Analysis of a parametrically driven exchange-type gate and a two-photon excitation gate between superconducting qubits

A current bottleneck for quantum computation is the realization of high-fidelity two-qubit quantum operations between two or more quantum bits in arrays of coupled qubits. Gates based on parametrically driven tunable couplers offer a convenient method to entangle multiple qubits by selectively activating different interaction terms in the effective Hamiltonian. Here, we theoretically and experimentally study a superconducting qubit setup with two transmon qubits connected via a capacitively coupled tunable bus. We develop a time-dependent Schrieffer-Wolff transformation and derive analytic expressions for exchange-interaction gates swapping excitations between the qubits (iswap) and for two-photon gates creating and annihilating simultaneous two-qubit excitations (bswap). We find that the bswap gate is generally slower than the more commonly used iswap gate, but features favorable scalability properties with less severe frequency-crowding effects, which typically degrade the fidelity in multiqubit setups. Our theoretical results are backed by experimental measurements as well as exact numerical simulations including the effects of higher transmon levels and dissipation.

Figure 1

(a) Optical micrograph and (b) circuit scheme of the device consisting of two fixed-frequency transmons (Q1, Q2) capacitively coupled to a flux tunable transmon (TB). The tunable coupler is controlled by a high-speed flux line (HSFL) providing a current $I\left(t\right)$ and a consequent flux $\mathrm{\Phi }\left(t\right)$ threading the superconducting quantum interference device (SQUID) loop of the coupler. Each of the fixed-frequency qubits is coupled to an individual readout resonator. (c) Level diagram of the device. Here, $|n_1{n}_2{n}_c\rangle$ denotes the state of the combined system with the qubit excitation number $n_{1,2}$ and the coupler excitation number $n_c$. The computational subspace is spanned by the states $\{|000\rangle ,|010\rangle ,|100\rangle ,|110\rangle \}$, with the coupler mostly residing in its ground state. The states $|100\rangle$ and $|010\rangle$ are separated by the difference frequency $\sim {\omega }_1-{\omega }_2$, whereas the states $|000\rangle$ and $|110\rangle$ are separated by the sum frequency $\sim {\omega }_1+{\omega }_2$. Additionally, the most important sideband transitions between qubits and coupler are shown.

Figure 2

Left panels: Experimentally measured state population of the (a) $|100\rangle$ state for the iswap gate and the (b) $|110\rangle$ state for the bswap gate, as a function of modulation frequency ${\omega }_{\mathrm{\Phi }}$ and gate time $t$. The modulation strength is fixed at (a) $\delta =0.063{\mathrm{\Phi }}_0$ and (b) $\delta =0.14{\mathrm{\Phi }}_0$. Note that as pulses of length zero are not generated, the depicted measurements start at a nonzero value of $t$. Right panels: Oscillation frequencies of the (c) iswap gate and the (d) bswap gate, as obtained from the state occupations shown in (a) and (b), respectively. The solid line is the best fit of the oscillation frequencies (circles) to an effective two-state model.

Figure 3

Oscillation frequencies of the population of (a) the $|010\rangle$ state and (b) the $|000\rangle$ state as a function of modulation frequency ${\omega }_{\mathrm{\Phi }}$. The results are obtained by a numerical simulation of the transmon Hamiltonian (A1) with the coupler frequency approximated by the expansion in Eq. (4) to second order in $\delta$. The system is initialized in (a) the $|100\rangle$ state and (b) the $|000\rangle$ state. Shown are the transitions that result in a population leakage $>{10}^{-5}$ out of the two-state subspace corresponding to (a) the iswap gate and (b) the bswap gate. The dc flux bias is $\theta =-0.108{\mathrm{\Phi }}_0$. The flux-modulation amplitude is fixed at (a) $\delta =0.13{\mathrm{\Phi }}_0$ and (b) $\delta =0.16{\mathrm{\Phi }}_0$.

Figure 4

(a) Gate strength of iswap (red) and bswap (blue) gates as a function of $\delta /{\mathrm{\Phi }}_0$. Experimental data is shown as triangles. Circles show the numerical simulation of the full transmon Hamiltonian (A1). In contrast to Fig. 3, the numerical simulations have been carried out using the complete transfer function in Eq. (2). Dashed lines show the analytic results to linear order in $\delta$, i.e., Eqs. (15) and (17). Solid lines show analytic results up to second order in $\delta$ (cf. Appendix pp3). The black-dashed line corresponds to the adiabatic approximation in (18). The values of $\delta$ for the experimental data points are calibrated using a fit between the measured dispersive shifts of the modulation frequencies and the prediction of the effective model (cf. Appendix pp4). (b) Modulation frequency ${\omega }_{\mathrm{\Phi }}$ for the iswap gate obtained from numerical simulations (red) and the effective model in Appendix pp4 (black dashed line). Additionally, the two closest sideband frequencies are shown (orange and green). For the latter, the coupler frequency ${\omega }_{\mathrm{c}}\left(t\right)$ is expanded to second order in $\delta$ and a time average is taken by replacing ${cos}^2\left({\omega }_{\mathrm{\Phi }}t\right)\to 1/2$ in Eq. (4). The dashed lines indicate that such a simple approximation breaks down when nearby resonances approach each other. (c) The analogous results for the bswap gate (blue) including a nearby sideband (orange) and the excitation of a higher-transmon level (gray).

Figure 5

Error $\epsilon =1-F$ of the iswap (red) and bswap (blue) gate as a function of gate time. The gate fidelity $F$ is calculated according to Eq. (20) from numerical simulations of the master equation (21). The circles in the main figure are obtained with a $T_2$ time of the coupler of $T_2^{\mathrm{c}}=7.3\mu \mathrm{s}$, whereas the coupler $T_2$ time for the triangles is $T_2^{\mathrm{c}}=500\mathrm{ns}$. The $\delta$ scale is valid for the triangles. The inset shows the dependence of the error on the $T_2$ time of the coupler for a fixed gate time of 220 ns (iswap) and 268 ns (bswap). The dashed lines in the inset correspond to the results with a dissipation-free coupler element.