocessor, in which billions of transistors are packed in a silicon die but not singularly addressed, in a quantum processor, each single qubit should be addressable, making the interconnections a formidable obstacle, among, of course, other obstacles. Initially suggested by Reilly [16] and first investigated by Charbon et al. [17], cryogenic CMOS integrated circuits seem to be a promising approach, because they make it possible to place the manipulation and read-out functionalities close to the qubits [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]. These circuits include ADC, DAC, multiplexers, microwave oscillators, PLL, LNA, and mixers, making a quantum microprocessor similar to a mixed-signal circuit rather than to a purely digital circuitry.

Because of this relevant analog content, the capability of integrating the qubit emulation into the Computer Aided Design (CAD) tool is crucial for proper design [36]. Although a CAD-external mathematical software tool (e.g., MATLAB, Mathematica) is a possible approach, a compact model of the qubit paves the way to closer co-integration with the circuit simulation tools usually employed by microelectronics engineers. Nevertheless, few papers in the literature report on CAD-oriented compact models of electron spin qubits [37,38]. Compact models can be classified into circuit-based models, in which equivalent electrical circuits mimic the device behaviour, look-up table models, in which the device response is stored in a set of discrete values of voltages and currents, and physics-based analytical models, in which the device’s mathematical equations are coded. The aim of the present work is to report a circuit-based compact model of an electron spin qubit.

The paper is organised as follows. Section 2 address the proposed compact model and compares it with the literature. Section 3 describes how the proposed model has been implemented in the Cadence^© circuit simulator. Section 4 describes the testing of the model in light of the analytical solution of the Schrödinger’s equation of the qubit. Finally, Section 5 ends the paper by drawing some conclusions.

2. The Qubit Model

As briefly cited in the previous section, a qubit can be obtained by exposing an electron confined in a QD to a magnetic field. Let B(t) = B_x(t)î + B_y(t)ĵ + B_z(t)ĥ be this magnetic field, where B_x, B_y, and B_y are the components along the three cartesian axes identified by the three unit vectors î, ĵ and ĥ. Usually, B(t) is the superposition of a strong static field with a weaker radiofrequency electromagnetic field. The static field makes the electron in the QD a qubit because it splits, by means of the Zeeman’s effect, the energy levels, generating two possible states for the electron: |ψ₀> and |ψ₁>. The electromagnetic field is useful in that it controls the electron state transitions. The state of the spin qubit |ψ> can be expressed as Dirac’s notation |ψ> = a(t)|ψ₀> + b(t)|ψ₁>, where a(t) and b(t) are complex valued functions, the squared amplitudes of which, |a(t)|² and |b(t)|², are the probability of the qubit being in the states |ψ₀> and |ψ₁>, respectively. As the qubit exhibits only two possible states |a(t)|² + |b(t)|² = 1 should be true.

The qubit state transitions can be described through the Schrödinger’s equation as follows [39]:

$$j\hslash \frac{\partial |\psi \rangle }{\partial t}=-\overrightarrow{M}·\overrightarrow{B}|\psi \rangle$$

(1)

where j is the imaginary unit and ħ is the reduced Planck’s constant. The magnetic momentum M of the electron can be expressed by adopting Pauli’s formalism [40]:

$$\overrightarrow{M}=-\frac{g{\mu }_B}{2}\left({\sigma }_x\widehat{𝚤}+{\sigma }_x\widehat{𝚥}+{\sigma }_x\widehat{h}\right)·\overrightarrow{B}$$

(2)

where g is the spin g-factor, μ_B is Bohr’s magneton, and σ_x, σ_y, σ_z are Pauli’s matrices:

$${\sigma }_x=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\hspace{1em}{\sigma }_y=\left(\begin{array}{cc}0& -j\\ +j& 0\end{array}\right)\hspace{1em}{\sigma }_z=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$$

(3)

After replacing Equations (2) and (3) into Equation (1), you obtain:

$$j\hslash \frac{\partial |\psi \rangle }{\partial t}=\frac{g{\mu }_B}{2}\left(\begin{array}{cc}{B}_z& B_x-j{B}_y\\ B_x+j{B}_y& -B_z\end{array}\right)|\psi \rangle$$

(4)

It is worth noting that Equation (4) describes an isolated electron spin qubit because it does not take into account any decoherence mechanism [41].

