Semiclassical Modeling of Circuit Quantum Electrodynamics Devices

Circuit quantum electrodynamics (cQED) architectures are one of the most popular approaches currently being pursued to achieve a quantum advantage with quantum computers. However, modeling methods to perform the detailed engineering design of these systems are still in their infancy. To help improve the design process of cQED systems, it will be necessary to develop a tighter integration between rigorous full-wave computational electromagnetics solvers and the quantum analyzes necessary to assess quantum effects. One area where this can be more readily achieved is in the semiclassical analysis of cQED systems. In this analysis approach, the dynamics of the qubits of the system are evaluated in a quantum mechanical manner while the electromagnetic fields are still treated as classical quantities. This type of numerical approach has found great use in the analysis of certain optical systems, but has never been expored in the context of cQED systems up to this point.

Although semiclassical models invariably contain approximations to the full physics taking place, they can still find use in analyzing certain important effects at a much lower computational cost than a fully quantum mechanical analysis. For instance, semiclassical models of cQED systems can be useful in more rigorously analyzing qubit state preparation, the fidelity of various quantum gates, and the measurement of qubit states. This is possible because all of these operations involve applying a classical microwave pulse to various transmission lines that are coupled to the cQED qubits. One advantage of semiclassical models is that they can provide a self-consistent description of how the applied fields modify the qubit's state and how changes in the qubit state go on to modify the fields in the transmission lines. 

To verify the importance of these self-consistent interactions, we have implemented a 1-D transmission line theory version of this modeling method. We found that our method agreed well with an approach that was not self-consistent when the coupling between the qubit and transmission lines was kept weak and the pulse width of control signals were wide, as shown in the figure below. However, when we broadened the design space to try and increase the speed of control pulses, we found that accounting for the self-consistent interactions became increasingly important. In particular, we found non-trivial dynamical effects, like the transmon re-biasing itself through the interaction with the transmission line. Modeling these effects with traditional approaches is difficult because it requires ~20 modes of the microwave resonator to be included in the model to capture the underlying wave propagation effect causing the non-trivial dynamics. This would require extracting eigenmodes to ~100 GHz, which is computationally prohibitive for a general 3-D model. In contrast to this, a semiclassical method that naturally describes the propagation of fields needs no special treatment to efficiently analyze these effects.

(a) Effective circuit model of typical quantum control analysis that is not self-consistent and (b) transmission line schematic of our self-consistent semiclassical analysis. Excited state probability in the weak coupling regime for (c) wide pulse width and (d) narrow pulse width. In (d) the transmon re-biases itself after the control pulse has exited the resonator, which is only captured through the self-consistent analysis.

State-dependent resonator transmission characteristics for dispersive readout with the transmon detuned (a) negatively and (b) positively from the cavity frequency. Theoretical estimates predict a state-independent shift of \chi_{12}/2 to the resonant frequency of the uncoupled resonator, which the state-dependent resonant peaks are then equally-displaced from by \chi. Our results agree well with these tehoretical predictions, with some slight deviations expected due to our model including more states of the transmon than are taken into account in the theoretical treatment and because our approach makes no asymptotic approximations.