Optimizing adiabatic quantum pathways via a learning algorithm

Designing proper time-dependent control fields for slowly varying the system to the ground state that encodes the problem solution is crucial for adiabatic quantum computation. However, inevitable perturbations in real applications demand us to accelerate the evolution so that the adiabatic errors can be prevented from accumulation. Here, by treating this trade-off task as a multiobjective optimization problem, we propose a gradient-free learning algorithm with pulse smoothing technique to search optimal adiabatic quantum pathways and apply it to the Landau-Zener Hamiltonian and Grover search Hamiltonian. Numerical comparisons with a linear schedule, local adiabatic theorem induced schedule, and gradient-based algorithm searched schedule reveal that the proposed method can achieve significant performance improvements in terms of the adiabatic time and the instantaneous ground-state population maintenance. The proposed method can be used to solve more complex and real adiabatic quantum computation problems.

Figure 1

Performance function values $F_1$ and $F_2$ vs different combinations of the adiabatic time $T$ and the weight factor $\alpha$ for a Landau-Zener Hamiltonian. (a),(b) The averaged results obtained by D-MORPH over five runs. The maximum iteration number was $G_{max}=1000$. The step size ${\lambda }^G$ was initialized as 0.02 and decreased by a factor 0.5 if the calculated $F$ was worse than the previous one, but with the maximum trial times 100. The control fields were all bounded in the range [0,1] during the optimization. (c),(d) The averaged results produced by DE over five runs. The maximum iteration number was $G_{max}=300$, and the initial guess was chosen as $u_1^g\left(s\right)=1-s,u_2^g\left(s\right)=s$. The algorithm parameters were $S=0.6,C=0.95,P=20,D=12,N_c=2$. Moreover, the controls were also constrained in the range [0,1] during the searching process.

Figure 2

Optimization results for the Landau-Zener Hamiltonian obtained by the Linear, D-MORPH, and DE methods. The controls fields $u_1\left(s\right)$ and $u_2\left(s\right)$, the instantaneous ground-state population $P_0\left(s\right)$, and the energy gap $g\left(s\right)$ are shown vs the scaled time $s$ optimized by (a)–(c) D-MORPH (solid red line) and Linear (dashed blue line) when $T=3,\alpha =0.1$; (d)–(f) DE (solid red line) and Linear (dashed blue line) when $T=3,\alpha =0.1$; (g)–(i) D-MORPH (solid red line) and Linear (dashed blue line) when $T=3,\alpha =0.5$; and (j)–(l) DE (solid red line) and Linear (dashed blue line) when $T=3,\alpha =0.5$.

Figure 3

Optimization results for the Grover search algorithm Hamiltonian using $n=1$ to 6 qubits obtained by the Linear, RC, D-MORPH, and DE methods. (a) The searched minimum adiabatic time $T$ vs the qubit number $n$ when $\alpha =0.1$, where $T$ was gradually increased and stopped when the difference between two successive $F$ was smaller than ${10}^{-3}$. (b),(c) The corresponding searched final performance function values $1-F$ and $1-F_1$ vs $n$. The control fields $u_1\left(s\right)$ and $u_2\left(s\right)$, the instantaneous ground-state population $P_0\left(s\right)$, and the energy gap $g\left(s\right)$ are shown vs the scaled time $s$ for (d)–(f) $n=1$; (g)–(i) $n=2$; (j)–(l) $n=4$; and (m)–(o) $n=6$. In all figures, the different methods are Linear (dotted black line), RC (dash-dotted green line), D-MORPH (dashed blue line), and DE (solid red line), and all the algorithm parameters were the same as the above Laudau-Zener Hamiltonian case.

Figure 4

Run time of D-MORPH and DE for the Grover search algorithm Hamiltonian. (a) The run time per iteration with respect to the number of qubits $n$. (b) The total run time regarding the number of qubits $n$.