Optimized pulse shapes for a resonator-induced phase gate

The resonator-induced phase gate is a multiqubit controlled-phase gate for fixed-frequency superconducting qubits. Through off-resonant driving of a bus resonator, statically coupled qubits acquire a state-dependent phase. However, photon loss leads to dephasing during the gate, and any residual entanglement between the resonator and qubits after the gate leads to decoherence. Here we consider how to shape the drive pulse to minimize these unwanted effects. First, we review how the gate's entangling and dephasing rates depend on the system parameters and validate closed-form solutions against direct numerical solution of a master equation. Next, we propose spline pulse shapes that reduce residual qubit-bus entanglement, are robust to imprecise knowledge of the resonator shift, and can be shortened by using higher-degree polynomials. Finally, we present a procedure that optimizes over the subspace of pulses that leave the resonator unpopulated. This finds shaped drive pulses that further reduce the gate duration. Assuming realistic parameters, we exhibit shaped pulses that have the potential to realize $\sim 212$ ns spline pulse gates and $\sim 120$ ns optimized gates with $\sim 6\times {10}^{-4}$ average gate infidelity. These examples do not represent fundamental limits of the gate and, in principle, even shorter gates may be achievable.

Figure 1

The bus resonance is shifted up in frequency by ${\chi }_{jk}$ when the qubits are placed in the state $|jk\rangle$. When the bus resonator is driven by a tone at ${\omega }_d$ that is detuned by $\Delta >0$ from the ground-state bus resonance, the resonator responds differently for each qubit state. The response is a function of the detuning $\Delta$ and the shifts ${\overline{\chi }}_{jk}={\chi }_{00}-{\chi }_{jk}$.

Figure 2

The bus resonator is driven by an unmodulated tone as described in the text. The mean photon number $\langle n\rangle$ of the bus is calculated from closed-form expressions and by direct numerical solution of the master equation. Both solutions are consistent with one another on this time scale.

Figure 3

A pair of qubits coupled to a driven bus resonator accumulate phase as described in the text. The closed-form and numerical solutions for the phase angle $\theta \left(t\right)$ are consistent over the time scale shown. The solid line shows a linear accumulation of phase at the steady-state rate.

Figure 4

A pair of qubits coupled to a driven bus resonator suffer measurement-induced dephasing as described in the text. The time evolution of the squared off-diagonal matrix element between $|00\rangle$ and $|11\rangle$ decays exponentially to zero over many microseconds $t\gg 1/\kappa$. The closed-form and numerical solutions are consistent over the time scale shown.

Figure 5

Pulse shapes constructed from piecewise polynomials rise from amplitude zero to $|{\tilde{\epsilon }}_{\max }|$ over time $t_r$ and are symmetric about $t=t_r+t_p/2$. Such pulse shapes have some number of vanishing derivatives at times $t=0$ and $t=2t_r+t_p$. The first three examples are shown here and we take $\mathrm{Im}\left[\tilde{\epsilon }\right]=0$ and $t_p=0$.

Figure 6

This plot of the drive detuning vs the peak drive amplitude (as a Rabi rate) shows roughly a square-root dependence of the detuning on the drive amplitude for splines of degree 3, 5, and 7. The composite gates in each case have pulse shapes that achieve ${10}^{-4}$ average gate infidelity. The corresponding curves are fits to the form $c\sqrt{|\tilde{\epsilon }|}$, where $c$ is a fitting parameter.

Figure 7

For the same set of pulse shapes as Fig. 6, this plot shows the rise time of each pulse in the composite sequence as a function of drive amplitude. The corresponding curves are fits to the functional form $c_0/(c_1+c_2|\tilde{\epsilon }{|}^{(d+1)/\left(2d\right)})$, where $c_j$ are fitting parameters that depend on $d$.

Figure 8

Example of a pulse shape produced by null-space optimization. On the left, the in-phase (real) part is plotted in (thick) red and the quadrature (imaginary) part is plotted in (thin) blue. The same pulse shape is plotted on the right in the complex (IQ) plane. System parameters are the “high” set given in Appendix pp5. The sequence has 240 discrete samples with $\Delta t=0.25$ ns. This example realizes a square root of controlled-z gate $\left(\theta =\pi /2\right)$ with angular error $\Delta \theta \approx 2\times {10}^{-3}$ and achieves cost components $1-\mathcal{F}\approx 6\times {10}^{-4}$, on-off penalty $\sim 9\times {10}^{-5}$, and bandwidth penalty $\sim 5\times {10}^{-5}$.

Figure 9

For each dressed basis state $|jk\rangle$, the pulse shape shown in Fig. 8 causes the bus resonator response $\langle {\alpha }_{jk}\left(t\right)\rangle$ to rapidly “ring up,” follow a nearly circular trajectory in phase space, and return to the origin. The pulse populates the bus with fewer than 10 photons at peak for any initial state of the qubits.

Figure 10

Spectrum of four different pulses found by null-space optimization starting from four random bounded initial pulses. Several spectral features (circled) are apparent across all of the solutions. The $B=300$ MHz bandwidth is visible, as is a notch at $\{\Delta +{\chi }_{jk}\}$ and oscillations at a frequency $\omega /2\pi \approx 60$ MHz.