Numerical analysis of effective models for flux-tunable transmon systems

Simulations and analytical calculations that aim to describe flux-tunable transmons are usually based on effective models of the corresponding lumped-element model. However, when a control pulse is applied, in most cases it is not known how much the predictions made with the effective models deviate from the predictions made with the original lumped-element model. In this work we compare the numerical solutions of the time-dependent Schrödinger equation for both the effective and the lumped-element models, for microwave and unimodal control pulses (external fluxes). These control pulses are used to model single-qubit (X) and two-qubit gate (iswap and cz) transitions. First, we derive a nonadiabatic effective Hamiltonian for a single flux-tunable transmon and compare the pulse response of this model to the one of the corresponding circuit Hamiltonian. Here we find that both models predict similar outcomes for similar control pulses. Then, we study how different approximations affect single-qubit (X) and two-qubit gate (iswap and cz) transitions in two different two-qubit systems. For this purpose we consider three different systems in total: a single flux-tunable transmon and two two-qubit systems. In summary, we find that a series of commonly applied approximations (individually and/or in combination) can change the response of a system substantially, when a control pulse is applied.

Figure 1

Sketches of the circuit architectures (a) I and (b) II. Both types of architectures use flux-tunable transmon qubits to activate two-qubit gate transitions (see Ref. [5] for (a) and Ref. [6] for (b)). We use the circuit and effective Hamiltonians given by Eqs. (7) and (31) and the device parameters listed in Tables 1 and 3 to perform simulations of iswap and cz two-qubit gate transitions for architecture I. Similarly, we use the circuit and effective Hamiltonians given by Eqs. (8) and (32) and the device parameters listed in Tables 2 and 2 to perform simulations of iswap and cz two-qubit gate transitions for architecture II.

Figure 2

External flux $\phi /2\pi$ as a function of time for two different flux control pulses. (a) Microwave pulse using Eq. (33), amplitude $\delta /2\pi =0.075$, drive frequency ${\omega }^D/2\pi =1.089\mathrm{GHz}$, a rise and fall time of $T_{\text{r/f}}=13\mathrm{ns}$, and pulse duration $T_{\mathrm{d}}=205.4\mathrm{ns}$. (b) Unimodal pulse using Eq. (33), amplitude $\delta /2\pi =0.297$, drive frequency ${\omega }^D/2\pi =0\mathrm{GHz}$, a rise and fall time of $T_{\text{r/f}}=20\mathrm{ns}$, and pulse duration $T_{\mathrm{d}}=84\mathrm{ns}$.

Figure 3

Ground-state probabilities $p^{\left(0\right)}$ as functions of the pulse duration $T_{\mathrm{d}}$ and the drive frequency ${\omega }^D$. We use the device parameters for a single flux-tunable transmon listed in Table 1, row $i=2$, and the pulse given by Eq. (33) with $T_{\mathrm{r}/\mathrm{f}}=T_{\mathrm{d}}/2$ and the pulse amplitude $\delta /2\pi =0.001$ [see Fig. 2] to obtain the results. The results in (a) are obtained by solving the TDSE for the circuit Hamiltonian in Eq. (2). Similarly, the results in (b) are obtained by solving the TDSE for the nonadiabatic effective Hamiltonian in Eq. (18). At time $t=0$ the systems are initialized in the corresponding eigenstates $p^{\left(0\right)}\left(0\right)=1$. Here we model Rabi transitions between the ground state and the first excited state. Note that (a) and (b) are centered around the frequency ${\omega }^{\left(0\right)}$ which corresponds to the energy difference $E^{\left(1\right)}-E^{\left(0\right)}$ in the corresponding model, i.e., the circuit or the effective model. We see that apart from the shift in the transition frequency both models show a similar qualitative and quantitative behavior. However, the effective model given by the Hamiltonian in Eq. (25) does not allow us to model these transitions.

