Introduction

Quantum computation has achieved remarkable progress over the past decade, demonstrating unprecedented computational capabilities and potential1,2,3,4,5,6,7. However, many challenges still stand in the way of fulfilling Feynman’s proposition of performing tasks outside the classical reach8, one of which is undoubtedly scaling quantum systems to the likely millions of physical qubits required to tackle problems of scientific, commercial, and societal impact9,10,11,12, as well as managing the classical signals necessary for their initialization, control, and readout13,14,15,16. To perform operations on a quantum processor, each qubit needs to interface with classical hardware, potentially operating at cryogenic temperatures, with demanding specifications in terms of accuracy and noise, which require careful design17,18,19,20. Moreover, transients and non-idealities in both the quantum and classical layers may have unexpected and difficult-to-predict effects and interactions, ultimately degrading the performance of the quantum processing21,22. In addition, there has been a growing interest in leveraging the properties of quantum devices to create implementations of traditional analog circuits and sensors with low power dissipation, a nanoscale footprint, and capable of working at cryogenic temperatures23,24,25,26,27,28,29,30.

To meet the stringent specifications and efficiently design quantum-enabled devices at scale, efficient co-simulation of the quantum and classical layers is of paramount importance. Until now, seminal efforts have shown the great potential of co-simulation31,32, showing how the quantum dynamics may be recast and expressed in a compatible way with standard analog circuit simulators33,34,35,36. In particular, the system’s quantum dynamics can be expressed in Verilog-A, the industry-standard hardware description language for behavioral modeling of analog and mixed-signal systems, which is supported by most modern circuit simulators37. In this work, we leverage the expressiveness of Verilog-A to map with no approximations the equations describing the quantum dynamics to equivalent electrical circuits and integrate them into a Verilog-A compact model of the quantum device. Our models retain all coherent behavior, while being able to include decoherence effects that lead to important phenomena such as the finite lifetime of quantum states and dynamical power dissipation38. The framework we discuss is based on the Lindblad master equation (LME) and is thus applicable to any arbitrary multilevel quantum system, although here we focus on systems based on quantum dots (QDs), which are particularly relevant to semiconductor- and Majorana-based quantum computing39,40,41,42. Overall, our framework enables quantum information processing technologies to leverage the many decades of development in electronic computer-aided design and automation responsible for the very large-scale integration achieved by the semiconductor industry for the purpose of simulating and optimizing hybrid quantum systems. This is schematically depicted in Fig. 1.

Fig. 1: Applications of quantum-classical co-simulation.
figure 1

Our compact models allow for the simultaneous simulation of quantum and electrical variables within standard circuit simulators, (i.e., gate current and state probabilities in the top diagrams) granting the ability to carefully design quantum-classical interfaces for quantum computation (left), and hybrid analog circuits containing classical and quantum circuit elements (right), including, for example, frequency multipliers, mixers and parametric amplifiers.

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This work is structured in three sections. Firstly, we discuss a systematic way to express the quantum dynamics within a compact model, with particular emphasis on QD-based devices, which may include charge, spin and Majorana qubits43. Then, we showcase three particular QD compact models, which represent the building blocks of any QD-based quantum device: (i) a QD exchanging single electrons with a reservoir, (ii) a double-QD (DQD) charge qubit, and (iii) a singlet-triplet spin qubit. We perform simulations of the quantum devices alone, which demonstrate excellent agreement with the theoretical expectations. Finally, we present the co-simulations of circuits formed of both classical and quantum components. In particular, (i) we exploit the nonlinear voltage-dependent impedance of the QD-to-reservoir transition to design a frequency multiplier suitable for operation at cryogenic temperatures, and (ii) we simulate qubit readout considering a charge qubit coupled to a high-Q microwave resonator, modeling the reflected signal in the adiabatic and resonant regimes. In this work, we mostly focus on showcasing quantum and co-simulation capabilities. Our models may also find use in the analysis of noise budgets, as well as dissipation effects, which are left for future work. All co-simulations presented in this work are performed using Cadence Spectre®—an industry-standard circuit simulator—and, wherever possible, compared with theoretical expectations obtained with standard Crank-Nicholson-based Lindblad simulations27,44,45,46.

Results

Quantum dynamics in analog compact models

Conceptually, co-simulating quantum dynamics within an electric circuit is an exercise in simultaneously simulating two worlds with different sets of variables, which evolve according to different laws, while providing a translation layer to allow for the exchange of signals at the boundary between the two (Fig. 2). Electronic components are generally described by a potential nature (voltage) and a flow nature (current), whose dynamics is described by Kirchhoff’s circuit laws. Quantum devices, on the other hand, do not natively fit this picture (i.e., notice how a circuit diagram is not expressive enough to indicate tunnel coupling between QDs in Fig. 2a). A better description may be found in terms of a density matrix ρ(t), which fully characterizes the state of an (open) quantum system47. In particular, the diagonal elements of the matrix (\({\rho }_{i,i}={\bf{Tr}}\,(\rho \left\vert i\right\rangle \left\langle i\right\vert)\)) are known as populations, and represent the probability of occupation of a (pure) state \(\left\vert i\right\rangle\), while the off-diagonal elements of ρ, the coherences, quantify the superposition between states48,49.

Fig. 2: Quantum-classical co-simulation.
figure 2

a The differential problem solved by the circuit simulator is conceptually spit into two realms: classical (blue), following Kirchhoff’s laws, and quantum (red), described by a Lindblad master equation. b The quantum evolution is recast into many equivalent subcircuits based on voltage-controlled current sources (VCCSs), while the model specifies how to convert between quantum and classical variables (c). αk,l represents the respective lever arm of each gate. Summation over repeated indices is implied.

