Trajectory-independent speed limits for controlled open quantum systems

Abstract

Existing quantum speed limits for controlled open quantum systems depend on the specified trajectory. For example, lower bounds on quantum annealing times in the presence of dissipation depend explicitly on the chosen annealing schedule. Recently, schedule-independent speed limits have been derived for annealing in the closed quantum system setting (SciPost Phys. 18, 159 (2025)). In this work, we generalize these results to open quantum systems, deriving schedule-independent lower bounds for quantum annealing times in systems described by a Lindblad master equation. We analyze the interplay between coherent control and dissipation in single- and two-qubit examples, demonstrating that the derived lower bounds capture key scaling behavior with respect to the strength of the dissipator. Finally, we apply the bound to thermal state preparation and show that the bound matches the expected asymptotic behavior for an Ising model in the high temperature limit.

Figures (4)

• Figure 1: Time $T$ to prepare a desired target state for the single-qubit system described by Eqs. \ref{['eq:singlequbitham']} and \ref{['eq:ampdamp']} as a function of the decay rate $\gamma$ for fixed $\omega=1$. The bound in Eq. \ref{['eq:twolevelineq']} is shown as a dashed red line. The blue markers show the time required to prepare the target state up to a trace distance of $0.1$ with a piecewise-constant control $f(t)$. The optimal time is numerically estimated using the method described in the introduction of Sec. \ref{['sec:casestudies']}. For reference, the green markers show the time required for the dissipator alone to drive the system to its fixed point up to the same trace distance as in the case assisted by coherent control.
• Figure 2: Controlled (blue) and uncontrolled (green) trajectories subject to amplitude damping with $\gamma = 1$ for the single-qubit system described by Eq. \ref{['eq:singlequbitham']} and \ref{['eq:ampdamp']}. In (a), no drift Hamiltonian is present, i.e., $\omega=0$, while in (b) we set $\omega=1$. Even though the green trajectory in (b) spirals around the target state due to the presence of the drift Hamiltonian, the total preparation time is still equal to the preparation time that correspond to the trajectory in (a), highlighting that the presence of the drift Hamiltonian does not change the required preparation time. The blue trajectory in (b) illustrates that drift, control and amplitude damping are able to work together to accelerate the preparation of the target state 2.7-fold. The green and blue markers are synchronized in time to help illustrate that the control is unable to accelerate the state preparation without the drift Hamiltonian. The red stars on the north poles illustrate the target states. These visualizations and the one-qubit dynamics are generated using QuTiP qutip5.
• Figure 3: Time $T$ to prepare a Bell state for the two-qubit system described by Eqs. \ref{['eq:hamtwoqubits']} and \ref{['eq:dissipatortwoqubits']} as a function of the dissipation rate $\gamma$ for fixed $\omega=1$. The bound in Eq. \ref{['eq:specineqweak']} is shown as a red dashed line. The blue markers show the time to prepare the target state up to trace distance 0.1 with a piecewise-constant control. The optimal time is numerically estimated from above. The green markers show the time required for the dissipator alone to drive the system to its fixed point up to the same trace distance as in the case assisted by coherent control.
• Figure 4: Time $T$ to prepare the thermal state for the four-qubit system described by Eqs. \ref{['eq:hamising']} and \ref{['eq:dissipatorising']}--i.e. an extensive all-to-all antiferromagnetic Ising model coupled to a bath of harmonic oscillators in thermal equilibrium--as a function of the temperature $1/\beta$. The bound in Eq. \ref{['eq:specineqweak']} is shown as a red dashed line. The blue markers show the time to prepare the target state up to trace distance 0.1 with a piecewise-constant control. The optimal time is numerically estimated from above. For reference, the green markers show the time required for the dissipator alone to drive the system to its fixed point up to the same trace distance as in the case assisted by coherent control.

Theorems & Definitions (2)

• Proposition 1
• proof