Figure 4

Probabilities (a) $p^{\left(0\right)}\left(t\right)$ and (b) $p^{\left(1\right)}\left(t\right)$ as functions of time $t$ obtained with the effective model (blue circles) and the circuit model (green squares). We use the pulse given by Eq. (33) with the pulse amplitude $\delta /2\pi =0.001$, the pulse duration $T_{\mathrm{d}}=200\mathrm{ns}$, the rise and fall time $T_{\mathrm{r}/\mathrm{f}}=100\mathrm{ns}$, and the device parameters for a single flux-tunable transmon listed in Table 1, row $i=2$, to obtain the results. We use the drive frequency ${\omega }^D=7.636\mathrm{GHz}$ to obtain the results with the circuit Hamiltonian, Eq. (2). Similarly, we use the drive frequency ${\omega }^D=7.643\mathrm{GHz}$ to obtain the results with the effective Hamiltonian, Eq. (18). The systems are initialized in the ground state $p^{\left(0\right)}=1$ at time $t=0$. Note that in (a) and (b) we use the frequencies which cut through the centers of the chevron patterns in Figs. 3 and 3. As one can see, (a) and (b) show qualitatively and quantitatively similar behavior with respect to the time evolution.

Figure 5

Probabilities $1-p^{\left(m\right)}$ at time $T_d$ as functions of the rise and fall time $T_{\mathrm{r}/\mathrm{f}}$ and the pulse amplitude $\delta$. We use the pulse given by Eq. (33) with ${\omega }^D=0$ GHz and $T_{\mathrm{d}}=50\mathrm{ns}$ [see Fig. 2] and the device parameters for a single flux-tunable transmon listed in Table 1, row $i=2$, to obtain the results. The results for (a) $m=0$, (b) $m=1$, (c) $m=2$, and (d) $m=3$ are obtained by solving the TDSE for the circuit Hamiltonian in Eq. (2). Similarly, the results for (e) $m=0$, (f) $m=1$, (g) $m=2$, and (h) $m=3$ are obtained by solving the TDSE for the nonadiabatic effective Hamiltonian given by Eq. (18). At time $t=0$ the systems are initialized in the corresponding eigenstates $p^{\left(m\right)}\left(0\right)=1$. The simulations test whether or not we have left the pulse parameter regime where the adiabatic approximation is valid; i.e., the bright areas indicate the parameters which lead to nonadiabatic transitions. Note that it is impossible to use the effective Hamiltonian in Eq. (25) to model such nonadiabatic transitions. Interestingly, for $m=0$, $m=1$, and $m=2$ the effective model given by the Hamiltonian in Eq. (18) shows a qualitatively similar behavior as the circuit model given by the Hamiltonian in Eq. (2).

Figure 6

(a) Chevron pattern for a pulse of the form of Eq. (33) with the drive frequency ${\omega }^D=6.183\mathrm{GHz}$ and the amplitude $\delta /2\pi =0.045$. (b) Chevron pattern for a pulse of the form of Eq. (33) with the drive frequency ${\omega }^D=5.092\mathrm{GHz}$ and the amplitude $\delta /2\pi =0.085$. We use the same rise and fall time $T_{\mathrm{r}/\mathrm{f}}=T_{\mathrm{d}}/2$ for both cases. These patterns show how the circuit Hamiltonian in Eq. (7) (with the parameters listed in Table 1) reacts to two different pulses, characterized by the different pulse parameters. The color bar shows the probability $p^{(0,0,0)}$ as a function of the pulse duration time $T_{\mathrm{d}}$. The chevron patterns are used to calibrate control pulses, which are then used to obtain the results in Table 4.

Figure 7

Effective interaction strengths $g$ [see Eq. (29)] in blue on the left $y$ axis and $\overline{g}$ [see Eq. (30)] in green on the right $y$ axis as functions of the external flux $\phi$. We use the energies listed in Table 1 ($i=2$) to obtain $g$. Similarly, for $\overline{g}$ we use the parameters listed in Table 1 ($i=1$).

Figure 8

Effective interaction strength as a function of time for two different flux control pulses. (a) Effective interaction strength $g\left(\phi \right(t\left)\right)$ [see Eqs. (29) and (33)] for architecture I. We use the energies listed in Table 1 ($i=2$) to obtain $g$ and the same control pulse parameters as in Fig. 2. These control pulse parameters are also listed in Table 5 (row 6). (b) Effective interaction strength $\overline{g}\left(\phi \left(t\right)\right)$ [see Eqs. (30) and (33)] for architecture II. We use the energies listed Table 1 ($i=1$) to obtain $\overline{g}$ and the same control pulse parameters as in Fig. 2. These control pulse parameters are also listed in Table 5 (row 9).

