Higher-order Zeno sequences

TL;DR

This work tackles speeding up convergence to quantum Zeno dynamics by introducing higher-order Zeno sequences that attain $\mathcal{O}(1/N^{2k})$ scaling, achieved through a correspondence with higher-order Suzuki–Trotter formulas. It develops explicit constructions for projective measurements and unitary kicks, and extends the approach to high-frequency periodic control fields, showing second-order improvements and practical means to shorten Zeno sequences. The authors also tailor efficient protocols for weak-coupling regimes using Uhrig DD–inspired sequences and randomized schemes that suppress odd-in-$H_{PQ}$ contributions, with rigorous error bounds in terms of $J=\|H_{PQ}\|$ and $\beta=\|H_Z\|$. Overall, the results offer a versatile toolkit for faster Zeno dynamics with reduced resource overhead, with implications for decoherence suppression, Hamiltonian simulation, and quantum control, while leaving open problems in continuous-control design and systematic short-sequence construction.

Abstract

The quantum Zeno effect typically refers to freezing the dynamics of a quantum system through frequent observations. In general, quantum Zeno dynamics is obtained with an error of order $\mathcal{O}(1/N)$, where $N$ is the number of projective measurements performed within a fixed evolution time. In this work, we develop higher-order Zeno sequences that achieve faster convergence to Zeno dynamics, yielding an improved error scaling of $\mathcal{O}(1/N^{2k})$, where $k$ describes the order of the Zeno sequence. This is achieved by relating higher-order Zeno sequences to higher-order Trotter formulas that achieve similar convergence behavior. We leverage this relation to develop higher-order Zeno sequences for different manifestations of the quantum Zeno effect, including frequent projective measurements and unitary kicks. We go on to discuss achieving quantum Zeno dynamics through periodic control fields of high frequency. We explicitly develop control fields that yield a second-order type improvement in the Zeno error scaling and present shorter Zeno sequences. Finally, we discuss the connection to randomized and Uhrig dynamical decoupling to develop more efficient implementations in the weak coupling regime.

Figures (2)

• Figure 1: Log–log plot of the UDD-based Zeno sequence error $\epsilon_k^{\rm{UDD}}$ given in \ref{['eq:udd_error']} as a function of $\Delta t$ for different $k = 1, \cdots, 6$. The dashed lines are linear fits that have slopes $\approx k + 1$, which confirms the predicted scaling in the weak coupling regime. For smaller $\Delta t$, the error is dominated by the subleading term $\mathcal{O}(J^{2}\beta\Delta t^{3})$ in \ref{['eq:udd_error']}, confirming a $\Delta t^3$ scaling.
• Figure 2: Trace distance error between the ideal state $e^{-iH_Z\Delta t}\rho_0 e^{iH_Z\Delta t}$ and the states produced by the deterministic $2k$th-order Zeno sequence $\mathscr{S}_{2k}$ (circles) and its randomized counterpart $\mathcal{E}_{2k}^{(N=1)}$ (triangles) for $k = 1, 2$. Deterministic errors scale as $\mathcal{O}(J\beta^{2k}\Delta t^{2k+1})$, whereas randomized errors scale as $\mathcal{O}(J^2\beta^{2k-1}\Delta t^{2k+1})$. (Left) For fixed $\Delta t = 0.1$ and varying $J$, the observed slopes confirm these scalings. (Right) For fixed $J = 0.01$, both protocols show identical $\Delta t$-scaling behavior.