Analytical modeling of parametrically modulated transmon qubits

Building a scalable quantum computer requires developing appropriate models to understand and verify its complex quantum dynamics. We focus on superconducting quantum processors based on transmons for which full numerical simulations are already challenging at the level of qubytes. It is thus highly desirable to develop accurate methods of modeling qubit networks that do not rely solely on numerical computations. Using systematic perturbation theory to large orders in the transmon regime, we derive precise analytic expressions of the transmon parameters. We apply our results to the case of parametrically modulated transmons to study recently implemented, parametrically activated entangling gates.

Figure 1

Circuits of a fixed-frequency transmon F (left) and a tunable transmon T (right) that are capacitively coupled (capacitance $C$). Transmons are characterized by the charging energy of their capacitance, $E_C=e^2/\left(2C\right)$, and the Josephson energy, $E_J$. The transmon regime of the Cooper-pair box corresponds to $E_C/E_J\ll 1$. Tunable transmons are composed of a SQUID and controlled with pulses on the flux bias line, ${\varphi }_{\mathrm{ext}}\left(t\right)$.

Figure 2

Error on the transmon frequency and anharmonicity as a function of perturbation theory order in $\xi$ compared to numerical simulation with 30 Fock states (absolute value of the difference in full lines). The corresponding terms of the perturbative expansion are plotted in dashed lines. Sub-kHz accuracy of the transmon frequency and anharmonicity is found at large orders, and expressions are reported in Appendix pp2. Parameters are $\xi =0.2$ and $E_C/h=200\mathrm{MHz}$, giving $\omega /\left(2\pi \right)=3788\mathrm{MHz}$ and $\eta /\left(2\pi \right)=230\mathrm{MHz}$.

Figure 3

Error on the tunable-transmon frequency and anharmonicity at $25\mathrm{th}$ order in $\xi$ as a function of parking flux bias compared to numerical simulation with 30 Fock states. Parameters are $E_C/\left(2\pi \right)=200\mathrm{MHz}, {\xi }_{\max }=0.16$, and ${\xi }_{\min }=0.2$, giving ${\omega }_{\max }/\left(2\pi \right)=4791\mathrm{MHz}, {\omega }_{\min }/\left(2\pi \right)=3788\mathrm{MHz}, {\eta }_{\max }/\left(2\pi \right)=222\mathrm{MHz}$, and ${\eta }_{\min }/\left(2\pi \right)=230\mathrm{MHz}$ (subscripts max, min refer to ${\varphi }_{\mathrm{ext}}=0,\pi$, respectively).

Figure 4

Dispersive shift of capacitively coupled transmons (a) and error on the frequencies, anharmonicities, and frequency shift (b) at $25\mathrm{th}$ order in $\xi$ for four transmon eigenstates as a function of coupling strength compared to numerical simulation with 30 Fock states for each transmon. The dashed lines correspond to the dispersive results, valid for small enough couplings. The first transmon is characterized by $E_C/h=200\mathrm{MHz}$ and ${\xi }_1=0.18$, giving ${\omega }_1/\left(2\pi \right)=4234\mathrm{MHz}, {\eta }_1/\left(2\pi \right)=226\mathrm{MHz}$. It is interacting with a coupling $g$ to the second transmon characterized by $E_C/h=200\mathrm{MHz}$ and ${\xi }_2=0.175$, giving ${\omega }_2/\left(2\pi \right)=4361\mathrm{MHz}, {\eta }_2/\left(2\pi \right)=225\mathrm{MHz}$.

Figure 5

Time evolution of the transmon frequency (a), anharmonicity (b), and error on the tunable-transmon frequency and anharmonicity (c) under modulation at $25\mathrm{th}$ order in $\xi$ as a function of time over a period of the modulation frequency, $2\pi /{\omega }_p$. The time evolution of $\omega \left(t\right)$ and $\eta \left(t\right)$ from ${\varphi }_{\mathrm{ext}}\left(t\right)={\overline{\varphi }}_p+{\tilde{\varphi }}_{p}cos({\omega }_{p}t+{\theta }_p)$ is compared to the Fourier series of the 50 first harmonics, Eq. (35). Parameters are the same as Fig. 3 with ${\overline{\varphi }}_p=0$ (parking at a maximum) and ${\tilde{\varphi }}_p=2\pi$ (modulating between three consecutive maxima).

Figure 6

Modulation frequency activating three kinds of two-qubit gates (a) and corresponding effective coupling (b) as a function of the modulation amplitude. A fixed transmon $[E_C/h=200\mathrm{MHz}, \xi =0.21$, giving $\omega /\left(2\pi \right)=3597\mathrm{MHz}, \eta /\left(2\pi \right)=232\mathrm{MHz}]$ is coupled to a tunable transmon $[E_C/\left(2\pi \right)=190\mathrm{MHz}, {\xi }_{\max }=0.16$, and ${\xi }_{\min }=0.2$, giving ${\omega }_{\max }/\left(2\pi \right)=4551\mathrm{MHz}, {\omega }_{\min }/\left(2\pi \right)=3599\mathrm{MHz}, {\eta }_{\max }/\left(2\pi \right)=211\mathrm{MHz}$, and ${\eta }_{\min }/\left(2\pi \right)=218\mathrm{MHz}]$ that is parked at a maximum of the energy spectrum (${\overline{\varphi }}_p=0$). The flux modulation amplitude ${\tilde{\varphi }}_p$ and frequency ${\omega }_p$ are then chosen from (a) to activate a desired entangling gate among iSWAP, ${\mathrm{CZ}}_{02}$ and ${\mathrm{CZ}}_{20}$. The bare coupling is renormalized by the modulation by a coefficient that is plotted in (b). The vertical lines indicate operating points for fast gates.