Figure 9

Probabilities $p^{\left(z\right)}\left(t\right)$ as functions of time for (a)-(b) $z=(0,0,1)$ and $z=(0,1,0)$ and (c)-(d) $z=(0,1,1)$ and $z=(0,2,0)$. In panels (a) and (c) we model the system without a time-dependent effective interaction strength [see Eq. (29)]. In panels (b) and (d) we include the time dependence. In all cases we use the Hamiltonian in Eq. (31), the parameters listed in Tables 1 and 2, and a pulse of the form of Eq. (33) to obtain the results. The pulse parameters are discussed in the main text. The $z=(0,1,1)\to z=(0,2,0)$ transitions are usually used to implement cz operations and the $z=(0,0,1)\to z=(0,1,0)$ transitions are often used to realize iswap operations (see Refs. [3, 13]). Interestingly, we observe a large shift in the pulse duration $T_{\mathrm{d}}$ if we model the system with a time-dependent effective interaction strength [see Fig. 8].

Figure 10

Functional sketch of the control pulses [see Eq. (33)] we use to determine the results in Figs. 11 and 11. We use the Hamiltonian given by Eq. (31) and the device parameters listed in Table 1 to model the dynamics of a system of type architecture I. The intention is to investigate the transition from a model with a static effective interaction strength $g_{j,i}\left(t\right)$ given by Eq. (29) with ${\delta }^{*}/2\pi =0$ to a model where the effective interaction strength oscillates with ${\delta }^{*}/2\pi \in (0,0.125]$ [see also Fig. 8]. Therefore, we keep the pulse amplitude $\delta$ for the tunable coupler frequency given by Eq. (20) constant. We use $\delta /2\pi =0.075$ in Fig. 11 to model the two-qubit iswap transitions and $\delta /2\pi =0.085$ in Fig. 11 to model the two-qubit cz transitions. Note that these are the same pulse amplitudes we use in Figs. 9 and 9 and Figs. 9 and 9, respectively. Furthermore, if ${\delta }^{*}=0$ we simulate the scenarios we show in Figs. 9 and 9 and if ${\delta }^{*}=\delta$ we simulate the scenarios we show in Figs. 9 and 9. However, in Figs. 11 and 11 the pulse duration $T_{\mathrm{d}}$ is set to 300 ns for all cases.

Figure 11

Probabilities $p^{\left(z\right)}\left(t\right)$ as functions of time $t$ for (a) $z=(0,0,1)$ and (b) $z=(0,1,1)$. In (a) we model transitions which might be used to implement iswap gates. Similarly, in (b) we model transitions which might be used to implement cz gates. Here we use the effective Hamiltonian given by Eq. (31), the device parameters listed in Table 1, and the pulse in Eq. (33) to obtain the results. The system is modeled with a time-dependent interaction strength $g\left(t\right)$ given by Eq. (29). Panels (a) and (b) show the route from the model where we use a static interaction strength, i.e., with pulse amplitude ${\delta }^{*}=0$, to the model where the interaction strength is dynamic, i.e., with pulse amplitude ${\delta }^{*}\ne 0$. Here ${\delta }^{*}$ denotes the amplitude we use to model the time-dependent $g\left(t\right)$ given by Eq. (29) [see also Fig. 8]. The procedure is graphically illustrated in Fig. 10. In order to better understand how a time-dependent $g\left(t\right)$ affects the dynamics of the system, we turn on the dynamic interaction strength ${\delta }^{*}/2\pi \in [0,0.125]$ while keeping the amplitude $\delta$ for the tunable frequency given by Eq. (20) fixed. We use (a) $\delta /2\pi =0.075$ to model the iswap transition and (b) $\delta /2\pi =0.085$ to model the cz transition. Note that these are the same amplitudes $\delta$ we use to obtain the results in Fig. 9. In this scenario, we need to slightly adjust the drive frequencies ${\omega }^D$ as we increase ${\delta }^{*}$. We use ${\omega }^D/2\pi =1.088\mathrm{GHz}$ (blue lines and open markers) and ${\omega }^D/2\pi =1.089\mathrm{GHz}$ (green lines and solid markers) to model the iswap transitions in (a). Similarly, we use ${\omega }^D/2\pi =0.807\mathrm{GHz}$ (blue lines and open markers) and ${\omega }^{D}2\pi =0.808\mathrm{GHz}$ (green lines and solid markers) to model the cz transitions in (b). All results are obtained with the rise and fall time $T_{\text{r/f}}=13\mathrm{ns}$ and the gate duration $T_{\mathrm{d}}=300\mathrm{ns}$.