By adopting Dirac’s notation for the qubit state, Equation (4) turns into a pair of coupled differential equations in the unknowns a(t) and b(t) [39,42]:

$$\{\begin{array}{c}j\hslash \frac{\partial a\left(t\right)}{\partial t}=\frac{g{\mu }_B}{2}\left[B_{z}a\left(t\right)+B_{x}b\left(t\right)-j{B}_{y}b\left(t\right)\right]\\ j\hslash \frac{\partial b\left(t\right)}{\partial t}=\frac{g{\mu }_B}{2}\left[B_{x}a\left(t\right)+j{B}_{y}a\left(t\right)-B_{z}b\left(t\right)\right]\end{array}$$

(5)

By writing a(t) = Re{a(t)} + jIm{a(t)} and b(t) = Re{b(t)} + jIm{b(t)}, where Re{a(t)}, Im{a(t)}, Re{b(t)}, and Im{b(t)} are the real and imaginary parts of a(t) and b(t), respectively, Equation (5) yields the following four coupled first-order differential equations in the four unknowns Re{a(t)}, Im{a(t)}, Re{b(t)}, and Im{b(t)}:

$$\{\begin{array}{c}\begin{array}{c}\frac{2\hslash }{g{\mu }_B}\frac{\partial Re\left\{a\left(t\right)\right\}}{\partial t}=B_z\left(t\right)Im\left\{a\left(t\right)\right\}-B_y\left(t\right)Re\left\{b\left(t\right)\right\}+B_x\left(t\right)Im\left\{b\left(t\right)\right\}\\ \frac{2\hslash }{g{\mu }_B}\frac{\partial Im\left\{a\left(t\right)\right\}}{\partial t}=-B_z\left(t\right)Re\left\{a\left(t\right)\right\}-B_x\left(t\right)Re\left\{b\left(t\right)\right\}-B_y\left(t\right)Im\left\{b\left(t\right)\right\}\end{array}\\ \frac{2\hslash }{g{\mu }_B}\frac{\partial Re\left\{b\left(t\right)\right\}}{\partial t}=B_y\left(t\right)Re\left\{a\left(t\right)\right\}+B_x\left(t\right)Im\left\{a\left(t\right)\right\}-B_z\left(t\right)Im\left\{b\left(t\right)\right\}\\ \frac{2\hslash }{g{\mu }_B}\frac{\partial Im\left\{b\left(t\right)\right\}}{\partial t}=B_y\left(t\right)Im\left\{a\left(t\right)\right\}+B_z\left(t\right)Re\left\{b\left(t\right)\right\}-B_x\left(t\right)Re\left\{a\left(t\right)\right\}\end{array}$$

(6)

These four equations can be described through the equivalent electrical circuit depicted in Figure 1, in which the voltages are dimensionless and the currents are expressed in Tesla. Four capacitors, C_QUBIT, all of an identical value equal to 2ħ/gμ_B = 11.37 Tps (h = 6.626 × 10⁻³⁴ Js, g = 2, and μ_B = 9.27 × 10⁻²⁴ J/T), are present in the circuit. The real and imaginary parts of a(t) and b(t) are the voltages across these C_QUBIT capacitors (e.g., Re{a(t)} is the voltage across capacitor C_QUBIT in the first differential equation) and the current flowing through these capacitors accounts for the left term of each differential equation. The twelve multiplying current sources implement the right terms in the differential equations.

The four DC generators in the model force the initial conditions Re{a(t = 0)}, Im{a(t = 0)}, Re{b(t = 0)}, and Im{b(t = 0)} under the control of clock CLK₁. Another clock, CLK₂, controls the application of the three magnetic field components B_x(t), B_y(t), and B_z(t), which are the inputs to the model (on the left) together with the CLK₁; the model outputs are Re{a(t)}, Im{a(t)}, Re{b(t)}, and Im{b(t)} (on the right). The model thus exhibits four inputs and four outputs. At the start of the simulation, when the initial conditions are to be applied, the CLK₁ closes the switches inside the model while the CLK₂ keeps open the external ones. The moment the magnetic field is applied, CLK₂ closes the external switches while CLK₁ opens the internal ones.

