Phase-Modulated Decoupling and Error Suppression in Qubit-Oscillator Systems

We present a scheme designed to suppress the dominant source of infidelity in entangling gates between quantum systems coupled through intermediate bosonic oscillator modes. Such systems are particularly susceptible to residual qubit-oscillator entanglement at the conclusion of a gate period that reduces the fidelity of the target entangling operation. We demonstrate how the exclusive use of discrete shifts in the phase of the field moderating the qubit-oscillator interaction is sufficient to both ensure multiple oscillator modes are decoupled and to suppress the effects of fluctuations in the driving field. This approach is amenable to a wide variety of technical implementations including geometric phase gates in superconducting qubits and the Molmer-Sorensen gate for trapped ions. We present detailed example protocols tailored to trapped-ion experiments and demonstrate that our approach has the potential to enable multiqubit gate implementation with a significant reduction in technical complexity relative to previously demonstrated protocols.

Figure 1

(a) Schematic plots of ${\alpha }_k(t)$, $0\le t\le \tau$, for two modes, $k=1,2$. Each path represents $N$ phase-space trajectories ${\alpha }_k^{\mu }(t)=f_k^{\mu }{\alpha }_k(t)$, for $\mu =1,\dots ,N$. Here $\tau =2\pi /{\delta }_1$, so that the path labeled $k=1$ closes. (b) An illustrative example of a single closed path, generated by an eight-interval, piecewise-constant phase-modulation sequence. (c) Timing schematic for piecewise-constant phase-modulation sequence, comprising a total of $n$ instantaneous phase shifts. For the general $M$ mode decoupling sequence (5) $n=2^M-1$.

Figure 2

(a) Raman laser geometry for a MS gate applied to 2 ions in a 5 ion chain in which only transverse ($x$-direction) phonon modes are excited. The red ($r$) and blue ($b$) Raman fields have frequencies ${\omega }_{r/b}$ and phases ${\varphi }_{r/b}$. (b) Detuning diagram for 5 excited TP modes, ${\tilde{\omega }}_k={\omega }_0\pm {\omega }_k$ ($+$ for ${\omega }_b$ and $-$ for ${\omega }_r$), where ${\omega }_0$ is the hyperfine qubit level splitting. (c) Closed paths, ${\alpha }_k^{,}\equiv |{\delta }_k|{\alpha }_k(t)$, $0\le t\le \tau$, (normalized by $|{\delta }_k|$) for the detunings shown in (b), generated by 7 discrete phase shifts.

Figure 3

Laser amplitude noise filter functions for 3 excited oscillator modes $k=1,2,3$. The effect of the noise on the coupling to each mode is suppressed to third order for $k=1$ [$F_1(\omega )\propto {\omega }^3$ as $\omega \to 0$], second order for $k=2$ [$F_2(\omega )\propto {\omega }^2$], and first order for $k=3$ [$F_3(\omega )\propto \omega$].