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In this work, we model the evolution of the quantum dynamics via the LME47,50,51, a Markovian approximation that allows for the modeling of the coherent (unitary) quantum behavior while also including decoherence caused by coupling of the quantum system with the environment27,52. The LME defines the evolution of the density matrix as

$$\dot{\rho }(t)={\mathcal{L}}\rho (t),$$
(1)

where \(\dot{\rho }\) indicates the time derivative of ρ, and \({\mathcal{L}}\) is the (super-)operator known as the Liouvillian of the system, reading

$${\mathcal{L}}\rho =-{\rm{i}}\left[H/\hslash ,\rho \right]+\sum _{l}{\Gamma }_{l}{\mathcal{D}}\left({L}_{l}\right)\rho ,$$
(2)
$${\mathcal{D}}\left({L}_{l}\right)\rho ={L}_{l}\rho {L}_{l}^{\dagger }-\frac{1}{2}\left\{{L}_{l}^{\dagger }{L}_{l},\rho \right\}.$$
(3)

Here we define H as the Hamiltonian of the quantum system, while Ll is the jump operator describing a particular decoherence process (i.e., relaxation or dephasing), occurring at rate Γl. The evolution described by Eq. (1) is Markovian and assumes that the environment bears no memory. However, the inclusion of non-Markovian noise may be important to correctly model the fidelity of quantum information processing. To this end, it is in principle possible to extend the Verilog-A models to non-Markovian quantum dynamics, such as post-Markovian master equations53,54,55 or quantum Langevin and stochastic Schödinger equations48,49,56 evaluated via Monte Carlo methods57. However, we would like to note that non-markovianity and memory effects of the quantum dynamics usually arises from non-equilibrium fluctuators (in solid-state qubits typically two-level fluctuators58,59 and/or 1/f noise59,60), which could—and, for the sake of simulation efficiency and accuracy, should—be implemented directly as noise sources in the simulator’s netlist, causing the inputs of the model (i.e., gate voltages) to noisily fluctuate, effectively offloading the computational cost of dealing with stochastic models to the (highly optimized) simulator’s noise-analysis algorithm. We believe that assessment of Monte Carlo methods is outside of the scope of this document, and will be object of future work.

It is interesting to point out how, for each element of the density matrix, the linearity of the LME allows Eq. (2) to be written as

$${\dot{\rho }}_{i,j}+{\gamma }_{i,j}{\rho }_{i,j}=\sum _{k,l\ne i,j}{{\mathcal{L}}}_{klij}{\rho }_{k,l}=\sum _{k,l\ne i,j}{I}_{k,l\to i,j}^{\rho }$$
(4)

which, in circuit terms, describes a capacitor and a resistor in parallel driven by voltage-controlled current sources (VCCSs) whose value (linearly) depends on all other elements of the density matrix (ρk,l≠i,j), see Fig. 2b. This description of the LME is particularly amenable to co-simulation33,34,35 as it allows one to leverage electronics simulators’ native capabilities of solving coupled differential equations37, while, most importantly, allowing the quantum and classical dynamics to be expressed simultaneously (i.e., within one single Jacobian) to avoid any numerical issues regarding convergence61. As an implementation note, we recall for the benefit of the reader that the density matrix is generally a complex Hermitian matrix (ρ† = ρ). Thus, one may need to exploit this symmetry in order to separate the real and imaginary parts to comply with the need for real-valued voltages and currents in the equivalent circuit.

Figure 2b can be used to gain an intuitive understanding of the quantum behavior. In particular, unlike previous descriptions in the literature34,35, our LME-based approach shows a resistive element in parallel to the capacitor representing the time derivative. This promptly finds a physical interpretation by defining a time constant (1/γi,j) over which excitations in the relevant branch will be exponentially damped. In fact, with a little algebra, it is possible to show that the RC constant of diagonal elements (i = j) is equivalent to relaxation times (T1), while for off-diagonal branches it corresponds to its dephasing time (\({T}_{2}^{* }\)), providing a direct link between the quantum properties and the circuit representation of the system. Our ability to model non-unitary quantum dynamics enables not only the simulation of realistic quantum devices, but is critical for the inclusion of transition with leads in open systems, which are relied upon by charge sensors in most spin qubit architectures62,63,64. An example is discussed in the subsequent section.

Having described how the quantum and classical worlds may be implemented within a single differential problem, we are left with the task of describing how signals behave when crossing the boundary between the two worlds (Fig. 2c). While this depends heavily on the physics of the devices being modeled, it is generally a matter of charge bookkeeping. Charge is globally conserved. Thus, every charge entering the domain of quantum mechanics must be paid for by classical currents, and, likewise, each charge exiting the quantum world must be sourced as current by the terminals to then propagate within the electrical network. Charge conservation is a foundational requirement (and typical pain point) of compact modeling37.

When it comes to QD systems, it is particularly important to draw the world boundary just outside the quantum device and include the screening charges arising from capacitive couplings between different elements of the nanostructure (i.e., gates and leads). Not only this guarantees the charge neutrality of the model, but also aids in ensuring charge conservation. When charges in the QD system redistribute as a consequence of the quantum dynamics, in fact, this movement must be followed by a redistribution of the screening charge accumulated on the gates to which the QDs are capacitively coupled, giving rise to a gate current. If we consider a QD array where the lth terminal (gate) is coupled to the kth site via a capacitance Ck,l, at equilibrium, the respective screening charge will read65

$$\frac{{Q}_{k,l}}{{C}_{k,l}}=\frac{e{P}_{k}}{{C}_{{\Sigma }_{k}}},$$
(5)

where \({C}_{{\Sigma }_{k}}\) is the QD’s total capacitance and Pk its probability of occupation66. Therefore, the current at each terminal arising from charge movements within the quantum system reads66,67

$${I}_{l}(t)=e\sum _{k}{\alpha }_{k,l}{\dot{P}}_{k}(t),$$
(6)

where we define the lever arm \({\alpha }_{k,l}={C}_{k,l}/{C}_{{\Sigma }_{k}}\). From the quantum perspective, the QD’s occupation may be written in terms of the electrical dipole operator of the kth QD as \({P}_{k}={\bf{Tr}}\,({\Pi }_{k}\rho)\) (see “Methods”). According to Kirchhoff’s current law, the current Il(t) sourced or sunk by the terminal must be provided by nets in the classical circuit, thus linking the quantum and classical evolution of charges in the circuit. This is the physical description of gate currents66, which will be one of the key observables in our subsequent showcasing of QD compact models.

