Mid-circuit measurements tools (MCMs)

Introduction

To improve initialization and readout fidelity, we can employ the MCMs. In particular, we will implement the post-selection (PS) and the maximum likelihood 3 readout (ML3).

Maximum-likelihood 3 readout and post-selction

The ML3 RO measurement procedure is implemented at the measurement step. Instead of reading out the state only once (ML1), we perform three measurements back to back. Depending on the 3 measurement outcomes, we assign the qubit's state. It allows us to gain additional information about the qubit state over an extended duration by tracking the trajectory of the state over the three measurements.

To correctly assign the qubit's state, we measure in which 3-measurement combinations you are more likely to be in $\left|0\right\rangle$ or $\left|1\right\rangle$. To characterize the readout, we use the circuit below (Figure 7). We prepare the qubits in $\left|0\right\rangle$ and $\left|1\right\rangle$ state and perform three back-to-back measurements. We calculate the probability of obtaining each of them for each combination of measurement outcomes considering the initial state of the qubit. From these statistics, we can conclude what combination most likely corresponds to what input state.

Post-selection is implemented by doing a qubit measurement right after the initialization. If the qubit is not in the desired state, we disregard the job.

Fig.%201.jpeg (Figure 7. ML-3 RO calibration procedure for one qubit. $m_{j}$,$m_{k}$,$m_{l}$ are the three measurements that characterize the ML-3 RO procedure. With only one job, we fully characterize the ML3 RO (TV mode). First, we prepare the qubit in the $\left|0\right\rangle$ state and we measure it three times. Then, the qubit is initialized in the $\left|1\right\rangle$ state and measured.)

You can find the utility functions that we use below to create the local directory and save the data in the public GitHub repository: https://github.com/DiCarloLab-Delft/QuantumInspireUtilities/tree/main

Since some measurement combinations occur very rarely, we acquire a lot of data to get a good statistic of the outcome.

The following function calculates the frequencies of the measurement combinations that are related to the different initial states, the return "i" from the function below is this threshold. This integer number tells us which qubit's state correspond to the 3 bit outcome.

The graphs for each qubit are shown below. If we consider only the first measurement (ML1), we will assign the outcome '011' erroneously to the state $\left|0\right\rangle$. Indeed, we see that we are more likely to measure outcome '011' when the qubit is initialized in $\left|1\right\rangle$. So, we move the threshold one step to the left. Different qubits can have different thresholds to asses the state $\left|0\right\rangle$ or $\left|1\right\rangle$. While the ML1 RO threshold is always between 0 and 1, in the ML3 RO procedure, it can shift over time.

The fractions on the Y-axis indicate the probability to obtain a certain measurement outocome, given the initial state of the qubit. The threshold in the picture discriminates between the $|0\rangle$ and $|1\rangle$ outcome.

Calculating increase in fidelity

To compare the ML3 and ML1 readout procedures we calculate the average assignment fidelity in the two cases. We use the equation

\begin{equation} \label{eq: Hadamard} F_{aa} = 1-\frac{\epsilon_{01} + \epsilon_{10}}{2} \end{equation}

The values of $\epsilon_{01}$ and $\epsilon_{10}$ are the probabilities that we wrongly assign the $\left|1\right\rangle$ and $\left|0\right\rangle$ state respectively.