To the best of the author’s knowledge, this is the first effort reported in the literature to model the behaviour of an electron spin qubit with such an approach. The qubit SPINE emulator proposed in [37,38] is indeed a physics-based compact model coding the qubit hamiltonian in Verilog-A. The equivalent circuit in Figure 1 is an innovative approach also with respect to the simulations of qubit gates based on C, Matlab or Python codes. Despite being few and not addressing the electron spin qubit, other qubit models are reported in the literature, in particular, to convert the behaviour of charge qubits into an equivalent circuit [43]. It is also worth remarking that in this paper, the authors identify a model at the core of which are voltage-controlled current sources. These sources are reported as time-varying, while in the model proposed in the present paper, the current sources generate currents proportional to the product of two voltages, one corresponding to a component of the magnetic field and the other to a component of the qubit state, as depicted in Figure 1. Equivalently, these sources may be described as time-varying voltage-controlled current sources in which the time-varying behaviour is forced by the magnetic field component while the qubit state component acts as the controlling input voltage. Interestingly, two different physical approaches, indipendently carried out on two different types of qubits, convergence on a set of voltage-controlled time-varying or equivalently multiplying current sources. In both the cases, the currents generated by these sources are integrated on a capacitor. In the present paper, the adopted mathematical physics approach allows us to trace back the capacitance of this capacitor to two fundamental constants: the Planck constant and the Bohr magneton. It is worth closing this section with a historical perspective. In the mid-fourties of the XX century, when computers were not yet available (in 1944, Alan Turing was designing vacuum-triodes-based Colossus to break the Enigma cipher), a physical analog approach was the only possible way to numerically solve the Maxwell [44] and the Schrödinger [45] differential equations. Kron used RLC bidimensional and tridimensional electrical circuits. Resorting to passive circuits was a necessity, as the transistor effect was yet to be discovered (which happened on the day before Christmas eve in 1947). In Kron’s approach, current sources are therefore not present. More recently, RLC circuits demonstrated their validity in quantum mechanics as an emulation tool of quantum gates [46]. In this approach, the voltage-controlled current sources are present but they are not time-varying or multiplying. Therefore, time-varying/multiplying current sources appear to be specific to the approach used in the present paper for the electron spin qubit and in [43] for the charge qubit.

3. Cadence^© Implementation

The proposed qubit compact model was implemented in Cadence^©. The upper left corner of Figure 2 depicts the implementation of the sub-circuit concerning the imaginary part of a(t) in Figure 1. In this implementation, the voltage signal V_ImagA, corresponding to Im{a(t)} in Figure 1, is generated by integrating, on a capacitor with capacitance equal to 11.37 pF, the current I_ImagA generated by the current source, which was coded in Verilog-A. It is worth noting that in other circuit simulators (e.g., Quite Universal Circuit Simulator), these sources can be implemented through so-called Equation Defined Device/Sources [47]. As illustrated in Figure 2, the magnetic field components applied to the qubit are implemented as the voltage signals V_Bx, V_By, and V_Bz, which are the input of the current source block. The current source also receives as input the voltage signal V_RealA, corresponding to Re{a(t)} in Figure 1, voltage signal V_RealB, corresponding to Re{b(t)} in Figure 1, and voltage signal V_ImagB, corrsponding to Im{b(t)} in Figure 1. Each of these three voltage signals, V_RealA, V_RealB, and V_ImagB, are generated by a circuit identical to that depicted in Figure 2 for V_ImagA. All the current source blocks receive as input voltages V_Bx, V_By, and V_Bz, but in agreement with the model in Figure 1, the current source generating current I_RealA receives voltage signals V_ImagA, V_RealB, and V_ImagB, the current source generating current I_ImagB the voltage signals V_ImagA, V_RealA, and V_RealB, and the current source generating current I_RealB the voltage signals V_ImagA, V_RealA, and V_ImagB. As for the voltage signal V_ImagA, the voltage signals V_RealA, V_RealB, and V_ImagB are also made available at the output of an ideal voltage buffer implented by using a Voltage-Controlled Voltage Source (VCVS). This buffer isolates the capacitor upper plate, the most delicate node of the circuit, from the rest of the circuitry. The initial condition on this node is forced by applying a DC voltage source, as highlighted in the left upper corner. The switches are implemented with an open (close) resistance of 1 TΩ (1 nΩ).