Equation (6) links the flow at the model’s terminals to the rate of change of quantum variables. However, we must also describe the effect of the terminal’s potential on the quantum dynamics. Once again, this is highly dependent on the physics of the system being modeled. To first order, however, and within the constant interaction model, the main effect of applying a voltage to a terminal is to vary the electrochemical potential of each site εk. This arises from the fact that the primary effect on a QD is the coupling of the electric field to the dipole of the system, neglecting any effect that alters the spatial envelope of the wave function. If the latter effect may not be ignored (e.g., if the QD system presents barrier gates), further modeling is necessary. In the simplest case, however, the on-site electrostatic energy reads27,68,69

$${\varepsilon }_{k}=-e\sum _{l}{\alpha }_{k,l}{V}_{l}.$$
(7)

Typically—as we shall see below—this quantity affects both the Hamiltonian and the decoherence processes in Eq. (3) (see Methods), making it possible for the classical signals to alter the quantum dynamics and thus inextricably coupling the differential equations governing the classical and quantum evolutions.

Quantum dot models

In this Section, we apply the co-simulation strategy discussed above and showcase the capability of compact models for three of the simplest basic blocks of QD circuits: (i) a QD exchanging single electrons with a reservoir, (ii) a DQD charge qubit, and (iii) a singlet-triplet spin qubit.

Single-electron box

The first model we discuss is the single-electron box (SEB), where a QD cyclically exchanges single electrons with a reservoir at thermal equilibrium at temperature T (Fig. 3). The QD is capacitively coupled to a gate and to the reservoir (CG and CR in Fig. 3a respectively), and charge tunneling events occur with a (total) tunnel rate Γ.

Fig. 3: Admittance of a single-electron box.
figure 3

Model of a SEB from the circuit (a) and quantum (b) perspectives. Lineshapes (left) and zero-detuning admittances (right) at f = 1 GHz in the thermal (c, d), lifetime (e, f), and power (g, h) broadening regimes. When not swept, the values of T = 100 mK, Γ = 0.5 GHz, and δVGR = 1 μV are assumed.

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From a Lindblad perspective, an SEB is best modeled as a two-level system, with the two states representing the state in which an electron either occupies or does not occupy the QD27,52. The Hamiltonian of this system reads, up to an arbitrary trace defining the zero of energy,

$$H=\frac{1}{2}\left(\begin{array}{cc}-{\varepsilon }_{{\rm{SEB}}}&0\\ 0&{\varepsilon }_{{\rm{SEB}}}\\ \end{array}\right)$$
(8)

where

$${\varepsilon }_{{\rm{SEB}}}/e={\alpha }_{{\rm{G}}}{V}_{{\rm{G}}}-(1-{\alpha }_{{\rm{R}}}){V}_{{\rm{R}}},$$
(9)

is the QD energy detuning with respect to the Fermi level of the reservoir. The parameters αG(R) indicate the QD-gate (reservoir) lever arm. The tunneling process in and out of the QD can then be represented by the jump operators27

$${L}_{{\rm{in}}}=\left(\begin{array}{cc}0\;1\\ 0\;0\\ \end{array}\right)\qquad {L}_{{\rm{out}}}=\left(\begin{array}{cc}0\;0\\ 1\;0\\ \end{array}\right),$$
(10)

and their respective tunnel rates then read52

$${\Gamma }_{{\rm{in}}}=\Gamma {\mathcal{F}}({\varepsilon }_{{\rm{SEB}}})\qquad\;\;{\Gamma }_{{\rm{out}}}=\Gamma \left(1-{\mathcal{F}}({\varepsilon }_{{\rm{SEB}}})\right).$$
(11)

The function

$$\begin{array}{lll}{\mathcal{F}}({\varepsilon }_{{\rm{SEB}}})\;=\;{{\mathcal{D}}}_{{\rm{QD}}}({\varepsilon }_{{\rm{SEB}}};\Gamma )* {f}_{{\rm{R}}}({\varepsilon }_{{\rm{SEB}}};{k}_{{\rm{B}}}T)\\\qquad\qquad\,=\;\displaystyle\frac{1}{2}+\frac{1}{\pi }{\psi }_{0}\left(\displaystyle\frac{1}{2}+\frac{\Gamma +{\rm{i}}{\varepsilon }_{{\rm{SEB}}}}{2\pi {k}_{{\rm{B}}}T}\right)\end{array}$$
(12)

represents the convolution of the (Lorentzian) effective density of states of the QD and the Fermi–Dirac distribution in the reservoir at thermal equilibrium. ψ0(z) represents Euler’s digamma function52.

To begin with, we investigate the small-signal properties of the compact model. We apply a small sinusoidal voltage (eαGδVGR ≪ kBT, hΓ) at the gate terminal and monitor the resulting gate current to determine the equivalent admittance seen by the gate. As shown in Fig. 3c–f, we find excellent agreement of the model with the theoretical expectations. Generally, the admittance as a function of gate-reservoir potential difference takes the form of a zero-centered single peak (Fig. 3c, e), whose width and height depend on the model’s parameters24,52,65,70. To showcase the model’s capabilities, we first vary the simulation temperature. This results in a thermal broadening of the peak (Fig. 3c), with the subsequent reduction in the maximum peak height (inversely proportional to the reservoir temperature24) shown in Fig. 3d.

