Engineering strong beamsplitter interaction between bosonic modes via quantum optimal control theory

In continuous-variable quantum computing with qubits encoded in the infinite-dimensional Hilbert space of bosonic modes, it is a difficult task to realize strong and on-demand interactions between the qubits. One option is to engineer a beamsplitter interaction for photons in two superconducting cavities by driving an intermediate superconducting circuit with two continuous-wave drives, as demonstrated in a recent experiment [Gao et al., Phys. Rev. X 8, 021073 (2018)]. Here we show how quantum optimal control theory (OCT) can be used in a systematic way to improve the beamsplitter interaction between the two cavities. We find that replacing the two-tone protocol by a three-tone protocol accelerates the effective beamsplitter rate between the two cavities. The third tone's amplitude and frequency are determined by gradient-free optimization and make use of cavity-transmon sideband couplings. We show how to further improve the three-tone protocol via gradient-based optimization while keeping the optimized drives experimentally feasible. Our work exemplifies how to use OCT to systematically improve practical protocols in quantum information applications.

Figure 1

Tripartite system consisting of two superconducting cavities, labeled A and B, coupled via an intermediate (driven) transmon, labeled C. The system interacts with its environment, modeled by individual decay rates for each subsystem.

Figure 2

Population dynamics for the three initial states $|\overline{0,1,0}\rangle ,|\overline{1,0,0}\rangle$, and $|\overline{1,1,0}\rangle$ with $|\overline{0,0,0}\rangle$ omitted as it does not give rise to any cavity excitations. (a) Dynamics under the original two-tone protocol of Ref. [63] using the parameters from Table 1. (b) Dynamics as in (a) but for a three-tone protocol with optimized but constant amplitude ${\mathrm{\Omega }}_3$ and frequency ${\omega }_3$. Note the different timescales. See main text for details. The term “levels” refers to the bare Fock states $|n_{\mathrm{a}}\rangle$ and $|n_{\mathrm{b}}\rangle$ in the case of cavities A and B and to the anharmonic ladder states $|n_{\mathrm{c}}\rangle$ of the bare transmon in the case of transmon C.

Figure 3

Mutual information $\mathcal{I}$ between all pairs of subsystems A, B, and C. Panels (a) and (b) correspond to the two- and three-tone protocols and the population dynamics shown in Figs. 2 and 2, respectively. Panel (c) corresponds to the three-tone protocol that has been further fine-tuned by a gradient-based optimization technique (see main text for details).

Figure 4

Panel (a) shows the time-dependent amplitudes ${\mathrm{\Omega }}_k\left(t\right)$ of the three-tone protocol (solid and dashed lines) and their fine-tuned versions (dashed-dotted and dotted lines), while panels (b) and (c) show a close-up view. The coherent errors for the three-tone protocol and its fine-tuned version are ${\varepsilon }_3=2.6\%$ and ${\varepsilon }_{3,\mathrm{grad}}=0.2\%$, respectively. Panels (d)–(f) show the spectra (the absolute value of the Fourier transform) of $\text{Re}\left\{{\mathrm{\Omega }}_k\left(t\right)\right\}$ and $\text{Im}\left\{{\mathrm{\Omega }}_k\left(t\right)\right\}$ for the three-tone protocol (solid lines) and its fine-tuned version (dashed-dotted and dotted lines), respectively. Note that $\text{Im}\left\{{\mathrm{\Omega }}_k\left(t\right)\right\}$ is zero for the three-tone protocol, and hence its spectrum is omitted. The dashed-dotted and dotted lines are heavily overlapping and are thus hard to distinguish visually. The base frequencies for the three tones are given in Table 1 and Eq. (21).

Figure 5

Comparison of the protocol times as obtained numerically (green dots) and from analytical calculations including one from Floquet theory in the spirit of Ref. [65] (red dots) and one using the effective theory from Eqs. (29a) and (29b) (black solid line). Note that approximated versions of $\mathrm{\Delta }$ and other Stark-shifted parameters have been used here as the exact Stark shifts are not known analytically. This is reflected by the error bars for the Floquet theory and the black dashed lines for the effective theory. The lower (upper) error bars are obtained using a value of $\mathrm{\Delta }/2\pi$ that is $-(+)1\mathrm{M}\mathrm{Hz}$ away from the value indicated on the horizontal axis.

Figure 6

Influence of decoherence on the error ${\varepsilon }_2$ of the original two-tone protocol (a) as well as its fine-tuned version (b) and the error ${\varepsilon }_3$ of the three-tone protocol (c) as well as its fine-tuned version (d). Note that the relaxation times, $T_1^{\mathrm{a}}$ and $T_1^{\mathrm{b}}$, and pure dephasing times, $T_{\varphi }^{\mathrm{a}}$ and $T_{\varphi }^{\mathrm{b}}$, for both cavities are fixed in all panels to those values given in Table 1 and only the times for the transmon are varied.

Figure 7

Bloch trajectories for the reduced state of cavities A and B. Only those components $r_k\left(t\right)={\langle A_k\rangle }_{{\rho }_{\mathrm{AB}}\left(t\right)}$ in the generalized Bloch vectors are depicted that show significant changes in time. The opaque, solid lines correspond to the Bloch vector trajectories of ${\rho }_{\mathrm{AB}}\left(t\right)$ obtained by tracing out transmon C for the dynamics under the two-tone protocol (left column) and the three-tone protocol (right column) as shown in Fig. 2. The dotted lines correspond to the Bloch vector trajectories obtained for a dynamics with the effective Hamiltonian (A2), which is defined only on the reduced system of the two cavities.

Figure 8

Error analysis for various two-tone protocols based on the difference ${\xi }_1-{\xi }_2$ of the normalized amplitudes ${\xi }_1$ and ${\xi }_2$ but conditioned by ${\xi }_1{\xi }_2=0.126$. Panel (a) shows the average bare ground state population of the transmon (without decoherence) for the respective protocol; cf. Eq. (C1). Panel (b) shows the protocol error ${\varepsilon }_2$ with and without decoherence, while panel (c) shows the corresponding protocol duration $T$.