Figure 12

Probabilities $p^{\left(z\right)}\left(t\right)$ as functions of time for (a)-(b) $z=(0,0,1)$ and $z=(0,1,0)$ and (c)-(d) $z=(0,1,1)$ and $z=(0,0,2)$. In panels (a) and (c) we model the system without a time-dependent effective interaction strength [see Eq. (30)]. In panels (b) and (d) we include the time dependence. In all cases we use the Hamiltonian in Eq. (32), the parameters listed in Tables 1 and 2, and a pulse of the form of Eq. (33) to obtain the results. The pulse parameters are discussed in the main text. The $z=(0,1,1)\to z=(0,2,0)$ transitions are usually used to implement cz operations and the $z=(0,0,1)\to z=(0,1,0)$ transitions are often used to realize iswap operations. We observe a modest shift in the pulse duration $T_{\mathrm{d}}$ if we model the system with a time-dependent effective interaction strength [see Fig. 8].

Figure 13

Deviations between the numerically exact spectrum and two different approximations of this spectrum as a function of the external flux $\phi$ for an asymmetry factor (a) $d=0.33$ and (b) $d=0.5$. First, we compute the numerically exact spectrum with 50 charge states. Then we use two different sets of expressions for the qubit frequency and anharmonicity to determine the approximated spectrum with Eq. (B1). Approximation I: $\omega \left(\phi \right)$ is given by Eq. (20) and $\alpha \left(\phi \right)=\text{const.}$ Approximation II: $\tilde{\omega }\left(\phi \right)$ is given by Eq. (B2) and $\tilde{\alpha }\left(\phi \right)$ is given by Eq. (B3). In the end, we use Eq. (B5) to determine the deviations between the numerically exact energies and the different approximants (see approximations I and II). We use the capacitive and Josephson energies as well as the qubit frequency and anharmonicity listed in Table 1, rows (a) $i=0$ and (b) $i=1$.

Figure 14

Probabilities $p^{\left(0\right)}\left(t\right)$ and $p^{\left(1\right)}\left(t\right)$ as functions of time $t$. We use three transmon basis states $N_m=3$ to model the dynamics of the system, a control pulse of the form of Eq. (33), and a drive frequency ${\omega }^D$ equal to the qubit frequency $\omega$ (see Table 1, row $i=2$). The rise and fall time $T_{\mathrm{r}/\mathrm{f}}$ is set to half the duration time $T_{\mathrm{d}}$. The system is initialized in the state $|{\psi }^{\left(0\right)}\rangle$. The pulse amplitude $\delta /2\pi$ is set to (a) $\delta /2\pi =0.001$ and (b) $\delta /2\pi =0.01$. We can observe that an increase in the pulse amplitude $\delta$ by a factor of 10 leads to a decrease of the pulse duration $T_{\mathrm{d}}$ by a factor of 10 (roughly). Note that these transitions cannot be modeled with the effective Hamiltonian in Eq. (25).

Figure 15

Probabilities $p^{\left(z\right)}\left(t\right)$ for (a) $z=(0,0,0)$ and $z=(0,0,1)$ and (b) $z=(0,0,0)$ and $z=(0,1,0)$ as a function of time $t$. In both cases we use three basis states $N_m=3$ to model the dynamics of the system, a control pulse of the form of Eq. (33), and a rise and fall time $T_{\mathrm{r}/\mathrm{f}}$ set to half the duration time $T_{\mathrm{d}}$. (a) We use the drive frequency ${\omega }^D=5.092\mathrm{GHz}$ and the pulse amplitude $\delta /2\pi =0.085$. (b) We use the drive frequency ${\omega }^D=6.183\mathrm{GHz}$ and the pulse amplitude $\delta /2\pi =0.045$. The initial state of the system is always $|{\psi }^{(0,0,0)}\rangle$. Note that we were not able to activate these transitions in the effective model of architecture I [see Hamiltonian in Eq. (31)].