Our inclusion of the effective density of state in Eq. (12) allows the compact model to further include a phenomenon known as lifetime broadening24,52. This effect is shown in Fig. 3e, f, where we sweep the tunnel rate for T = 100 mK. When first increasing Γ, we see a simple increase in peak height (green and purple traces in Fig. 3e). This arises from the fact that faster tunneling results in a larger number of tunneling events per ac cycle, thus increasing the gate current52. As the tunnel rates become faster, however, we notice the peak start to broaden and the height drops (red trace). This effect originates from the metastable nature of the electron level as the QD is coupled to the reservoir. The finite lifetime of electrons in the QD causes a Heisenberg broadening of the energy level with width hΓ. When the tunnel rates become comparable to (or faster than) the reservoir temperature, the short electron lifetime leads to a further broadening and lowering of the admittance peak due to the smearing of the effective electron density of states24,27,52,71.

Finally, we explore the properties of the model in the large-signal regime (Fig. 3g, h). To do so, we fix Γ = 2 GHz and kBT = 100 mK and increase the amplitude of the ac stimuli. From the resulting gate current, we extract the average admittance, defined as the first harmonic component of the gate current divided by the voltage amplitude52. As shown in Fig. 3g, when increasing the amplitude of the ac excitation, the gate current (δVGR∣Y∣) increases, resulting in a higher and broader peak—a phenomenon known as power broadening27. Observing the maximum peak height (Fig. 3h), we notice how the increase is at first linear with respect to the voltage amplitude (small-signal regime), and saturates at large amplitudes. This is easily understood by noting that, once the voltage swing is comparable to the lifetime and thermal broadening, the QD completely empties and fills once per ac cycle27. Therefore, increasing the voltage amplitude only affects the tails of the peak, but leads to progressively negligible increase of the gate current at zero detuning. This interpretation also implies nonlinearities in the gate current at large powers27, which we shall investigate and exploit for the purpose of frequency multiplication later in this work.

Notably, we see how all the effects discussed in this section are captured by our SEB compact model and are in excellent agreement with the expectations from theory52,62,66.

Double quantum dot

As a second example of a QD device, we demonstrate a compact model for a DQD charge qubit (Fig. 4). This is composed of two QDs coupled capacitively (with capacitance Cm) and through tunneling (with overlap tc), each also capacitively coupled to its own gate. The model allows for cross-capacitances between QDs and gates (not shown), as well as a stray capacitance (Cb) to ground. From a quantum perspective, the DQD is modeled as a two-level system, whose Hamiltonian reads66:

$$H=\frac{1}{2}\left(\begin{array}{cc}-{\varepsilon }_{{\rm{DQD}}}&2{t}_{{\rm{c}}}\\ 2{t}_{{\rm{c}}}&{\varepsilon }_{{\rm{DQD}}}\\ \end{array}\right),$$
(13)

where

$${\varepsilon }_{{\rm{DQD}}}/e={\alpha }_{{{\rm{G}}}_{1}}{V}_{{{\rm{G}}}_{1}}+{\alpha }_{{{\rm{G}}}_{2}}{V}_{{{\rm{G}}}_{2}}$$
(14)
Fig. 4: Admittance of a double quantum-dot charge qubit.
figure 4

Model of a DQD from circuit (a) and quantum (b) perspectives. Admittance lineshapes (left) and value at zero-detuning (right) at f = 1 GHz when varying temperature (c, d) and tunnel coupling (e, f). When not specified, the values T = 100 mK, tc = 8 GHz, Γcr = 0.5 GHz, and Γϕ = 0 are assumed.

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is the DQD detuning. The presence of a finite tunnel coupling between the two QDs gives rise to an avoided crossing at zero detuning between the two levels (Fig. 4b). Therefore, the eigenstates of the system (\(\left\vert \pm \right\rangle\)) are now dependent on detuning (and hence on voltage). This is in stark contrast to the previous case of the SEB, and, as we shall demonstrate, allows for coherence phenomena in response to changes in voltage.

Our compact model includes the presence of decoherence processes, treated in a Born approximation known as the instantaneous eigenvalue approximation52,72, which assumes stochastic processes to be well-described in the instantaneous eigenbasis of the (time-dependent) Hamiltonian51,73,74,75. In particular, we follow a standard model in the literature for the DQD66,68,69, where relaxation processes are dominated by coupling with the phonon bath in the lattice, reading

$${L}_{\uparrow }^{{\prime} }=\left(\begin{array}{cc}0\;1\\ 0\;0\\ \end{array}\right)\qquad {L}_{\downarrow }^{{\prime} }=\left(\begin{array}{cc}0\;0\\ 1\;0\\ \end{array}\right),$$
(15)

where the primes indicate that these are defined in the instantaneous eigenbasis. Their respective relaxation rates read

$${\Gamma }_{\uparrow }={\Gamma }_{{\rm{cr}}}n({\varepsilon }_{{\rm{DQD}}})\qquad {\Gamma }_{\downarrow }={\Gamma }_{{\rm{cr}}}\left(n({\varepsilon }_{{\rm{DQD}}})+1\right),$$
(16)

where Γcr is the charge relaxation rate quantifying the charge-phonon coupling, and

$$n({\varepsilon }_{{\rm{DQD}}})={\left(1-\exp \left(\frac{\sqrt{{\varepsilon }_{{\rm{DQD}}}^{2}+4{t}_{{\rm{c}}}^{2}}}{{k}_{{\rm{B}}}T}\right)\right)}^{-1}$$
(17)

is the Bose–Einstein statistics of the phonon bath at the (instantaneous) \(\left\vert \pm \right\rangle\) energy separation. We also include the possibility of pure dephasing processes, described by the jump operator

$${L}_{\phi }^{{\prime} }=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1&0\\ 0&-1\\ \end{array}\right),$$
(18)

with (energy-independent) rate Γϕ. The total decoherence rate reads \(\gamma /2\pi ={\Gamma }_{{\rm{cr}}}\left(n({\varepsilon }_{{\rm{DQD}}})+1/2\right)+{\Gamma }_{\phi }\)66.

