Resonator reset in circuit QED by optimal control for large open quantum systems

We study an implementation of the open GRAPE (gradient ascent pulse engineering) algorithm well suited for large open quantum systems. While typical implementations of optimal control algorithms for open quantum systems rely on explicit matrix exponential calculations, our implementation avoids these operations, leading to a polynomial speedup of the open GRAPE algorithm in cases of interest. This speedup, as well as the reduced memory requirements of our implementation, are illustrated by comparison to a standard implementation of open GRAPE. As a practical example, we apply this open-system optimization method to active reset of a readout resonator in circuit QED. In this problem, the shape of a microwave pulse is optimized such as to empty the cavity from measurement photons as fast as possible. Using our open GRAPE implementation, we obtain pulse shapes, leading to a reset time over 4 times faster than passive reset.

Figure 1

Schematic of a gradient-based optimization step with GRAPE-type controls update and Gaussian filtering to account for experimental constraints. Starting from the initial controls shown in (a), we calculate the filtered experimental pulse shape in (b). From this filtered field, the gradient $\partial \mathrm{\Phi }/\partial u_k$ is calculated using the chain rule (see Ref. [9] or Appendix pp1 for details) and the controls are updated in (c), which leads to a new filtered field in (d). The boundary conditions of the field are taken into account by fixing the first and last control.

Figure 2

(a) Average runtime per evaluation of the performance index and its gradient ($T_{\mathrm{grad}}$). Points are numerical data obtained from the averaging over 100 optimization iterations. In the matrix exponentiation case, a first-order GRAPE implementation based on the control module of QuTiP is used [51]. (b) Average memory usage (RAM) during a GRAPE iteration normalized by the $d=6$ value. Points are numerical data obtained using the “memory profiler” package (version 0.47). For both panels, solid lines are power-law fits. A constant background due to program overhead is included in the fits of the second panel (see text). The matrix-exponential calculations were done for a Hilbert space size of only $d\le 24$ due to limitations of RAM. From an initial state $|\alpha \rangle \otimes |e\rangle$, with the cavity in a coherent state with $\alpha =\sqrt{d/8}$ and the qubit in the excited state, we use the two quadratures of the cavity drive to perform an active reset (target state $|0\rangle \otimes |e\rangle$) in a time $T=\pi /g$ separated in $N=200$ steps. The qubit and cavity are coupled through a standard Jaynes-Cummings interaction with coupling strength $g=100\kappa$. Additional parameters are ${\omega }_d={\omega }_r$ and $\mathrm{\Delta }=10\kappa$ (see Sec. 4a for definitions).

Figure 3

(a) Average photon number during resonator reset procedures following a readout process with drive power $P_{\mathrm{norm}}=4$. The solid (dashed) lines indicate results for the qubit in the ground (excited) state. See panel (b) for legend. The inset shows the average photon number for $T_{\kappa }>1$ on a logarithmic scale to allow for better comparison of the reset schemes. (b) Pulse shapes for the resonator reset procedures used in panel (a). System parameters and the Hamiltonian are described in Sec. 4a. Additional parameters for the GRAPE algorithm include a control duration $\mathrm{\Delta }t=1$ ns and Gaussian filtering with bandwidth ${\omega }_B/2\pi =100$ MHz and subpixel duration $\delta t=0.1$ ns. (See Appendix pp1 for parameter definitions.)

Figure 4

Average photon number at the end of the active reset pulse as a function of the readout power $P_{norm}$. The results are shown for pulses of duration $T=150$ ns, 110 ns, 70 ns, and 40 ns, corresponding to $T\approx 1.04T_k,0.76T_{\kappa },0.48T_{\kappa },\text{and}0.28T_{\kappa }$. The solid (dashed) line is the average final photon number for the qubit in the ground (excited) state. The vertical gray lines indicate the failure points for the 70-ns and the 40-ns optimizations.

Figure 5

The green dots are the numerical speed limit extracted from the open GRAPE optimizations. We define here the speed limit as the time where the optimization fails, corresponding to the branching points indicated by gray lines in Fig. 4 for the 40-ns and 70-ns curves. The dashed gray line is a power-law fit $\propto {\left(P_{\mathrm{norm}}\right)}^{\xi }$ to the data with $\xi =0.65$.

Figure 6

Photon number as a function of time during the active reset pulse for a pulse duration of 110 ns and 70 ns. The solid (dashed) line is for the qubit in the ground (excited) state. The inset shows the corresponding Gaussian filtered drives. The solid lines of the inset are the $X$ drive while the dotted lines are the $Y$ drive.

Figure 7

Photon number as a function of time for optimized drives with (light orange curves) and without (dark blue curves) the photon number penalty ${\mathrm{\Phi }}_p$ included. The solid (dashed) curves are for the qubit in the ground (excited) state. The inset displays the same data with a logarithm photon number axis. The parameters are the same as Fig. 3. We use the penalty weight $\beta =0.2/T$.