Approximations in Transmon Simulation

Classical simulations of time-dependent quantum systems are widely used in quantum control research. In particular, these simulations are commonly used to host iterative optimal control algorithms. This is convenient for algorithms that are too onerous to run in the loop with current-day quantum hardware, as well as for researchers without consistent access to hardware. However, if the model used to represent the system is not selected carefully, an optimized control protocol may be rendered futile when applied to hardware. We present a series of models, ordered in a hierarchy of progressive approximation, which appear in quantum control literature. The validity of each model is characterized experimentally by designing and benchmarking control protocols for an IBMQ cloud quantum device. This result demonstrates error amplification induced by the application of a first-order perturbative approximation. Furthermore, the emergence of errors that cannot be corrected by simple amplitude scaling of control pulses is demonstrated in simulation, due to an underlying mistreatment of noncomputational dynamics. Finally, an evaluation of simulated control dynamics reveals that despite the substantial variance in numerical predictions across the proposed models, the complexity of discovering local optimal control protocols appears invariant in the simple control scheme setting.

Figure 1

A hierarchy of system Hamiltonians, ordered by level of approximation, is proposed. Here $H_{\mathrm{CPB}}$ represents the model that most accurately describes a coupled transmon-resonator system. Following an arrow indicates the application of an approximation that obfuscates the system dynamics in some manner.

Figure 2

Effective circuit diagram of the Cooper pair box. A Josephson junction is placed in parallel to a capacitance $C_s$, and capacitively coupled to a voltage source. In the transmon architecture, this Josephson junction is further shunted by a large secondary capacitance.

Figure 3

Spectral features of a coupled transmon-resonator system, under a range of modeling schemes and $E_J/E_C$ ratios. Panel (a) demonstrates significant qubit frequency deviations between modeling approaches for an experimentally relevant regime ($n_{\exp }=1$) with ${\omega }_{01}$ in the GR model held constant, while (b) demonstrates further deviations in the qubit anharmonicity across models with $E_C$ (equivalent to $\alpha$ in the GR model) held constant. In each case, the model spectra are seen to converge as $E_J/E_C\to \mathrm{\infty }$. Inset graphs illustrate how $E_C$ and $E_J$ are individually varied to increase the $E_J/E_C$ ratio; in each case, optimized to maintain the GR spectral feature of interest.

Figure 4

The dynamics induced in each model by a Gaussian pulse with period 142.2 ns, over a set of pulse amplitudes from 0 to 75 MHz. The first excited level population is taken to be representative of the dynamics in each case. There is a clear divergence in population between the family of models in red (CPB) and orange (${\mathrm{DO}}_3$), and the family of models in dark blue (GR) and light blue ($R$). This result implies that the efficacy of optimized control fields may depend markedly on the model used in the optimization process.

Figure 5

Randomized benchmarking results for a single-qubit Clifford gate set, transpiled using RX($\pi /2$) pulses. The pulse parameters used in (a) and (b) are optimized using ${\mathrm{DO}}_3$ and GR models, respectively. The EPC (error per Clifford gate), established by exponential curve fitting, increases by over a factor of 2 from ${\mathrm{DO}}_3$ to GR. This result substantiates the earlier inference that the efficacy of an optimized control field depends markedly on the model used in the optimization process.

Figure 6

Deviations in simulated control between the ${\mathrm{DO}}_3$ and GR models. The $x$ axis is detuning of a 5 ns drive pulse from the qubit frequency. The $y$ axis is $E_J/E_C$ values for defining the GR Hamiltonian. The $E_J$ and $E_C$ values of a comparable ratio are chosen to precisely match the qubit frequency and anharmonicity for the ${\mathrm{DO}}_3$ Hamiltonian. Amplitudes around 190 MHz are chosen for the drive, scaled to maximally match dynamics. Both models undergo evolution via 5 ns square pulses. The $z$ axis represents the maximum infidelity experienced between models from a subset of times during the evolution of the 5 ns square pulse, beginning in a computational superposition state. As can be seen, maximum infidelities lie within the range of frequencies associated with higher-level transitions, and exist across all $E_J/E_C$ values. Infidelity is artificially increased at around $E_J/E_C\sim 50$ due to the qubit frequency briefly nearing the resonator frequency.

Figure 7

(a) The control landscape of the CPB model under application of a Gaussian control pulse. The $z$ axis shows the population of the coupled system state associated with computational $|1⟩$, after being driven by a Gaussian control pulse with parameters defined on the $x$ and $y$ axes. (b) A visualization of the deviations in simulated control between the ${\mathrm{DO}}_3$ and GR models. The axes represent pulse parameters in the same way as (a). The color intensity of the figure is then the absolute difference in first-level populations after application of the relevant Gaussian control pulse. This figure illustrates the magnitude of model deviation that can occur for high-energy control fields.

Figure 8

The endpoint distribution of control trajectories generated by GOAT. Initial points in parameter space sampled from a multivariate Gaussian distribution around a previously known maximum are provided to the GOAT algorithm for each model. The blue and red represent the trajectory endpoints of the ${\hat{H}}_{\mathrm{CPB}}$ and ${\hat{H}}_{{\mathrm{DO}}_3}$ models, respectively, the distributions of which are near identical in location and in shape. Models ${\hat{H}}_{\mathrm{GR}}$ and ${\hat{H}}_R$, in gold and black, respectively, differ significantly in parameter-space location, but share a similar elliptical distribution. These results demonstrate consistency in the nature of solutions across models, but considerable differences in the parameter-space regions at which they occur.

Figure 9

The behavior of the metric ${\mathrm{\Delta }}_{M_2}^{M_1}$ under application of a 142.2 ns Gaussian driving pulse, plotted with respect to the amplitude of said driving pulse. This metric measures the difference in simulated first energy level population between models $M_1$ and $M_2$. In blue, substantial differences between models ${\mathrm{GR}}_3$ and ${\mathrm{DO}}_3$ can be observed, with the difference between these models being the application of a first-order perturbative approximation. In comparison, the red points display the relatively small differences between models ${\mathrm{GR}}_3$ and $\mathrm{GR}$, attributable to omission of the sextic perturbative term in simulation. This justifies the statement that the first-order perturbative correction in deriving ${\hat{H}}_{\mathrm{GR}}$ from ${\hat{H}}_{{\mathrm{DO}}_3}$ is responsible for the magnitude of divergence in simulated dynamics between these models.

Figure 10

Deviations in simulated control between frequency-matched and amplitude-scaled ${\mathrm{DO}}_3$ and GR models at $E_J/E_C\sim 100$. The $x$ axis is detuning of a 5 ns square drive pulse from the qubit frequency. Amplitudes around 190 MHz are chosen for the drive, scaled to maximally match dynamics. The system begins in a 50–50 superposition between the ground and excited states. The $y$ axis represents the maximum infidelity experienced between models from a subset of times during the evolution of the 5 ns square pulse. Significant infidelity peaks exist at frequencies that correspond to higher-level transitions and two-photon transitions, as well as some small deviation at the qubit frequency. In particular, comparison of the models at the two-photon transition $|1⟩\to |3⟩$ yields an infidelity in excess of 50%.

Figure 11

Normalized simulation times for Gaussian control pulses of varying periods. The most faithful model (CPB) is seen in blue. The proposed transmon basis model of highest precision (${\mathrm{DO}}_3$) is seen in green, while the generalized quantum Rabi (GR) and two-level quantum Rabi ($R$) models offer significant accelerations in computation time as seen in red and black, respectively. This result communicates that the merit of the Rabi family of models (GR and R) lies in their relatively low computational demand.