Qubit reset beyond the Born-Markov approximation: optimal driving to overcome polaron formation

Abstract

Qubits are typically reset into a known state by coupling them to a low-temperature environment. When treated in the Born-Markov approximation such couplings produce exponential relaxation to equilibrium, giving high reset fidelities limited only by temperature. We investigate qubit reset beyond this approximation, using numerically exact tensor network methods and the time-dependent variational principle, focussing on a spin-boson model describing a transmon qubit coupled to a resistor. Beyond the Born-Markov approximation the reset fidelity becomes limited by the buildup of system-environment correlations which corresponds to the formation of a polaron. We implement numerical optimal control to find time-dependent qubit Hamiltonians which overcome this limitation by steering the dynamics of the correlated system-environment state. The optimal controls becomes more effective when the environment is filtered to span a smaller spectral range, and remain effective when the multilevel nature of the transmon is considered. A related paper [C. Ortega-Taberner, E. O'Neill and P. R. Eastham, arXiv:XXXX.XXXX] addresses the complementary case of control via a time-dependent system-environment coupling. Our results show how limitations on reset speed and fidelity can be overcome, and how time-dependent driving can steer system-environment correlations and reverse polaron formation.

Figures (6)

• Figure 1: Excited state population $P_+$ of the qubit at the end of a reset protocol of duration $t_f$ for coupling strengths $\alpha=0.3$ (top two curves), $0.03$ (middle two curves), and $0.003$ (lower two curves). Dashed lines: protocol with a constant qubit splitting. Solid lines and points: protocols obtained by numerical optimization of the time-dependent splittings $\omega_q(t)$. Crosses: predictions of the polaron ansatz.
• Figure 2: (a) Dynamics of the qubit population for the optimized $\omega_q(t)$ with $\alpha=0.03$ and $t_f=11$ ns. Black: numerical results using OQuPy. Grey: TDVP approximation of the dynamics starting from the polaron state. (b) Time-dependent frequency control, $\omega_q(t)$, found by the optimizer. (c) Power spectrum of the optimal protocol, $\omega_q(t)$, showing that the oscillation frequency matches the initial constant value $\omega_q^0$.
• Figure 3: (a) Complex displacement $f_k(t)$ in the polaron state for a bath mode with frequency $\omega_k = \omega_C/2$, for $\alpha = 0.03$, and the optimal frequency control shown in Fig. \ref{['fig:fig2']}(b). Increasing oscillations in the control result in oscillations in the displacement which approach zero. In red we highlight one period of the oscillations, tracing an ellipse in the complex plane with an aspect ratio of $1.45 \approx \omega_k'/\omega_q^0$. (b) Time-dependent phases of the oscillator displacements from equilibrium, $q_k(t) = f_k(t) - f_{k0}$. The optimal frequency control rephases the oscillators such that at final protocol time all oscillators approach the ground state simultaneously. (c) Phase of the oscillators for the optimal protocol at the second to last minima in Fig. \ref{['fig:fig2']}(a), while the weak driving assumption is still valid. In a dashed line we have shifted $\omega_q^0$ from the value in the optimal protocol to showcase the predicted $\pi$ phase shift in the oscillations across the resonance.
• Figure 4: (a) Dynamics of the qubit population for the optimal protocol with a filtered bath at the end of the process. The inset shows the initial dynamics. Black (grey) curve shows the results from TEMPO (TDVP). (b) Time-dependent frequency control found by the optimizer which oscillates with two frequencies. (c) Power spectrum of the optimal protocol showing two dominant frequencies to either side of the center of the distribution of shifted bath frequencies (red). $\alpha = 0.013, \omega_C/2\pi = 5$ GHz.
• Figure 5: Phase profile of the oscillators at the final time for the optimal protocol with a filtered bath, as a function of the shifted bath frequency $\omega_k' = \omega_q^0 + \omega_k$. In the frequency region of the filter, centered on $\omega_q^0+\omega_C = 10$ GHz, the phase is approximately constant, but varies strongly outside this region.