Figure 16

Probabilities $p^{(0,0,1)}\left(t\right)$ and $p^{(0,1,0)}\left(t\right)$ as functions of time $t$. We use (a) $N_m=3$, (b) $N_m=4$, (c) $N_m=6$, and (d) $N_m=15$ basis states to model the system. We use a control pulse of the form of Eq. (33), with the pulse parameters ${\omega }^D=1.089\mathrm{GHz}$, $T_{\mathrm{r}/\mathrm{f}}=13\mathrm{ns}$, and $\delta /2\pi =0.075$. The pulse duration is $T_{\mathrm{d}}=209.40\mathrm{ns}$. The system we simulate is defined by Eq. (7) and Table 1. The $z=(0,0,1)\to z=(0,1,0)$ transition is often used to implement iswap operations (see Ref. [3]). We find that numerical accurate modeling of the dynamic behavior of the system seems to require at least $N_m=6$ transmon basis states.

Figure 17

Probabilities $p^{(0,1,1)}\left(t\right)$ and $p^{(0,2,0)}\left(t\right)$ as functions of time $t$. We use (a) $N_m=3$, (b) $N_m=4$, (c) $N_m=8$, and (d) $N_m=15$ basis states to model the system. We use a control pulse of the form of Eq. (33), with the pulse parameters ${\omega }^{\mathrm{D}}=0.809\mathrm{GHz}$, $T_{\mathrm{r}/\mathrm{f}}=13\mathrm{ns}$, and $\delta /2\pi =0.085$. The pulse duration is $T_{\mathrm{d}}=297.55\mathrm{ns}$. The system we simulate is defined by Eq. (7) and Table 1. The $z=(0,1,1)\to z=(0,2,0)$ transition is usually used to implement cz operations (see Refs. [3, 13]). We find that numerical accurate modeling of the dynamic behavior of the system seems to require at least $N_m=8$ transmon basis states.

Figure 18

Probabilities $p^{(0,0,1)}\left(t\right)$ and $p^{(0,1,0)}\left(t\right)$ as functions of time $t$. We use (a) $N_m=3$, (b) $N_m=4$, (c) $N_m=14$, and (d) $N_m=25$ basis states to model the system. We use a control pulse of the form of Eq. (33), with the pulse parameters ${\omega }^D=0\mathrm{GHz}$, $T_{\mathrm{r}/\mathrm{f}}=20\mathrm{ns}$, and $\delta /2\pi =0.289$. The pulse duration is $T_{\mathrm{d}}=100.00\mathrm{ns}$. The pulse is supposed to perform an iswap gate. The system we simulate is defined by Eq. (8) and Table 1. The $z=(0,0,1)\to z=(0,1,0)$ transition might be used to implement iswap operations. Note that solutions in panels (a) and (b) do not have much in common with the reference solutions in panels (c) and (d).

Figure 19

Probabilities $p^{(0,1,1)}\left(t\right)$ and $p^{(0,0,2)}\left(t\right)$ as functions of time $t$. We use (a) $N_m=3$, (b) $N_m=4$, (c) $N_m=16$, and (d) $N_m=25$ basis states to model the system. We use a control pulse of the form of Eq. (33), with the pulse parameters ${\omega }^{\mathrm{D}}=0\mathrm{GHz}$, $T_{\mathrm{r}/\mathrm{f}}=20\mathrm{ns}$, and $\delta /2\pi =0.3335$. The pulse duration is $T_{\mathrm{d}}=125.00\mathrm{ns}$. The pulse is supposed to perform a cz gate. The system we simulate is defined by Eq. (8) and Table 1. The $z=(0,1,1)\to z=(0,0,2)$ transition can be used to implement cz operations (see Ref. [4]). Note that solutions in panels (a) and (b) do not have much in common with the reference solutions in panels (c) and (d).