Push-pull optimization of quantum controls

Optimization of quantum controls to achieve a target process is centered around an objective function comparing the realized process with the target. We propose an objective function that incorporates not only the target operator but also a set of its orthogonal operators whose combined influence leads to an efficient exploration of the parameter space, faster convergence, and extraction of superior solutions. The push-pull optimization, as we call it, can be adopted in various quantum control scenarios. We describe adopting it for gradient based and variational-principle based approaches. Numerical analysis of quantum registers with up to seven qubits reveals significant benefits of the push-pull optimization. We describe applying the push-pull optimization to prepare a long-lived singlet order in a two-qubit system using NMR techniques.

Figure 1

(a) Piecewise-constant control parameter $u_{jk}$ versus the segment number $j$. (b) Infidelity $1-F$ versus $u_{jk}$.

Figure 2

Performance analysis on a two-qubit system for GRAPE GC (first column), GRAPE SC (second column), Krotov GC (third column), and Krotov SC (fourth column). (a)–(d) Infidelity $1-F$ of two-qubit controls versus iteration number ($i$) and number ($L$) of orthogonal operators for GRAPE and Krotov as indicated. Black lines represent mean infidelities. (e)–(h) Mean infidelity versus $i$. Curves for $L=0$ (red) and for $L$ leading to the maximum final fidelity (green) are highlighted. (i)–(l) Mean final infidelity (left axis) and relative computing time (right axis) versus $L$. Error bars represent one standard deviation. (m)–(p) Probability versus advantage factor.

Figure 3

Infidelities for 40 random guesses (thin lines) and their mean (thick lines) versus iteration number $i$ with Krotov (red) and with PP-Krotov (blue) ($L=1$ and $\alpha =0.2$) for QFT on qubit registers of varying sizes ($n$ as indicated).

Figure 4

Mean infidelities (thick lines) of GRAPE sequences implementing a two-qubit cnot gate. (a) Performance with/without conjugate gradients. (b) Pull-only method ($L=0$) with adoptive step size $\epsilon$ (red) and push-pull method ($L=15$) with simultaneously adopted step size and push weight $\alpha$ (blue).

Figure 5

(a) Thermal and LLS spectra of TCP (molecule in inset). (b) PP-Krotov SC sequence ($L=5$) preparing LLS directly from the thermal state. (c) LLS fidelity evolution during the sequence in (b) at different rf inhomogeneity levels. (d) The $T_1$ values measured by the inversion recovery experiment and the $T_{\text{LLS}}$ measured by storage under spin lock.

Figure 6

Shown on top are the $X$ and $Y$ amplitudes for a two-qubit cnot gate with pull-only GRAPE (red) and push-pull PP-GRAPE (green) ($L=5$) algorithms. Shown on the bottom are the evolutions of the $X$ and $Y$ gradients versus iteration number for one particular segment (segment number 78). Notice how the mean push gradients (blue) from the orthogonal operators modulate the effective push-pull gradients (green).

Figure 7

Infidelity versus the push weight $\alpha$ for $L=6$. Error bars indicate one standard deviation. The black point at $\alpha =0$ corresponds to the standard pull-only algorithms. The green and red regions indicate, respectively, superior and inferior performances of PPOQC with respect to the pull-only algorithm.