Quantum optimal control using phase-modulated driving fields

Quantum optimal control represents a powerful technique to enhance the performance of quantum experiments by engineering the controllable parameters of the Hamiltonian. However, the computational overhead for the necessary optimization of these control parameters drastically increases as their number grows. We devise a variant of a gradient-free optimal-control method by introducing the idea of phase-modulated driving fields, which allows us to find optimal control fields efficiently. We numerically evaluate its performance and demonstrate the advantages over standard Fourier-basis methods in controlling an ensemble of two-level systems showing an inhomogeneous broadening. The control fields optimized with the phase-modulated method provide an increased robustness against such ensemble inhomogeneities as well as control-field fluctuations and environmental noise, with one order of magnitude less of average search time. Robustness enhancement of single quantum gates is also achieved by the phase-modulated method. Under environmental noise, an XY-8 sequence constituted by optimized gates prolongs the coherence time by $50\%$ compared with standard rectangular pulses in our numerical simulations, showing the application potential of our phase-modulated method in improving the precision of signal detection in the field of quantum sensing.

Figure 1

A comparison of optimal results of SFB, SFB-P, SFB-P2, and PM under the conditions $T=100$ ns, $W/2\pi =10$ MHz, and amplitude constraint ${\mathrm{\Omega }}_{\text{max}}/2\pi =10$ MHz. (a) Optimal value of $F_{\text{obj}}$ as a function of $N$. (b) Average field amplitude of the optimal fields as a function of $N$. (c) Average number of function evaluations over 120 optimal results as a function of $N$. Under the same constraints, the PM method reaches the best $F_{\text{obj}}$ using the least number of parameters, showing the efficiency of the PM method on the premise of limited optimization resources.

Figure 2

Infidelity of the final state as a function of the number of objective function evaluations ($n_f$). Red lines and green lines represent for the PM method with three parameters and the SFB-P2 method with four parameters, respectively, both with 120 different random starting points. The parameters used here are $N=1, T=100$ ns, $W/2\pi =10$ MHz, and ${\mathrm{\Omega }}_{\text{max}}/2\pi =10$ MHz.

Figure 3

Optimal fidelity as a function of the FWHM ($W$) of the distribution of $\delta$. The red line with triangles is the result of the PM method with $N=1, N_p=3$. The purple line with squares is the result of the SFB-P2 method with $N=5, N_p=20$. The green line with circles is the result of the SFB-P2 method with $N=1, N_p=4$. Other parameters used here are $T=100$ ns, $W/2\pi =10$ MHz, and ${\mathrm{\Omega }}_{\text{max}}/2\pi =10$ MHz.

Figure 4

(a)–(c) Optimal control field in the interaction picture, given by the SFB-P2 ($N=1, N_p=4$), SFB-P2 ($N=5, N_p=20$), and PM method ($N=1, N_p=3$), respectively. Solid line, $x$-direction field; dot-dashed line, $y$-direction field. (d)–(f) The corresponding frequency spectrum of the control fields in (a)–(c), respectively. (g)–(i) The corresponding number of frequency components with amplitude $⩾5$ MHz during the optimization process.

Figure 5

(a)–(c) Results of 120 optimization processes with different initial values, ranked by the value of $F_{\text{obj}}$. (a) The SFB-P2 method with $N=1, N_p=4$ and its parameter randomized version SFB-P2-r where ${\omega }_i$ are randomly taken from the range $2\pi \times [0,5/T]$ and not optimized. (b) The SFB-P2 method with $N=5, N_p=20$ and its parameter randomized version SFB-P2-r where ${\omega }_i$ are randomly taken from the range $2\pi \times [0,5/T]$ and not optimized. (c) The PM method with $N=1, N_p=3$ and its parameter randomized version PM(r) where ${\nu }_i$ are randomly taken from the range $2\pi \times [0,5/T]$ and not optimized. (d)–(f) Corresponding number of function evaluations (dot) and the average value (solid line, nonrandomized methods; dashed line, randomized methods).

