Performance evaluation of adiabatic quantum computation via quantum speed limits and possible applications to many-body systems

The quantum speed limit specifies a universal bound of the fidelity between the initial state and the time-evolved state. We apply this method to find a bound of the fidelity between the adiabatic state and the time-evolved state. The bound is characterized by the counterdiabatic Hamiltonian and can be used to evaluate the worst case performance of the adiabatic quantum computation. The result is improved by imposing additional conditions and we examine several models to find a tight bound. We also derive a different type of quantum speed limit that is meaningful even when we take the thermodynamic limit. By using solvable spin models, we study how the performance and the bound are affected by phase transitions.

Figure 1

QSL for a two-level system. (a), (b) $|{\dot{\theta }}_{\mathrm{ad}}\left(t\right)|$ (black solid), $\mathrm{\Delta }{E}_1\left(t\right)$ (red dotted), and $\mathrm{\Delta }{E}_2\left(t\right)$ (blue dashed) for a protocol $\theta \left(t\right)$ given in the inset of each panel. We set the annealing time $T=50$ in units of $h$. (c) Distribution of $\left({\int }_0^{T}dt\mathrm{\Delta }{E}_2\left(t\right),{\int }_0^{T}dt\left|{\dot{\theta }}_{\mathrm{ad}}\left(t\right)\right|\right)$ for randomly generated protocols $\theta \left(t\right)$. The protocols are generated as decreasing functions from $\pi /2$ to 0. The solid line represents the bound. (d) Annealing time dependence for the protocol in the top left-hand panel. ${\dot{\theta }}_{\mathrm{ad}}\left(t\right)$ (black), $|{\dot{\theta }}_{\mathrm{ad}}\left(t\right)|$ (blue), $\mathrm{\Delta }{E}_1\left(t\right)$ (red), and $\mathrm{\Delta }{E}_2\left(t\right)$ (green) are integrated from 0 to $T$.

Figure 2

QSL for a spin chain model. (a) $|{\dot{g}}_{\mathrm{ad}}\left(t\right)|$ (black solid line) and the right-hand side of Eq. (14) (red dotted) for a linear protocol $A\left(t\right)=A\left(0\right)(1-t/T)$, $B\left(t\right)=A\left(0\right)t/T$. We set $N=1000$ and $T=1000$ in units of $A\left(0\right)$. The inset represents a blowup around the peak. (b) Size dependence of the peak height obtained in (a). The black solid line with the symbol $•$ denotes ${max}_t\left|{\dot{g}}_{\mathrm{ad}}\left(t\right)\right|$ and the red dashed line with $\circ$ the corresponding quantity in the QSL. They show the same power-law behavior denoted by the blue dotted line at large $N$. (c) Protocol dependence. We choose two types of $A\left(t\right)$ shown in the insets and $B\left(t\right)=A\left(0\right)-A\left(t\right)$. The solid lines denote $|{\dot{g}}_{\mathrm{ad}}\left(t\right)|$ and the dashed lines the QSL. We set $N=1000$ and $T=1000$ in units of $A\left(0\right)$. (d) Annealing time dependence. We use the linear protocol used in (a) and set $N=200$.

Figure 3

QSL for a quantum quench system with dynamical phase transitions. We set $J=h$. The black solid line represents $\dot{g}\left(t\right)$, the blue dashed line $|\dot{g}\left(t\right)|$, and the red dotted line the QSL specified by the right-hand side in Eq. (13). Inset: Time-integrated quantities.