Speed limits of two-qutrit gates

TL;DR

The paper investigates how fast a two-qutrit CZ gate can be implemented in capacitively coupled superconducting transmons. It combines fidelity-optimized protocols from optimal control theory (QuOpt) to determine an empirical speed limit and quantum-speed-limit techniques to derive a theoretical lower bound, finding the empirical limit around the qubit CZ time $T_{min}$ while the bound is $T_{min}^* \ge \pi/(6g) = (2/3)T_{min}$, revealing a gap that invites tighter bounds. The results extend to higher-dimensional qudits and show that ORTs can substantially slow gates in realistic models, underscoring the importance of considering practical constraints. Together, these findings suggest that multi-level gates can approach, but are limited by, fundamental and experimental factors, with implications for achieving deeper quantum circuits on near-term hardware.

Abstract

The speed of elementary quantum gates sets a limit on the speed at which quantum circuits can be applied and, as a result, the size of the computations that can be performed on a quantum computer. This limitation stems from the fact that present-day quantum hardware systems have finite coherence times that limit the total computation time. The speeds of qubit gates in various hardware settings have been well studied over the past few decades. The recent interest in multi-level quantum systems naturally creates a need for similar investigations of the speeds of multi-level or qudit gates. In this work, we perform an empirical study of the speed limit for the three-level or qutrit CZ gate. Our analysis focuses on a theoretical model for capacitively coupled superconducting transmons but can be extended to other systems. We generate CZ gate protocols using optimal control theory techniques and observe when the fidelity crosses certain thresholds. In addition to the empirical approach, we derive an analytical speed limit for the qutrit CZ gate using traditional quantum speed limit techniques. We compare the speed limits derived using these two different approaches and discuss the gap that remains between them. We also compare the time needed to implement the qutrit CZ gate with its qubit counterpart.

Figures (9)

• Figure 1: An example of the optimized pulse $\Omega_1^{x,1}$ representing the $x$ drive for the control pulse that is resonant with the $\ket{0}\leftrightarrow\ket{1}$ transition for the first qutrit. We plot the first 20 out of 40 segments of our protocol for the point $T/T_{min} = 1.1$ for the qutrit CZ gate (see Fig. \ref{['fig:qutritCZ_plot']}).
• Figure 2: The fidelity $F$ of the qutrit CZ gate plotted vs. the control pulse time, expressed in terms of the speed limit of the qubit CZ gate. In other words, the time $T/T_{min} = 1$ is the minimum time needed to generate the qubit CZ gate in a two-qubit system with the same coupling strength $g$.
• Figure 3: Infidelity $1-F$ of the qutrit CZ gate plotted versus the time to generate the gate in terms of the qubit speed limit.
• Figure 4: The fidelity $F$ of the ququart CZ gate plotted vs. the control pulse time, expressed in terms of the speed limit of the qubit CZ gate. In other words, the time $T/T_{min} = 1$ is the minimum time needed to generate the qubit CZ gate in a two-qubit system with the same coupling strength $g$.
• Figure 5: Linear-log plot of the infidelity $1-F$ of the ququart CZ gate versus the control pulse time. This figure uses the same data plotted in Fig. \ref{['fig:CZ4']}.