The voltage sources generating signals V_Bx and V_By are the interface with the CMOS RFIC if voltage V_Bz accounts for the static component of the magnetic field. Physically speaking, the radiofrequency magnetic field is generated by radiofrequency current I_RF circulating into a metallization line, operating as a microwave antenna placed close to the qubit (see for instance [48]). This current I_RF is forced by a radiofrequency circuit that usually is constituted of an envelope generator, a vector modulator, and a radiofrequency local oscillator. During the design of an RFIC, the voltage sources for V_Bx and V_By are therefore replaced by the schematic of the RFIC cascaded with a block describing how the voltage signals generated by the RFIC circuit translate into the radiofrequency magnetic field.

4. Qubit Model Test

The model has been tested in the case of component B_z(t) being static and components B_x(t) and B_y(t) describing an electromagnetic field rotating in the plane normal to the static field. This circularly polarized magnetic field enables solving the differential equation system in the laboratory frame without invoking the rotating wave approximation. With B_z(t) = B₀, B_x(t) = B₁cosωt, and B_y(y) = B₁sinωt from Equation (5), you obtain [49]:

$$\{\begin{array}{c}\frac{\partial a\left(t\right)}{\partial t}=-j{\omega }_{0}a\left(t\right)-j{\omega }_{1}b\left(t\right)e^{-j\omega t}\\ \frac{\partial b\left(t\right)}{\partial t}=-j{\omega }_{1}a\left(t\right)e^{j\omega t}+j{\omega }_{0}b\left(t\right)\end{array}$$

(7)

where ω₀ = gμ_BB₀/2ℏ and ω₁ = gμ_BB₁/2ℏ. In order to induce prompt changes in the qubit state, frequency ω of the electromagnetic field should be set at the resonance frequency 2ω₀, known as the Larmor frequency ω_Larmor. As detailed in Appendix A, the solution of system (7) under the Larmor resonance condition obtains the following simple mathematical expressions for a(t) and b(t):

$$a\left(t\right)=j{e}^{-j{\omega }_{0}t}sin{\omega }_{1}t$$

(8)

$$b\left(t\right)=-e^{-j{\omega }_{0}t}cos{\omega }_{1}t$$

(9)

The probabilities of P₀ and P₁ of the qubit being in state |ψ₀> and|ψ₁>, respectively, are as follows:

$$P_0={\left|a\left(t\right)\right|}^2=si{n}^2{\omega }_{1}t=\frac{1-cos2{\omega }_{1}t}{2}$$

(10)

$$P_1={\left|b\left(t\right)\right|}^2=co{s}^2{\omega }_{1}t=\frac{1+cos2{\omega }_{1}t}{2}$$

(11)

Each probability oscillates at frequency 2ω₁, known as the Rabi frequency ω_Rabi, whose value depends on magnitude B₁ of the rotating electromagnetic field and not on magnitude B₀ of the static magnetic field. Figure 3 compares the oscillating P₀ and P₁ probabilities predicted by Equations (10) and (11) with the probabilities obtained from the Cadence^© simulations of the proposed qubit compact model under resonance conditions for B₀ = 1.4 mT and B₁ = 10.6 μT. The simulations were carried out by using a trapezoidal integration method. The Euler integration method was avoided because it causes severe numerical damping. The convergence accuracy was set conservatively. In this way, the self-adaptive numerical algorithm adapts the actual time step for the best accuracy. It set the time step shorter (longer) when current or voltage variations in the circuit are large (small). The maximum time step was up-limited to 1ns. In addition, the amplitude of B₁ was chosen in order to obtain typical values for ω_Rabi [50,51]. If the difference between B₀ and B₁ amplitudes is too large, possible numerical limitations in the simulation engine may make the terms multiplied by B_x and B_y negligible in the right terms of the differential equation system (6), leaving B₀ as the only significant component. Under these conditions, the differential equation system is reduced to a couple of identical differential equations, each describing an oscillator at frequency ω₀. Several numerical tests led to the conclusion that the amplitude difference between B₀ and B₁ should not be higher than two orders of magnitude in order to avoid the decoupling of the differential equations. Nevertheless, low values of B₀, which are useful to avoid differential equations decoupling, may lead to ω_Larmor lower than expected in a real case. In the co-simulation, this issue can be solved by means of an ideal frequency conversion of the signals generated by the RFIC circuit. It is worth nothing that, in the case of hot qubits, [51,52], the values of B₀ found in the literature are in the order of 0.1 T, corresponding to ω_Larmor in the range of a few GHz. Figure 3 shows that the qubit compact model correctly reproduces the sinusoidal behaviour predicted by the analytical solution. In particular, it is worth remarking that the model generates probability curves oscillating between 0 and 1, out of phase of π, and intersecting at 0.5, thus meeting the constraint P₀ + P₁ = 1. The best (worst) agreement between the compact model and the analytical solution of about 0.05% (1.4%) is achieved at 3.40 μs (1.70 μs). The Rabi frequency generated by the compact model is about 296.7 kHz, close to the 296.6 kHz value predicted by the mathematical expression of 2ω₁.

