Adiabatic leakage elimination operator in an experimental framework

Adiabatic evolution is used in a variety of quantum information processing tasks. However, the elimination of errors is not as well developed as it is for circuit model processing. Here, we present a strategy to improve the performance of a quantum adiabatic process by adding leakage elimination operators (LEOs) to the evolution. These are a sequence of pulse controls acting in an adiabatic subspace to eliminate errors by suppressing unwanted transitions. Using the Feshbach $PQ$ partitioning technique, we obtain an analytical solution for a set of pulse controls. The effectiveness of the LEO is independent of the specific form of the pulse but depends on the average frequency of the control function. By observing that the evolution of the target eigenstate is governed by a periodic function appearing in the integral of the control function, we show that control parameters can be chosen in such a way that the instantaneous eigenstates of the system are unchanged, yet a speedup can be achieved by suppressing transitions. Furthermore, we give the exact expression of the control function for a counter unitary transformation to be used in experiments which provides a clear physical meaning for the LEO, aiding in the implementation.

Figure 1

Example 1: Fidelity versus parameter ${\omega }_{1}t$ for (a) different integrals $S$, with $\mathrm{\Delta }=\xi$, $\mathrm{\Delta }/\tau =1/1$, $T=1/{\omega }_1$, and the time step length is taken to be $\xi =0.005/{\omega }_1$. (b) Different ratios of $\mathrm{\Delta }/\tau$, with $\mathrm{\Delta }+\tau =10\xi$, $S=0.03$.

Figure 2

Example 1: Fidelity versus parameter ${\omega }_{1}t$ for different pulse intervals, (a) $\mathrm{\Delta }=\xi$, (b) $\mathrm{\Delta }=10\xi$, (c) $\mathrm{\Delta }=25\xi$, (d) $\mathrm{\Delta }=50\xi$; $\mathrm{\Delta }/\tau =1/1$. (e) The fidelity for different $\mathrm{\Delta }$.

Figure 3

Example 1: Different types of pulses. (a) Regular rectangular pulses, $\mathrm{\Delta }/\tau =5/5$, $\mathrm{\Delta }+\tau =10\xi$, $S=0.05$. (b) Random rectangular pulses, $\mathrm{\Delta }/\tau \in [0,1]$, $\mathrm{\Delta }+\tau =20\xi$, $S\in [0,0.1]$. (c) The average values of noise for a thousand trials with $S\in (0.047,0.053)$. (d) A fast sine signal, $f\left(t\right)=10{sin}^2\left(50t\right)$.

Figure 4

Example 1: For four types of pulses plotted in Fig. 3, (a) average control frequency versus ${\omega }_{1}s$ and (b) the corresponding fidelity versus ${\omega }_{1}t$. For randomness and noise, we calculate (for a thousand trials) average density matrices and obtain the corresponding average control frequency and fidelity.

Figure 5

Example 2: (a) The average control frequency $\langle {\omega }_2\rangle$ as functions of parameter ${\omega }_{1}s$. (b) The corresponding fidelity versus parameter ${\omega }_{1}t$ for different integrals $S$. $\mathrm{\Delta }=5\xi$, $\mathrm{\Delta }/\tau =1/1$, and $T=1/{\omega }_1$. The time intervals are taken to be $\xi =0.001/{\omega }_1$.

Figure 6

Example 1: The fidelity versus parameter ${\omega }_{1}t/\pi$ for different cases: free evolution; zero-energy-change control ($n=60,m=1,{\omega }_1\mathrm{\Delta }=\pi /30$ and $n=120,m=2,{\omega }_1\mathrm{\Delta }=\pi /30$); and nonzero energy change ($n=60,m=1,{\omega }_1\mathrm{\Delta }=\pi /90$). $T=\pi /\left(3{\omega }_1\right)$.