General Bound on the Performance of Counter-Diabatic Driving Acting on Dissipative Spin Systems

Counter-diabatic driving (CD) is a technique in quantum control theory designed to counteract nonadiabatic excitations and guide the system to follow its instantaneous energy eigenstates, and hence has applications in state preparation, quantum annealing, and quantum thermodynamics. However, in many practical situations, the effect of the environment cannot be neglected, and the performance of the CD is expected to degrade. To arrive at general bounds on the resulting error of CD in this situation we consider a driven spin-boson model as a prototypical setup. The inequalities we obtain, in terms of either the Bures angle or the fidelity, allow us to estimate the maximum error solely characterized by the parameters of the system and the bath. By utilizing the analytical form of the upper bound, we demonstrate that the error can be systematically reduced through optimization of the external driving protocol of the system. We also show that if we allow a time-dependent system-bath coupling angle, the obtained bound can be saturated and realizes unit fidelity.

Figure 1

Schematic diagrams of (a) the concept of levels of controllability and (b) the setup. (a1) Controllability of a general many-body system. (a2) Controllability of the spin-boson model. Unit fidelity is achieved by the exact STA protocol [Eq. (3)] which requires a time-dependent control of the interaction. The lower bound on the fidelity $F$ set by the CD on the system alone is characterized by ${\cos }^2{l}_{\mathrm{BD}}$ via the inequality in Eq. (9). (b) We consider the spin-boson model to study the controllable limit of the spin system via CD under the influence of the environment. Here, controllability is measured by the ground state fidelity of the system.

Figure 2

Numerical demonstration of the bound Eq. (9) by varying $\mathrm{\Delta }$. The solid curves show the fidelity $F[|{\psi }_g(\tau ),{\rho }_S(\tau )]$, and the dashed curves are the lower bound ${\cos }^2{l}_{\mathrm{BD}}$ [Eq. (8)] for different values of $a$. The inset shows the functional form of the external driving $q(t)=\sinh [a(t-\tau /2)]/\sinh (a\tau /2)$, which is designed to better approximate the optimal drive $q(0)=-1$, $q(t)=0$ ($0<t<\tau$), and $q(\tau )=1$, as $a$ increases. Therefore, the lower bound ${\cos }^2{l}_{\mathrm{BD}}$ and also the fidelity become closer to unity as $a$ becomes larger, demonstrating the effectiveness of our inequality in Eq. (9). The parameters are $\phi =\pi /4$ (${\sigma }_x$ coupling), $\beta =1$, $\gamma =0.1$, $w_0=1$, $\lambda =0.1$, $\tau =2$.