In order to investigate further possible numerical issues, in addition to the previously discussed B₀/B₁ ratio, further simulation tests were carried out under off-resonance condition, by introducing a given amount of frequency offset Δω. The goal is to plot the qubit state probability, P₀ or P₁, versus Δω, when the qubit is excited with a microwave pulse of duration τ_p corresponding to one of the peak in the Rabi oscillation (e.g., for the probability P₀, τ_p = 5.067 μs corresponds to the second peak in Figure 3).

The mathematics in the Appendix A shows that the probability P₀(τ_p) obtained after having applied, to the electron spin qubit, a microwave pulse with a frequency offset Δω for a duration of τ_p = Nt_π, where N is an integer number and t_π is the pulse duration making the probability transits from 0 to 1, or from 1 to 0, under resonance conditions, exhibits the following mathematical expression:

$$P_0\left(\mathsf{\Delta }\omega ,N\right)=\frac{{\omega }_1^2}{{\omega }_1^2+{\left(\frac{\mathsf{\Delta }\omega }{2}\right)}^2}si{n}^2\left[\sqrt{{\omega }_1^2+{\left(\frac{\mathsf{\Delta }\omega }{2}\right)}^2}\frac{\pi N}{2{\omega }_1}\right]$$

(12)

which is known as the Rabi formula [53].

These curves are the cross sections on the discrete set of pulse durations πN/2ω₁ of the chevron pattern, which is the surface generated by plotting P₀ versus Δω and τ_p. Therefore, from this point on, these cross-section curves will be referred to as the chevron cross-sections. Figure 4 compares the chevron cross-sections predicted by Equation (12) with those generated by the qubit compact model for τ_p = 5.067 μs and for Δω up to 600 kHz. The figure shows that the simulated and analytically calculated cross-sections are frequency shifted.

In order to investigate the nature of this frequency shift, simulations were carried out by adopting different integration methods. Figure 4 shows that the degree of frequency shift depends on the adopted integration method, suggesting its numerical nature. Transient simulations are known indeed to be highly sensitive to the numerical method adopted to integrate nodal equations [54]. Figure 4 shows that the trapezoidal method leads to the lowest frequency shift, estimated as 35 kHz, while the gear (mixed trapezoidal gear) integration method leads to a frequency shift of about 68 kHz (50 kHz). This finding is in agreement with the fact that the trapezoidal integration method is usually the most efficient for a high accuracy [55]. In particular, the replacement of the continuous time derivative with a finite-difference approximation leads to a numerical error known as a local truncation error. This depends on the integration method and how it accumulates in the time depends on the circuit under simulation. In the case of an oscillator, it may cause phase drift. Since the chevron cross-sections come from oscillating probability waveforms computed through transient simulations where a microwave pulse is applied, under a given amount of frequency offset, for a fixed amount of time, a phase drift may turn out into a frequency shift in the chevron cross-sections.

Figure 5 shows that when the simulated chevron cross-sections are compensated forwith a 35-kHz frequency shift, the simulated trapezoidal chevron cross-section perfectly fits the analytical solution, which is not true for the other two cases. Indeed, the chevron cross-sections simulated with the gear and the gear-trapezoidal integration method not only still suffer from frequency detuning but also from being of lower local maxima than those predicted by the analytical solution. The disagreement is about 16.7% for the first peak and in the range of 6% for the second one. Since this disagreement is observed only when a gear component in the integration method is present, it appears to be reasonable to ascribe this disagreement in the local maxima to numerical damping, from which the gear integration method suffers even if it is as severe as in the case of the Euler integration method.

