Improving qubit readout with hidden Markov models

We demonstrate the application of pattern recognition algorithms via hidden Markov models (HMM) for qubit readout. This scheme provides a state-path trajectory approach capable of detecting qubit-state transitions and makes for a robust classification scheme with higher starting-state assignment fidelity than when compared to a multivariate Gaussian or a support vector machine scheme. Therefore, the method also eliminates the qubit-dependent readout time optimization requirement in current schemes. Using a HMM state discriminator we estimate fidelities reaching the ideal limit. Unsupervised learning gives access to transition matrix, priors, and $IQ$ distributions, providing a toolbox for studying qubit-state dynamics during strong projective readout.

Figure 1

Running average (colored line) of the heterodyned signal of a single shot in a two-qubit four-state system. The different colors correspond to time evolution. The star marker denotes the demodulated $IQ$ value over the entire measurement time. Bayes classifier and contour lines representing probability distributions shown for reference.

Figure 2

Summary of qubit readout. (a) Factors determining readout fidelity begin with the quantum hardware, Hamiltonian parameters, and SNR. (b) Apart from hardware, fidelity may be improved at the demodulation stage. Conventional demodulation scheme requires tuning to an optimal integration time in which the distributions are approximately Gaussian (c). (d) Regression analysis can be used to train models for improved classification performance. (e) Classification errors are extracted from the misclassification probabilities.

Figure 3

State-path prediction with hidden Markov models for readout of a single $20\text{−}\mu \mathrm{s}$ shot. Each shot is discretized into a sequence of 80-ns observations ($O_i$), and each observation is an ($I,Q$) pair emitted from the hidden state having an emission probability distribution $B_d$. The forward-backward algorithm computes the probability of being in the ground (blue) or excited (red) state $\{|0\rangle ,|1\rangle \}$, respectively. The probability the qubit was in the excited state at the start of the measurement was 99.98%.

Figure 4

Single-shot state discrimination with a hidden Markov model (HMM) initial-state classifier for three different readout times ($T$). A HMM is used to classify each shot into one of three cases: (blue) shots predicted to have started and remain in the ground state during the readout, (red) shots predicted to have started and remain in the excited state, and (yellow) shots in which a relaxation transition occurred during the readout. The table lists the counts.

Figure 5

(a) Classifier fidelity in units of the effective relaxation time $T_{1,\mathrm{eff}}$ for the excited state. The HMM scheme achieves a maximum assignment fidelity of $96.5\%\pm 0.4\%$, while the SVM and MVG methods achieve a fidelity of $95.9\%\pm 0.4\%$, and $96.1\%\pm 0.4\%$, respectively. Shaded regions indicate errors. The HMM scheme is robust against qubit relaxation time, eliminating the need for readout time optimization. (b) Total classification error determined with a simulated data set in which the starting state is prepared with 100% preparation fidelity demonstrates that the HMM scheme has a lower total classification error.

Figure 6

(a) $IQ$ scatter plot and equal-variance Gaussian fits of ground- (blue) and excited-state (red) shots filtered with the HMM state discriminator for a 1.2-$\mu \mathrm{s}$ demodulation time. Shots predicted to start in the excited state which transitioned to the ground state during the readout are shown in yellow points on the left panel. Color-matched markers on the right panel represent the projected distributions, solid lines represent the corresponding fits. Fitting errors are smaller than the markers. (b) Fidelity may be improved by rejecting low-probability shots via a threshold parameter, but only at the expense of readout efficiency.

Figure 7

Statistical variations extracted by the bootstrap method on the learned values of the means of the Gaussian $IQ$ distributions for both the ground and excited states.

Figure 8

Statistical variations extracted by the bootstrap technique on the learning values of the transition matrix ($\mathbf{A}$).

Figure 9

Simulated data sets varying $T_1$ from 1 to $16\mu \mathrm{s}$ were used to train several HMM's via unsupervised learning using the Baum-Welch algorithm. The $y$ axis represents the $T_1$ values extracted from HMM and they are plotted against the actual simulated values.

Figure 10

The effective $T_1$ time extracted from the unsupervised learning of the starting-state distributions by sweeping the start time of the integration during the demodulation.