To begin with, we consider the small-signal (ac) admittance of the model seen by one of the gates at frequency f = 1 GHz. We consider slow relaxation (Γcr = 0.5 GHz) with no dephasing. Firstly, we explore the system’s response to an increase in temperature (Fig. 4c, d) in the adiabatic regime tc = 5 GHz ≫ f. Similarly to the SEB, the admittance takes the form of a zero-centered peak, whose height decreases as the temperature increases, in accordance with the theoretical expectations. The physical origin of this phenomenon is that, in this regime, the admittance is dominated by the system’s quantum capacitance, deriving from the redistribution of electron occupation within the QDs due to the avoided crossing68. This quantity is generally proportional to the second derivative of the eigenenergies69, and thus is opposite in sign for ground and excited states. As temperature increases, the system depolarizes and there is (exponentially) more probability of occupation of the excited state. For temperatures larger than the energy splitting, the \(\left\vert \pm \right\rangle\) quantum capacitances cancel out, leading to a vanishing admittance. However, from Fig. 4c we see that the peak not only shrinks, but also broadens. This is due to the energy dependence of the relaxation rates in Eq. (16), which gives rise to a tunneling capacitance and Sisyphus resistance65,68,69. The faithful reproduction of this phenomenon in the Spectre® simulation demonstrates the ability of our model to include Sisyphus phenomena and the full capabilities of the presented approach to implement Liouvillians within the instantaneous-eigenvalue approximation.

Secondly, we fix the simulation temperature at T = 100 mK, and explore the response for varying tunnel coupling (Fig. 4e, f). As tc becomes smaller, the anticrossing becomes narrower and sharper, as does the DQD admittance. This trend, however, only holds while the system is in the adiabatic regime. As 2tc decreases below the excitation frequency, the excitation is able to coherently drive the charge qubit. This leads to the admittance splitting into two peaks (Fig. 4e), with sharp features at the detuning, where the charge qubit is resonant with the ac stimuli66. This can be understood as a resonant driving of the charge qubit by the sinusoidal excitation76,76,77,78,79. Decreasing tc further decreases the dipole of the system, thus leading to a sharp drop in the zero-detuning admittance (Fig. 4f)80. Figure 4 is a demonstration of the capability of a compact model to reproduce resonant quantum behavior within a standard classical electronics simulator.

We now explore the large-signal regime (Fig. 5), further discussing the description of coherent phenomena. From the quantum perspective, the biggest difference compared with the SEB is the presence of an anticrossing, which, when rapidly changing detuning in its proximity, may give rise to diabatic transitions into other states, known as Landau–Zener transitions, with probability PLZ (Fig. 5b)81,82. If the voltages are changed periodically, even if the system is initialized in its ground state, periodic fast passages across the anticrossing (Fig. 5b) may generate a coherent superposition of the ground and excited states. This leads to the accumulation of a (dynamical) quantum phase in the system, which in turn leads to self-interference of the electron, in a process known as Landau–Zener–Stückelberg–Majorana (LZSM) interference83,84,85. In Fig. 5c, we explore this phenomenon in our compact model. In particular, we set 1/Γcr = 2 ns and neglect dephasing, to allow the coherence time to be longer than the period of the sinusoidal excitation (f = 1 GHz). As shown, this results in the emergence of LZSM interference fringes in the equivalent DQD admittance, corresponding to constructive and destructive interference of the accumulated phase, in excellent agreement with the theory. To explore the role of dephasing, we now increase Γϕ such that the coherence time of the system is shorter than the ac period. As shown in Fig. 5d, this results in the disappearance of the interference fringes, which blur into one single broad peak, in accordance with the theory and experimental findings27,71,83,84. This further demonstrates the possibility of fully capturing coherent quantum phenomenon in a traditional mixed-signal electronic simulation.

Fig. 5: LSZM interference in a charge qubit.
figure 5

Model of LSZM interference from the circuit (a) and quantum (b) perspectives: the gate of a charge qubit is strongly driven to cause diabatic transitions and accumulation of a dynamical phase. First-harmonic gate current (δVG1∣Y∣) in the case where the coherence time 2π/γ is much longer (c) and much shorter (d) than the excitation frequency f = 1 GHz.

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Finally, we explore the coherent evolution of our model in the time domain (Fig. 6). To do so, we first initialize the system in its ground state far from the anticrossing, so that the charge is strongly localized in one of the QDs. We then pulse one of the gates within 500 ps to a (variable) voltage near the charge transition, where we allow the system to evolve freely (Fig. 6a, b) while recording the gate current of one of the gates and the occupation probability of the associated QD. Figure 6 shows the characteristic pattern of the Rabi chevron86 recreated in the gate current (Fig. 6c, d) while Rabi oscillations are clearly visible in the population of the QD (Fig. 6e, f). These are due to the fast pulse diabatically creating a superposition of ground and excited states, outside the z-axis of the (instantaneous) Bloch sphere at VRabi. This state then naturally precesses around the z-axis, generating Rabi oscillations within the charge occupation, exponentially damped in the Lindblad formalism by the finite coherence rate. As shown in Fig. 6, the simulation results are in excellent agreement with the theory, showcasing the ability to reproduce coherent effects in time-domain transients.

Fig. 6: Damped Rabi oscillations of a charge qubit.
figure 6

Circuit (a) and quantum (b) model for generating Rabi oscillations: the gate of a charge qubit is pulsed towards the anticrossing (up to VRabi) to generate a coherent superposition of ground and excited state. Circuit simulations (left) and theory expectations (right) of the gate current (c, d) and QD occupation probability (e, f) after the pulse.