With the aim of testing the general validity of the 35 kHz frequency shift previously obtained for a maximum frequency offset of 600 kHz on the second Rabi peak, this frequency shift was applied to chevron cross-sections simulated for a larger frequency offset, up to about 1.5 MHz, and for different peaks of the Rabi oscillations. In all simulations, the adopted integration method was trapezoidal.

Figure 6, Figure 7 and Figure 8 depict the chevron cross-sections simulated for the first, second, and third peaks of the Rabi oscillations, corresponding to a τ_p of 1.689 μs, 5.067 μs, and 8.445 μs, respectively.

All the figures reveal a good agreement between the analytical solution and the compact model, proving the general validity of the 35 kHz frequency shift and supporting the hypothesis that the frequency shift stems from numerical issues.

5. Conclusions

The present paper proposes a CAD-oriented circuit-based compact model for the electron spin qubit. Its equivalent circuit approach is innovative with respect to the very few examples reported in the literature of electron spin qubit physics-based compact models coded in Verilog A code and with respect to the simulations of qubit gates based on C, Matlab or Python codes. At the core of the proposed model for the electron spin qubit are voltage-controlled current sources, as in the case of a recently published compact model of a charge qubit. Interestingly, two different mathematical formal approaches to the qubit modeling, applied to two different types of qubits, convergence on the same core. This suggests that voltage-controlled time-varying or multiplying sources may be at the core of a circuit compact model also in the case of a transmonic qubit. The proposed circuit-based compact model was implemented in the Cadence^© CAD suite and tested in the case of a circularly polarized radiofrequency magnetic field. The compact model was able to correctly reproduce the qubit state probabilities, which are counter-phase oscillating between 0 and 1. The comparison with the analytical solution of the qubit Schrödinger equation shows a discrepancy between the simulated and the analytically calculated probabilities in the range of 0.5%–1.4%, low enough to reasonably allow microelectronics designers to address the impact on the qubit control of the addressed circuit solutions, of the chosen technology design kit and of the fabrication tolerances (corner cases) [56]. In this regard, it is worth noting that the required threshold for the fidelity for quantum error correction using surface codes is in the range of 1% [57].

The simulations revealed two main limitations in the use of the proposed qubit compact model, both related to numerical issues. First, the difference between the magnitudes of the static magnetic field and that of the rotating magnetic field should be kept lower than two orders of magnitude. This limitation is not very compelling because the Rabi oscillation, which is the typical physical observable of interest, depends on the amplitude of the rotating magnetic field and not on that of the static field, whose amplitude can be kept lower during the simulation. On the other hand, the static magnetic field amplitude fixes the Larmor frequency. In a case where the adopted low value of the static magnetic field led to a Larmor frequency lower than the frequency generated by the designed electronic circuit, an ideal frequency conversion could be envisaged at the interface between the circuit and the qubit compact model. The second limitation revealed was a frequency shift between the chevron cross-sections predicted by the analytical solution and those generated by the compact model. It was observed that the magnitude of this frequency shift depends on the chosen integration method. The lowest shift of 35 kHz was achieved with a trapezoidal integration method. Both of these numerical limitations are worth of further investigation by implementing the qubit compact model in several CAD tools, where integration methods and convergence algorithms may differ. As the proposed compact model is based on an equivalent circuit, one can expect indeed that it can be implemented in several CAD tools.

In summary, the present paper demonstrates a circuit-based compact model for an electron qubit with potentialities that are good enough to be considered a promising approach in the design of RFIC for quantum computing applications. Eventually, it will be worth making a few last considerations about the validation of the proposed qubit model in the design of an RFIC. Beyond the comparison with the analytical solution of the Schrödinger equation, reported in the present paper, a validation of the proposed qubit model in the design of a RFIC needs, as briefly described in the last paragraph of Section 3, a magnetic field interface between the model of the qubit and the schematic of the RFIC, so that it is possible to relate the signals generated by the RFIC with the observe qubit behaviour.