Figure 6

Fidelity of the final state under different values of the detuning and the relative variations of the control-field amplitude. (a) Results under a control field optimized using the SFB-P2 with $N=1, N_p=4$. (b) Results under a control field optimized using the SFB-P2 method with $N=5, N_p=20$. (c) Results under a control field optimized using the phase-modulated method with $N=1, N_p=3$. (d) A comparison of the three optimal methods and the rectangular $\pi$ pulse when there are no variations of the control amplitude. The ratio of areas of $f>0.9$ in (a) and (c) is approximately $1:1.98$. The markers in (d) represent the $M=15$ sample points used in the objective function during the optimization process. Other parameters used here are $T=100$ ns, $W/2\pi =10$ MHz, and ${\mathrm{\Omega }}_{\text{max}}/2\pi =10$ MHz. The amplitude of the rectangular $\pi$ pulse is $2\pi \times 10$ MHz, with the pulse length of 50 ns.

Figure 7

Average fidelity $F$ of the ensemble system as a function of the dephasing rate, using the same optimized control fields as in Fig. 6.

Figure 8

Fidelity of the final state with dephasing rate $\gamma =2$ MHz, using the same control fields and parameters as in Fig. 6. (a) Results under a control field optimized using the SFB-P2 with $N=1, N_p=4$. (b) Results under a control field optimized using the SFB-P2 method with $N=5, N_p=20$. (c) Results under a control field optimized using the PM method with $N=1, N_p=3$. (d) A comparison of the three optimal methods when there are no variations of the control amplitude. The ratio of areas of $f>0.9$ in (a) and (c) is approximately $1:1.74$. The markers in (d) represent the $M=15$ sample points used in the objective function during the optimization process.

Figure 9

Optimal results of the robust gate fidelity given by PM ($N=1,2$) and SFB-P2 ($N=1,5$) methods, with the targeted Hadamard gate, Pauli-$X$ gate, Pauli-$Y$ gate, and Pauli-$Z$ gate, respectively. Here we use $K={10}^5$ random detunings ${\delta }_l$ obeying the Gaussian distribution (2) with $W/2\pi =10$ MHz, to approximate the ensemble fidelity by $F_G\approx {\sum }_{l=1}^K{f}_g\left({\delta }_l\right)/K$. Parameters used in the optimization process are $T=100$ ns, $W/2\pi =10$ MHz, and ${\mathrm{\Omega }}_{\text{max}}/2\pi =10$ MHz.

Figure 10

Gate fidelity under different values of the detuning and the relative variations of the control-field amplitude. (a)–(d) Results of the control fields optimized using the phase modulated method with $N=2$, with the target gate being the Hadamard gate, Pauli-$X$ gate, Pauli-$Y$ gate, and Pauli-$Z$ gate, respectively. Parameters used in the optimization process are $T=100$ ns, $W/2\pi =10$ MHz, and ${\mathrm{\Omega }}_{\text{max}}/2\pi =10$ MHz. (e)–(h) Results of the standard rectangular pulse with amplitude $2\pi \times 10$ MHz and pulse length 50 ns, with the target gate being the Hadamard gate, Pauli-$X$ gate, Pauli-$Y$ gate, and Pauli-$Z$ gate, respectively. (i)–(l) A comparison of the optimized PM pulse and the rectangular pulse when there are no variations of the control amplitude.

Figure 11

Simulation of the coherence time $T_2$ using one XY8 series. The initial state is taken as ${\psi }_0=|0\rangle$. (a) Schematic pulse location during one simulation process with evolution time $T$. The slim green lines at the beginning and the end represent the initialization pulse rotating the state around the $X$ axis by $\pi /2$ and the readout pulse rotating the state around the $X$ axis by $3\pi /2$, respectively, where both are presumed to be instantaneous. The black and gray sticks represent Pauli-$X$ and Pauli-$Y$ gates in the XY8 series, respectively, with finite pulse length $T_{\text{pulse}}$. The pulse separation ${\tau }_{\text{pulse}}$ refers to the time period from the end of one pulse to the beginning of the next pulse. The pulse number and pulse length are fixed for different evolution times $T$, and $T_2$ is recognized as the value of $T$ at which the population of $|0\rangle$ drops to $(1+1/e)/2$. (b) The population $P_0$ of $|0\rangle$ as a function of $T$. Each point averages the results of 1200 evolutions. PM optimal control fields give $T_2\approx 39\mu \mathrm{s}$, improving the coherence time with rectangular pulse $T_2\approx 24\mu \mathrm{s}$ by half.