Improving transmon qudit measurement on IBM Quantum hardware

The Hilbert space of a physical qubit typically features more than two energy levels. Using states outside the qubit subspace can provide advantages in quantum computation. To benefit from these advantages, individual states of the $d$-dimensional qudit Hilbert space have to be discriminated during readout. We propose and analyze two measurement strategies that improve the distinguishability of transmon qudit states. Based on a model describing the readout of a transmon qudit coupled to a resonator, we identify the regime in hardware parameter space where each strategy is optimal. We discuss these strategies in the context of a practical implementation of the default measurement of a ququart on IBM Quantum hardware whose states are prepared by employing higher-order $X$ gates that make use of two-photon transitions.

Figure 1

Overview of qudit parameters of the IBM Quantum devices listed in the legend. The qubit resonance frequency and anharmonicity are denoted by ${\omega }_{0,1}$ and ${\alpha }_1$; see Appendix pp1. The energy dispersion ${\epsilon }_3$ of the third excited state given in Eq. (A5) follows from these device specifications. The labeled straight black lines denote constant values of $E_J/E_C$.

Figure 2

(a) Drive-frequency-dependent phase-space positions $A_j^{\left(d\right)}$ and $A_j^{\left(m\right)}$ of the coherent state of the resonator given the qudit prepared in $|j\rangle$; see Eqs. (7) and (12). For ${\omega }_m={\omega }_d$, the trajectories of all states $A_j^{\left(d\right)}$ match, denoted by the black circle. The colored lines denote $A_j^{\left(m\right)}$ for ${\omega }_m={\omega }_{d,0}^{(0,1)}$. Crosses highlight the positions at ${\omega }_d={\omega }_m$ where both models match, $A_j^{\left(d\right)}=A_j^{\left(m\right)}$. (b),(c) Error measures ${\xi }_j$ and $\xi$ in the frame of ${\omega }_d$ and ${\omega }_m$ respectively; see Eqs. (17) and (18). Following Appendixes pp1 and pp2-s1, for these plots, we determine $E_J/E_C$ by the qubit parameters ${\omega }_{0,1}$ and ${\alpha }_1$ from ibm_lagos Q4. Moreover, we choose $g/2\pi =100$ MHz, $\mathrm{\Omega }/2\pi =100$ MHz, $\kappa /2\pi =5$ MHz, $T=0.35\mu \mathrm{s}$, ${\sigma }_j=0.13\mathrm{\Omega }/\kappa$, $\varphi =0$, and $n_g=0$.

Figure 3

Ratio of the standard deviation ${\mathrm{SD}}_m$ for the multifrequency strategy and the standard deviation ${\mathrm{SD}}_s$ of the single-frequency strategy applied to an equal-superposition state taking $N=1000$ shots. The gray region indicates where both standard deviations exceed ${\mathrm{SD}}_{s/m}\ge 0.1$. The straight lines denote constant values of $\sigma \kappa /\mathrm{\Omega }$. We take the same qudit parameters as in Fig. 2 and choose $g/2\pi =100$ MHz and $\mathrm{\Omega }/2\pi =100$ MHz.

Figure 4

(a) Drive-frequency-dependent ququart measurements on ibm_lagos Q4, the experimental equivalent of Fig. 2. The colored curves (with white shadows) correspond to the centers of Gaussian fits to the Fock states $|j\rangle$. Black straight lines indicate the boundaries of regions assigned to individual Fock states. These boundaries are constructed using the minimum distance estimator; see Eq. (16). For the drive frequency ${\omega }_0=7.2463$ GHz, we show the measurement results of all $N=2000$ shots for each prepared Fock state. Black crosses mark the centers of their Gaussian fits. (b) Measurement errors ${\xi }_j=1-M_{j,j}$ based on all measured shots of the data presented in (a). Here, the elements $M_{i,j}$ of the assignment matrix equal the relative number of shots, $N_i/N$, that are classified as $|i\rangle$ even if $|j\rangle$ is prepared. The horizontal gray line denotes the measurement error ${\xi }_{\text{def}}$ obtained using the default measurement pulse. (c) Measurement errors ${\xi }_j$ based on Gaussian fits to the data presented in (a) and the assignment matrix $M$ defined in Eq. (14). Using the centers and average $\sigma$ of Gaussian fits for each readout resonator drive frequency, we calculate $M$ numerically. The ${\xi }_j$ shown in (b) are larger than those in (c) since they do not only represent assignment errors, but also include additional errors such as qudit decay, leakage, and imperfect state preparation. In both (b) and (c), the minimum of the average assignment error $\xi$ will be smaller than ${\xi }_{\text{def}}$ obtained by the default pulse if the modulation frequency is chosen such that ${\omega }_m={\omega }_d$; cf. Figs. 2 and 2.

Figure 5

Numerical prediction of the energy levels $E_n$ of ibm_lagos Q4 based on Eq. (A1) and $E_J/E_C\approx 45.6$. The transition frequencies ${\omega }_{i,j}$ are displayed in units of $\left(2\pi \right)$ GHz.

Figure 6

(a) Qudit resonator spectroscopy of transitions $|i\rangle \leftrightarrow |j\rangle$. Colored markers denote measured data and solid curves correspond to Gaussian fits. We plot each resonance spectrum centered around the predicted transition frequency ${\tilde{\omega }}_{i,j}$ [see Eq. (B14)] using $g/2\pi =65\mathrm{MHz}$. (b) Rabi oscillations $|i\rangle \leftrightarrow |j\rangle$ depending on the drive amplitude for fixed pulse duration. We sweep the readout resonator drive amplitude while keeping all other parameters of the drive fixed. For first-order qudit state transitions, we fit a sinusoidal dependence on a linear function of the drive amplitude in the interval [0,0.5]; see the lines connecting crosses and, respectively, diamonds. For second-order qudit state transitions, we fit a sinusoidal dependence on a quadratic function of the drive amplitude in the interval [0,1]; see the lines connecting dots and, respectively, triangles. The vertical dashed lines indicate the locations of the first maxima obtained from the fits.