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Singlet-triplet spin qubit

Lastly, we showcase the capability of the modeling framework by going beyond simple charge dynamics and including spin degrees of freedom. In particular, we model the (2, 0) ↔ (1, 1) transition of a DQD (Fig. 7a) because of its relevance to quantum technologies due to the phenomenon of Pauli Spin Blockade (PSB)65. The Pauli exclusion principle states the electron wave function must be completely antisymmetric; thus, to occupy the same QD level (symmetrical orbital degrees of freedom), two (spinful) electrons must have antisymmetric spin orientation. This spin state takes the name of a singlet (\(\left\vert S\right\rangle =(\vert\! \uparrow \downarrow \rangle -\vert\! \downarrow \uparrow \rangle )/\sqrt{2}\)) and is the only state allowed in the (2, 0) charge configuration80. When the electrons are in separate QDs, on the contrary, the orbital state is already different and any of the four spin states (one singlet \(\vert S\rangle\) and three triplets \(\vert {T}_{-}\rangle =\vert\! \downarrow \downarrow \rangle ,\vert {T}_{0}\rangle =(\vert\! \uparrow \downarrow \rangle +\vert\! \downarrow \uparrow \rangle )/\sqrt{2},\vert {T}_{+}\rangle =\vert\! \uparrow \uparrow \rangle\)) is allowed (Fig. 7b). For the sake of simplicity in the model, we neglect spin-orbit coupling80, hence charge transitions may not affect spin degrees of freedom. Consequently, the transition (1, 1) → (2, 0) is allowed to occur only for the \(\vert {S}_{11}\rangle\) single state, while for the triplets it is blockaded by the Pauli principle (even if would be energetically favorable). This is of great technological importance as it allows the conversion between spin and charge degrees of freedom, and is at the core of enabling fast high-fidelity readout in most spin-qubit architectures62,63.

Fig. 7: Pauli spin blockade and its lifting due to fast relaxation.
figure 7

a Model of a singlet-triplet spin qubit from circuit perspective. b System eigenenergies as a function of detuning for zero (solid black lines) and non-zero magnetic field (dashed green lines). c Admittance for varying gate voltages and magnetic field at zero (ground state) and finite temperature for slow (center) and fast (right) relaxation, showing PSB and its lifting. The dashed line shows the signet-triplet crossing where \(2g{\mu }_{{\rm{B}}}B=\sqrt{{(2{t}_{{\rm{c}}})}^{2}+{\varepsilon }_{{\rm{DQD}}}^{2}}\). The system is probed at 500 MHz with tc = 7 GHz.

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From the modeling perspective, the Hamiltonian H in the \(\{\left\vert {S}_{20}\right\rangle ,\left\vert {S}_{11}\right\rangle ,\left\vert {T}_{-}\right\rangle ,\left\vert {T}_{0}\right\rangle ,\left\vert {T}_{+}\right\rangle \}\) basis reads68

$$\left(\begin{array}{ccccc}-{\varepsilon }_{{\rm{DQD}}}/2&{t}_{{\rm{c}}}&0&0&0\\ {t}_{{\rm{c}}}&{\varepsilon }_{{\rm{DQD}}}/2&0&0&0\\ 0&0&-g{\mu }_{{\rm{B}}}B+{\varepsilon }_{{\rm{DQD}}}/2&0&0\\ 0&0&0&{\varepsilon }_{{\rm{DQD}}}/2&0\\ 0&0&0&0&g{\mu }_{{\rm{B}}}B+{\varepsilon }_{{\rm{DQD}}}/2\\ \end{array}\right),$$
(19)

where B is the applied magnetic field, μB is Bohr’s magneton, g the system’s g factor (here we take g = 2 as a typical value for electrons in silicon87), and all other parameters are defined as above. It is notable that the (tunnel) coupling is only between the singlets because of the Pauli principle. The energy spectrum is shown in Fig. 7b for zero and nonzero field.

For the sake of simplicity, relaxation is modeled phenomenologically as in ref. 68, where we include jump operators for all pairs of states k ≠ l of the form \({L}_{kl}=\left\vert k\right\rangle \left\langle l\right\vert\) with rates66

$${\Gamma}_{kl}=\left\{\begin{array}{ll}{\Gamma}_{r}n\left(\frac{{E}_{k}-{E}_{l}}{{k}_{B}T}\right)\quad &{\rm{if}}\quad {E}_{k}-{E}_{l} > 0\\ {\Gamma}_{r}\left[n\left(\frac{{E}_{k}-{E}_{l}}{{k}_{B}T}\right)+1\right]\quad &{\rm{if}}\quad {E}_{k}-{E}_{l} < 0\\ \quad \end{array}\right.$$
(20)

where \(n(x)={[1-\exp (-x)]}^{-1}\) is the Bose–Einstein distribution as above. Notably, thanks to the detailed balance principle, this ensures that at equilibrium the system always approaches the Gibbs state \(\rho =\exp (-H/{k}_{B}T)\)66. The study of dephasing is left for future work.

Figure 7c shows the small-signal admittance of the model as a function of gate voltages (detuning) and magnetic field. For B = 0, the ground state is the singlet thanks to the effect of the tunnel coupling, giving rise to a similar peak to the charge qubit. As magnetic field increases, the \(\vert {T}_{-}\rangle =\vert\! \downarrow \downarrow \rangle\) state decreases its energy thanks to the Zeeman contribution, becoming degenerate with the singlet along the hyperbola where \(2g{\mu }_{{\rm{B}}}B=\sqrt{{(2{t}_{{\rm{c}}})}^{2}+{\varepsilon }_{{\rm{DQD}}}^{2}}\) (dashed cyan line), to the left of which (i.e., towards the (1, 1) configuration) it becomes the system’s ground state. As mentioned above, charge movement is blockaded in the \(\left\vert {T}_{-}\right\rangle\), suppressing the system’s admittance. As a result, ∣Y∣ sharply drops asymmetrically in the candlestick shape68, a clear signature of PSB. The expected response of the ground state (left) is maintained even at finite temperature in the regime of slow relaxation (center), where the main effect is a smearing of the peak, particularly notable in the sharp drop at the edge of PSB. As Γr ≪ ω (the system is probed at 500 MHz), relaxation is unable to occur within a cycle, preserving the blockade. The response is qualitatively difference in the fast relaxation regime (right) where Γr ≫ ω. In this case, the system may be pictured always in the instantaneous thermal equilibrium, causing a spin flip upon crossing of the hyperbola (see inset), effectively lifting PSB as expected from theory68. We stress that relaxation-based lifting of PSB is intrinsically of dynamical nature thanks to Sisyphus processes66. Thus, this demonstrates not only the capability of modeling non-charge quantum degrees of freedom, but also to faithfully include dynamical relaxation. Thus, our compact modeling methodology may find use in design and optimization of qubit initialization and readout schemes15,65.

