Evaluating three levels of quantum metrics on quantum-inspire hardware

Abstract

With the rise of quantum computing, many quantum devices have been developed and many more devices are being developed as we speak. This begs the question of which device excels at which tasks and how to compare these different quantum devices with one another. The answer is given by quantum metrics, of which many exist today already. Different metrics focus on different aspects of (quantum) devices and choosing the right metric to benchmark one device against another is a difficult choice. In this paper, we aim to give an overview of this zoo of metrics by grouping established metrics in three levels: component level, system level and application level. With this characterization, we also mention what the merits and uses are for each of the different levels. In addition, we evaluate these metrics on the Starmon-5 device of Quantum Inspire through the cloud access, giving the most complete benchmark of a quantum device from an user experience to date.

A Component-level metrics details

In this appendix we discuss the specific experiments used for the component-level metrics. All experiments are performed by executing a quantum circuit while varying one or more of the circuit parameters.

1.1 A.1 Experiment settings

Unless stated otherwise, each data point was run with 4, 096 shots. In addition, the following setting were used for each experiment:

• ${T_1}$ Number of different wait-durations: 32, linearly spaced between 0 and the maximum wait-duration of 60 $\mu s$;

• ${T_2^*}$ Number of different wait-durations: 32, linearly spaced between 0 and the maximum wait-duration of 24 $\mu s$, artificial oscillation frequency 0.125 MHz;

• ${T_2^\textrm{Hahn}}$ Number of different wait-durations (total wait-duration of circuit): 32, linearly spaced between 0 and the maximum wait-duration of 120 $\mu s$;

• Single-qubit RB Lengths of Cliffords sequences: $N \in \left[ 1, 20, 40, 80, 120\right]$. Number of sequences per length: 10.

1.2 A.2 Randomized benchmarking

For each qubit of the Starmon-5 device, the single-qubit fidelity is computed via standard randomized benchmarking. The standard randomized benchmarking protocol [10] consists of circuits that contain a sequence $C_j$ of N random single-qubit Clifford gates. For each sequence of random Clifford gates $C_j$, there exists a single-qubit Clifford gate $C_{N+1}=(\prod C_j)^{-1}$, due to the definition of Clifford gates. For an ideal quantum system, the combined circuit with gates $C_j$, $j=1, \ldots , N+1$ acts as the identity. In practice, due to errors in the gates, errors will increase exponentially with the number of Cliffords N, resulting in incorrect measurement results. For each value of N we generate multiple sequences of random Cliffords and apply them to the qubit initialized in state $|0\rangle$. For each sequence, the resulting qubit state is measured. The following model can then be fitted to the data

$$\begin{aligned} F(N) = A \alpha ^N + B, \end{aligned}$$

(3)

where F(N) is the fraction of $|1\rangle$ measurements and A and B are constants to be determined. Notice that F(N) exactly counts the fraction of incorrect measurements. From the parameter $\alpha$ one can calculate the error per Clifford r (EPC) using the equation

$$\begin{aligned} r = \frac{1}{2}(1-\alpha ). \end{aligned}$$

(4)

The single-qubit gate fidelity is determined from the average number of native gates required for a random Clifford, which is 1.875 on Starmon-5. The single-qubit gate fidelity can hence be computed as $\textrm{F}_{\textrm{1Q}}=(1-r)^{(1/1.875)}$.

Fig. 10

Quantum circuit diagrams for the single-qubit coherence measurements

1.3 A.3 Single-qubit coherence time experiments

In this work, three different single-qubit coherence times are computed: the $T_1$, $T_2^\textrm{Hahn}$ and $T_2^*$ time. These values are constants used in a function fitted to the data, computed as the number of $|1\rangle$ measurements after applying a certain quantum circuit. The quantum circuit for each coherence time experiment can be found in Fig. 10.

For the $T_1$ measurement, the qubit is prepared in the $|1\rangle$ state by applying an X-gate to the $|0\rangle$ state. The qubit is then measured after a variable waiting duration, called the wait-duration. Typically, a wait-duration of up to a few times the $T_1$ time is used. The system remains idle during the waiting stage, but the qubit can decay from the $|1\rangle$ state to the $|0\rangle$ state. This decay results in an exponential decay of the number of measured $|1\rangle$ states. The number of measured $|1\rangle$ states for variable wait-durations t is fitted to the function $F(t)=A +B e^{-t/T_1}$, from which the $T_1$ time can be computed.

To measure the $T_2^*$ time of the system, an $\sqrt{X}$-gate is applied to the $|0\rangle$-state to bring the system to an equal superposition of the 0- and 1-state. Afterward, an $R_Z(\phi )$-rotation is applied, again after a waiting stage with variable waiting time. The $R_Z(\phi )$ is applied with an angle of $\phi (t) = \sin (\phi _R t)$ that depends on the wait-duration t and the artificial frequency shift $\phi _R$. Lastly, a $\sqrt{X}$-gate is applied to map the system back. The result of the circuit is a damped oscillation with oscillation frequency $\phi _R$. The oscillation frequency is chosen such that in the total measurement a few oscillations occur so that we can properly fit the model. The artificial frequency shift [34] is added to the circuit to prevent confounding between the damping of the signal (due to the dephasing) and a very low frequency oscillation (due to a frequency mismatch between the driving signal and the qubit resonance frequency). After performing all the experiments, the number of $|1\rangle$ states for variable wait-durations is fitted to the functions $F(t) = A + B e^{-t/T_2^*} \sin (\omega t + \phi )$, from which the $T_2^*$ time can be computed.

The $T_2^\textrm{Hahn}$ time is measured in a similar fashion as the $T_2^*$ time. Again, a $\sqrt{X}$-gate is applied to the $|0\rangle$ state, after which the system remains idle for a variable waiting time. Then, an X gate is applied as a single refocusing pulse. Then, the system remains idle for the same variable wait-duration, after which another $\sqrt{X}$ is finally applied. An noiseless device would always measure the resulting state as $|0\rangle$, but due to noise the number of $|1\rangle$ measurements for increasing wait-durations t follows an exponential increase. By fitting this number to the function $F(t)=A +B e^{-t/T_2^\textrm{Hahn}}$, the $T_2^\textrm{Hahn}$ time can be computed.