Bang-bang shortcut to adiabaticity in trapped-ion quantum simulators

We model the bang-bang optimization protocol as a shortcut to adiabaticity in the ground-state preparation of a trapped-ion quantum simulator. Compared to a locally adiabatic evolution, the bang-bang protocol typically produces a lower ground-state probability, but its implementation is so much simpler than the locally adiabatic approach, that it can become a competitive choice to use for maximizing ground-state preparation in systems that cannot be solved with conventional computers. We describe how one can optimize the shortcut and provide specific details for how it can be implemented with current trapped-ion quantum simulators. However, when frustration is strong enough, no method appears to work well for adiabatic state preparation within the experimental time frames, and one must confront the issue of dealing with diabatic excitations within the simulation.

Figure 1

(a) Comparison of the different ramp protocols. The magnetic interactions are held fixed, while the field is ramped from an initial value about five times the average nearest-neighbor exchange to zero. Three ramp profiles are shown—the locally adiabatic ramp, which strives to have a uniform rate of diabatic excitation throughout the ramp, the bang-bang protocol, with sharp steps, and the exponential ramp with a time constant one fifth of the experimental run time. (b) The low-lying spectra of an $N=10$ Ising chain with spin-spin interactions for the trap used in Ref. [4] vs the ground-state energy as a function of the magnetic field (here, the exchange coefficients decay with an approximate power law of $\alpha =1.05$). Since the transverse-field Ising model has both spatial parity and spin-reflection parity, the red line denotes the lowest-energy state that is coupled to the ground state and hence plots $\mathrm{\Delta }\left(B\right)$, as described in the text.

Figure 2

False-color plots of the ground-state probability for the bang-bang shortcut to adiabaticity as a function of hold time and quench magnetic field for $\alpha =1.05$. Different panels correspond to different $N$ values: (a) $N=4$; (b) $N=8$; (c) $N=12$; and (d) $N=14$. Note the interesting plateaus that form, and remain at specific times for a range of magnetic quench fields. The light green circle marks the optimized value for the time interval of $t_{\exp }<6$ ms. Note that the false-color scale changes in each panel, as indicated on the accompanying legend.

Figure 3

Vertical cuts (perpendicular to the plateaus) through the false-color optimization plots for the bang-bang shortcut with three different power laws for the exchange coefficients: $\alpha =0.75$, 1.05, and 1.25. Different panels correspond to different size ion chains: (a) $N=4$; (b) $N=8$; (c) $N=10$; and (d) $N=12$ (only the larger two power laws are stable here).

Figure 4

Comparison of the final ground-state probability for the bang-bang shortcut vs the locally adiabatic ramp vs the exponential ramp. The horizontal axis is the number of ions in the chain. (a) is for the case $\alpha =0.5$, where the bang-bang protocol occasionally beats the locally adiabatic one. This also holds for (b) with $\alpha =0.75$. (c) is for $\alpha =1.05$. In that panel, the inset is the ratio of the ground-state percentage for the bang-bang shortcut to the locally adiabatic ramp (black) and to the exponential ramp (green). One can clearly see that the bang-bang shortcut produces about 80% of the locally adiabatic probability for the ground state and it does not appear to improve as the number of ions increases. (d) shows the case $\alpha =2$, where the bang-bang protocol always beats the exponential ramp, but lags significantly behind the locally adiabatic ramp. This same situation holds for larger $\alpha$ (not shown here). The lines are guides to the eye.