Co-simulation of quantum dots and analog circuits

Having investigated the behavior of the compact models, validating them against theory, we now showcase our co-simulation capabilities by implementing QD-based analog circuits composed of a QD compact model and linear circuit elements. In particular, we demonstrate (i) a SEB-based frequency multiplier, leveraging the nonlinear properties of the quantum system, and (ii) a charge qubit coupled to an RLC resonator, investigating the frequency response in the adiabatic and resonant regimes.

Single-electron box frequency multiplier

The first hybrid quantum-classical device we discuss is a frequency multiplier based on the QD-to-reservoir transition of a SEB (Fig. 8). Our implementation is based on traditional designs of analog circuits for frequency multiplication88,89,90, and it can be understood as essentially composed of two independent resonant current loops. On the input side (left Fig. 8a), we find a resistor-inductor-capacitor (RLC) resonator tuned for the input frequency (in our design f0 ~ 0.5 MHz), with the aim of providing a monochromatic driving for the SEB gate, while simultaneously increasing the amplitude of the oscillating gate voltage (Fig. 8b, g). A bias tee provides the ability to vary the dc detuning of the gate. On the output side (right Fig. 8a), we find a similar design, where the SEB reservoir is connected to a parallel RLC resonator featuring a variable capacitor. If the same inductor is used in both loops, setting the output resonator capacitor to CR/N2 causes the output loop to resonate at the Nth harmonic of the input frequency f0, filtering out unwanted Fourier components. Both the input and the output are coupled to 50 Ω lines via coupling capacitors.

Fig. 8: Frequency multiplier circuit based on a single-electron box.
figure 8

a Circuit schematic of the frequency multiplier. b–f The panels show the input, gate (b), and output voltages (e) in the time domain, and the time and frequency domains of the reservoir current (c, f). e The characteristic two-lobed fan of the output voltage with respect to SEB detuning and input amplitude27. g–m The same quantities when the circuit is tuned for N = 3.

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The frequency-multiplication capabilities of this circuit stem from the inherent nonlinearity of the SEB. In particular, when the amplitude of the gate-reservoir voltage swing is larger than any other broadening (δε ≫ hΓ, kBT), the excitation will produce one tunneling event per cycle. Thus, the probability of occupation of the QD will resemble a square wave (or, more generally, one combined with a sawtooth, depending on the tunnel rates)27,28,91. From Eq. (6), the reservoir current IR is proportional to the time derivative of the QD occupation, causing it to have a rich Fourier decomposition (Fig. 8c, f, h, m). Most importantly, from the above discussion, it is clear how the duty cycle of the QM occupation square wave depends on the dc detuning of the QD with respect to the reservoir, thus rendering the reservoir current electrically tunable.

In Fig. 8, we explore the frequency-multiplication performance of the circuit, analyzed via harmonic-balance simulations. In Fig. 8b–f, we tune the circuit for a multiplication factor of N = 2. Panel b shows both the passive amplification of the input resonator and the addition of the dc voltage, generating the highly nonlinear reservoir current in panel c. Panel f shows the Fourier decomposition of IR and the impedance Z22 seen by the output line, which drops to ~50 Ω for 2f0. The output resonator thus efficiently acts as a band-pass filter, ensuring high harmonic purity of the output, shown in Fig. 8e. In Fig. 8d, we sweep input amplitude and dc detuning of the system, showing the characteristic N-lobed fan experimentally observed in similar SEB-base systems27,28. We show similar results in Fig. 8g–m, in which we vary the values of the capacitors for the output branch to be resonant and 50 Ω-matched at N = 3, resulting in excellent harmonic purity of the third harmonic at the output (Fig. 8l). Moreover, in Fig. 8i, we show the expected three-lobed fan in the output voltage when sweeping detuning and input amplitude, similarly to the N = 2 case.

Circuit quantum electrodynamics with a charge qubit and a high-Q resonator

As a last example, we discuss the case of a DQD charge qubit coupled to a microwave resonator (f0 = 2 GHz). Similarly to the previous circuit, we consider a parallel RLC resonator (Fig. 9a) with a dc bias tee to vary the DQD detuning. The system is coupled to a 50 Ω lead via a coupling capacitor, mimicking a traditional radio-frequency reflectometry setup62,65, and we study the reflected signal (∣S11∣) while varying DQD parameters, comparing the simulated resonant frequency shift with the expectations from circuit quantum electrodynamics (cQED)66,76,92, shown as black dashed lines in Fig. 9.

Fig. 9: Adiabatic and resonant dispersive sensing of a charge qubit.
figure 9

a Circuit schematic of a DQD charge qubit coupled to a high-Q (Q = 1000) microwave resonator (f0 = 2 GHz). b–e ∣S11∣ parameter for the circuit as a function of frequency and detuning, showing the dispersive shift of the RLC resonator caused by the qubit in the adiabatic (hf0 ≫ 2tc, top) and resonant (hf0 < 2tc, bottom) regimes, in the presence of decoherence that is slow (ℏγ ≪ 2tc, left) or fast (ℏγ ≫ 2tc, right).

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Firstly, we consider the case in which the charge qubit has a long coherence time compared to the cavity (2π/γ = 5 ns ≫ 1/f0). In the adiabatic regime, where 2tc = 10 GHz (Fig. 9b), we see the traditional result of a pull of the resonance towards lower frequencies, due to the quantum capacitance of the DQD in parallel to CR92. If we lower 2tc below the resonator frequency, however, we see a starkly different response. In Fig. 9c, we observe a large shift in the resonant frequency where the qubit frequency is resonant with the cavity, characterized by a change in sign of quantum capacitance. This leads to a pull of the resonance towards higher frequencies in the region where the qubit frequency drops below f0, in excellent agreement with cQED66,92. The behavior in Fig. 9c may be understood effectively as a vacuum Rabi splitting caused by the qubit–resonator coupling, which we show faithfully reproduced by the Spectre® simulation.

