Open-System Adiabatic Quantum Search under Dephasing

Abstract

Adiabatic quantum algorithms must evolve slowly enough to suppress non-adiabatic transitions while remaining fast enough to be practical. In open systems, this trade-off is reshaped by decoherence. For Hamiltonians subject to dephasing Lindbladians, Avron et al. [1] showed that a unique timetable exists that maximizes the fidelity with a target state. This optimal schedule is characterized by a constant tunneling rate along the adiabatic path. In this work, we revisit their analysis and apply it to the adiabatic Grover search framework, obtaining closed-form expressions for the optimal evolution schedule, the minimum runtime, and the resulting achievable fidelity. Moreover, by invoking an energy-time uncertainty argument, we identify a critical dephasing threshold, beyond which further noise-assisted acceleration is prohibited, thereby defining the physically realizable boundaries for dephasing-based adiabatic quantum search protocols.

Figures (5)

• Figure 1: Dynamics of a 10-qubit adiabatic Grover problem under different control protocols for a fixed total evolution time $T=T_{\textbf{RC}}$ as defined in Eq. \ref{['eq:tRC']} (with $c=\epsilon_{\rm adiab}$).
• Figure 2: Contour plot of the minimum infidelity $\mathscr{T}_{\min}$ of Eq. \ref{['eq:Imin']} versus the system size and dephasing strength for a fixed simulation time $T=200$, in the spirit of Ref. trajectories. System sizes are $N = 2^n$ up to 11 qubits. Small values of $\mathscr{T}_{\min}$ correspond to a large overlap with the ground state.
• Figure 3: Minimal runtime $T_{\min}$ to reach $90\%$ fidelity for $n=30$ qubits as a function of $\gamma$. Shaded regions denote approximate weak (green), moderate (yellow), and strong (red) dephasing regimes.
• Figure 4: Scaling of the adiabatic quantity $C$ from Eq. \ref{['eq:adiabaticTheoremC']} with qubit number $n$ for dephasing $\gamma\propto N^{-a/2}$ (blue $a=0.1$, green $a=1$, orange $a=1.5$). (a) Linear schedule $q(s)=s$: $C\sim N$ (weak/matched) and $C\sim\sqrt N$ (strong dephasing). (b) Optimized schedule $q_{\rm opt}(s)$: $C\sim\sqrt N$ (weak/moderate) and $C\sim\mathrm{const}$ (strong). Grey lines show $N$ and $\sqrt N$.
• Figure 5: Minimum simulation time to reach 90% fidelity under dephasing Lindblad evolution (Eq. \ref{['eq:dephasing']}) as function of qubit number $n$ for $\gamma = g_{\min}$ and $\gamma = \kappa_{\max} g(t)$ (Eq. \ref{['eq:gamma_max_paper']}, $\varepsilon=e^{-1}$, other constants set to one), compared with unitary Roland–Cerf dynamics. The unitary QSL from Eq. \ref{['eq:DL_QSL']} is shown for reference. Dashed lines are color-matched $\sqrt{N}$ fits.

Theorems & Definitions (2)

• Theorem 1: Avron–Fraas avron
• Remark 1