Dynamics of qudit gates and effects of spectator modes on optimal control pulses

TL;DR

This work addresses fidelity degradation of high-dimensional qudit gates caused by spectator-mode crosstalk in cQED systems. By deriving an effective Hamiltonian $H_{\text{eff}} = H_0 + \varepsilon V$ with $H_0 = -\tfrac{\xi}{2}(\hat{n}\hat{n}-\hat{n})$ and $\varepsilon = \sum_j \xi_j n_j$, the authors show cross-Kerr couplings shift the target’s transition frequencies by $\sum_j \xi_j n_j$. They analytically predict a quadratic fidelity decay $F \simeq 1 - \frac{\big[\mathrm{Tr}(\bar{V}^2) - \mathrm{Tr}(\bar{V})^2\big] T^2}{\hbar^2 d^2} \varepsilon^2$ and confirm this scaling numerically for multiple SWAP gates, revealing a practical bound: useful single-qudit gates require spectator-induced shifts to be $\lesssim 0.1\%$ of the qudit nonlinearity. The results provide a quantitative criterion for designing qudit pulses in the presence of spectator modes and motivate mitigation strategies like dynamical decoupling and robust control to preserve high fidelities in scalable quantum processors.

Abstract

Qudit gates for high-dimensional quantum computing can be synthesized with high precision using numerical quantum optimal control techniques. Large circuits are broken down into modules and the tailored pulses for each module can be used as primitives for a qudit compiler. Application of the pulses of each module in the presence of extra modes may decrease their effectiveness due to crosstalk. In this paper, we address this problem by simulating qudit dynamics for circuit quantum electrodynamics (cQED) systems. As a test case, we take pulses for single-qudit SWAP gates optimized in isolation and then apply them in the presence of spectator modes each of which are in Fock states. We provide an experimentally relevant scaling formula that can be used as a bound on the fidelity decay. Our results show that frequency shift from spectator mode populations has to be $\lesssim 0.1\%$ of the qudit's nonlinearity in order for high-fidelity single-qudit gates to be useful in the presence of occupied spectator modes.

Figures (1)

• Figure 1: (Top) Infidelity for the labeled SWAP operation arising from a frequency shift $\varepsilon=\sum_j \xi_j n_j$ from the presence of $n_j$ photons in the $j$th spectator mode with cross-Kerr strength to the target mode $\xi_j$ relative to the ideal/target gate. We exclude zero occupation as the x-axis value would be $0$, but the infidelity for that case is that of the optimal control pulse without spectator modes, namely $\order{10^{-4}}$ to $\order{10^{-3}}$ for each gate, as seen at the smallest $\varepsilon$. The slope for small $\varepsilon/\xi=\order{10^{-4}}$ is $\approx 2$, meaning that infidelity scales quadratically with $\varepsilon$. The flat region at very small $\varepsilon$ is the region when the perturbation is negligible, while for larger $\varepsilon$ higher order terms (in part due to saturation near infidelity $\approx 1$) take effect. (Bottom) Rescaled fidelity curves such that the value at the $\varepsilon=10^{-4}$ for each curve is equal, so as to highlight the similarity of the slope in that region.