Finally, we explore the influence of DQD dephasing on the reflected signal (Fig. 9d, e). To do so, we increase the dephasing rate Γϕ to 2π/γ = 1 ps ≪ 1/f0. In the adiabatic regime (Fig. 9d), we merely see a reduction in the frequency pull, caused by the response of the qubit becoming increasingly resistive because of the increased losses66,76. The effect is far more dramatic in the resonant case (Fig. 9e), where we observe the peak splitting and the region of negative capacitance disappear, in favor of a zero-centered peak of positive capacitance resembling the adiabatic case. The Spectre® simulations correctly reproduce the behavior predicted by our unified linear-response theory of quantum systems66, demonstrating the full capabilities of our Lindblad-based compact models.

Discussion

We have shown that Verilog-A compact models of QD devices—a SEB and DQD charge and spin qubits—are able to reproduce coherent phenomena while remaining fully compatible with industry-standard analog circuit simulators. Furthermore, we have demonstrated the utility of these models for the design of two quantum-classical hybrid circuits, a SEB-based frequency multiplier and a charge qubit embedded in an LC resonator for quantum state readout, both circuits fully simulated in Cadence Spectre®. Our work proposes a systematic method to produce a compact model for any arbitrary multilevel quantum system, as long as it can be described by a LME, which can then be co-simulated alongside classical circuit components in most available circuit simulators. We believe our models and methodology will become a useful tool to explore the interplay between classical and quantum information, and between classical and quantum hardware, as well as for critically assessing the impact of information leakage and back-actions from the classical world and the quantum systems. Moreover, our models may be used for the purpose of designing qubit control and readout schemes, as well as quantitatively assessing the available noise budget. The ability to co-simulate quantum systems together with realistic hardware—including non-idealities and performance trade-offs—may also guide the development of qubits that operate with less complex electronics, enhancing the system’s scalability93. Lastly, the future inclusion of decoherence in our models may be leveraged to explore the boundaries of classical-quantum transitions, especially when such effects are brought on by coupling with classical subsystems. Overall, our methodology bridges the existing gap in simulation capabilities between quantum and classical electronics, and enables quantum circuits to leverage the electronic design and automation tools that have enabled the very large-scale integration achieved by classical electronics, ushering in a new paradigm for the design and optimization of the quantum-classical interfaces, and unlocking the design of novel hybrid quantum-classical analog circuits.

Methods

Derivation of the coupling between quantum and classical subsystems

Here, we give a rigorous derivation from the quantum perspective of the equations linking the LME governing the quantum dynamics to the voltages and currents at the model’s inputs and outputs.

Let us begin with the discussion of gate currents. As discussed in the main text, this keeps track the fact that charge redistribution events in a QD system will also necessarily appear as a redistribution of the screening charge accumulated on the gates to which said QDs are capacitively coupled. If we consider a QD array where the lth gate is coupled to the kth QD with a capacitance Ck,l, at equilibrium, this charge will read

$$\frac{{Q}_{k,l}}{{C}_{k,l}}=\frac{e{P}_{k}}{{C}_{{\Sigma }_{k}}},$$
(21)

where Pk(t) is the probability of the kth QD to be occupied, \({C}_{{\Sigma }_{k}}\) is its self-capacitance, and we use l to index over the gates and k over the QD sites. It is therefore natural to write65,68,69

$${I}_{k,l}(t)={\alpha }_{k,l}e\frac{d}{dt}{P}_{k}(t)$$
(22)

as the gate current which is caused on the lth gate of the array by a charge redistribution event on the kth QD. In Eq. (22) appear the lever arm \({\alpha }_{k,l}={C}_{k,l}/{C}_{{\Sigma }_{k}}\) as in the main text.

From the quantum perspective, the probability of occupation is an observable, and thus may be considered as the expected value of a (dimensionless) charge operator Πk, such that66,94

$${P}_{k}={\bf{Tr}}\,({\Pi }_{k}\rho) ,$$
(23)

and, thus, the total gate current on the lth gate reads, summing over all sites

$${I}_{l}(t)=e\frac{d}{dt}\left(\sum _{k}{\alpha }_{k,l}\,\left({\bf{Tr}}{\Pi }_{k}\rho\right) \right)=e\sum _{k}{\alpha }_{k,l}\,\left({\bf{Tr}}{\Pi }_{k}\dot{\rho }\right)$$
(24)

as discussed in Eq. (6) in the main text.

We now consider the effect of terminal potentials. As discussed in the main text, in this work we mostly focus on systems which couple to the gates’ electric field via their electric dipole. The interaction Hamiltonian from the lth gate and the kth QD reads66,77,95,96

$${H}_{k,l}^{\varepsilon }(t)=-{\alpha }_{k,l}e{\Pi }_{k}{V}_{l}(t),$$
(25)

where Πk is the same charge operator in Eq. (23)66. Thus, the on-site electrostatic Hamiltonian for the kth QD reads (up to potentially an unimportant trace)

$${H}_{k}^{\varepsilon }={\Pi }_{k}{\varepsilon }_{k},$$
(26)

where εk is the on-site electrostatic energy defined in Eq. (7) of the main text. This shows how the terminal voltages enter in the system’s Hamiltonian and thus the time evolution. Moreover, often both the jump operators and the tunnel rates (e.g., Eq. (11) for the SEB and Eq. (16) for the DQD) depend on the instantaneous on-site energies. Thus, at each timestep, the input potentials enter in the definition of the Linbladian \({\mathcal{L}}(t)\), thus coupling the quantum and classical (voltages and currents) dynamics. It is important to note how this formalism is entirely general, and can be expanded to voltage dependencies other than (plunger-defined) on-site energies, such as dynamical control of tunnel couplings via barrier gates97.