# Quantum Analyzer Result Tab

The Quantum Analyzer Result tab is the interface to the Result Logger unit of the Instrument and displays processed data after the qubit measurement unit (see Functional Overview for an overview block diagram). It is available on all SHFQA Instruments.

## Features

• Acquisition and display of measurement data

• Multiple probe points: after Integration and Threshold

## Description

Table 1. App icon and short description
Control/Tool Option/Range Description

QA Result

Configure the Result Logger.

The Quantum Analyzer Result tab (see Figure 1) is divided into a display section on the left and a configuration section on the right.

Figure 1. LabOne UI: Quantum Analyzer Result Tab.

This tool allows users to acquire, average, and analyze large sets of data sourced at various points of the signal processing chain. The data source setting is listed in Table 2.

The data are stored in a vector with a length of up to $$2^{19}$$ points and displayed in the plot area on the left once the acquisition is complete. The Result Logger supports averaging of multiple vectors which is enabled by setting the Averages to any number larger than 1.

Note that the QA Result Logger will not get data if the number of acquired data is less than the product of the Length and the Averages set in this tab. This could happen if the number of readouts configured in the Sequencer is less than the product, or the readout repetition rate exceeds 1/(440 ns) (including minimum integration hold off time of 20 ns).

### Amplitude and Phase of Input Signal

Data in QA result logger downloaded from Data Server can be used to calculate the amplitude and phase of input signal.

The analog signal $$E_{\mathrm{RF}}(\omega_{\mathrm{RF}},\ t)$$ after frequency down-conversion before ADC is expressed as

\tag{1} \begin{aligned} E_{\mathrm{RF}}(\omega_{\mathrm{RF}},\ t) &= \mathrm{Re}[A_{\mathrm{in}}(t)e^{i(\omega_{\mathrm{RF}}t+\phi_0 + \phi_1(t))}]\\ & = \mathrm{Re}[(A_{\mathrm{in}}(t)\cos(\phi_1(t)) + i A_{\mathrm{in}}(t)\sin(\phi_1(t)))e^{i(\omega_{\mathrm{RF}}t+\phi_0)}]\\ & = A_{\mathrm{in, \ I}}(t)\cos(\omega_{\mathrm{RF}}t+\phi_0) - A_{\mathrm{in, \ Q}}(t)\sin(\omega_{\mathrm{RF}}t +\phi_0),\\ \end{aligned}

where $$A_{\mathrm{in}}(t)$$ is the amplitude of the input signal, $$A_{\mathrm{in, \ I}}(t)=A_{\mathrm{in}}(t)\cos(\phi_1(t))$$ ($$A_{\mathrm{in, \ Q}}(t)=A_{\mathrm{in}}(t)\sin\phi_1(t))$$) is the in-phase (quadrature) component of the amplitude, $$\phi_0$$ is a global phase of the input signal, $$\phi_1(t)$$ is the time-dependent phase term related to relative amplitude between I and Q components. This signal is then digitized to integer value as

\tag{2} \begin{aligned} E_{\mathrm{RF}, \ n} & = g[A_{\mathrm{in, \ I}}(t_n)\cos(\omega_{\mathrm{RF}}t_n+\phi_0) - A_{\mathrm{in, \ Q}}(t_n)\sin(\omega_{\mathrm{RF}}t_n+\phi_0)],\\ \end{aligned}

where $$i$$ is integer, $$g$$ is the conversion factor depending on gain factor, ADC range and bit resolution. The digital signal is mixed with an internal digital oscillator at 2 GHz and down-converted to

\tag{3} \begin{aligned} E_{\mathrm{IF}, \ n} & = g[A_{\mathrm{in, \ I}}(t_n)\cos(\omega_{\mathrm{RF}}t_n+\phi_0) - A_{\mathrm{in, \ Q}}(t_n)\sin(\omega_{\mathrm{RF}}t_n+\phi_0)]e^{-i\omega_{\mathrm{LO}}t_n},\\ & = \frac{g}{2}[A_{\mathrm{in, \ I}}(t_n)(e^{i(\omega_{\mathrm{RF}}t_n+\phi_0)}+e^{-i(\omega_{\mathrm{RF}}t_n+\phi_0)}) +i A_{\mathrm{in, \ Q}}(t_n)(e^{i(\omega_{\mathrm{RF}}t_n+\phi_0)}-e^{-i(\omega_{\mathrm{RF}}t_n+\phi_0)})]e^{-i\omega_{\mathrm{LO}}t_n},\\ & = \frac{g}{2}[A_{\mathrm{in, \ I}}(t_n)(e^{i(\omega_{\mathrm{RF}}-\omega_{\mathrm{LO}})t_n+i\phi_0}+e^{-i(\omega_{\mathrm{RF}}+\omega_{\mathrm{LO}})t_n-i\phi_0})\\ &\ \ \ \ +i A_{\mathrm{in, \ Q}}(t_n)(e^{i(\omega_{\mathrm{RF}}-\omega_{\mathrm{LO}})t_n+i\phi_0}-e^{-i(\omega_{\mathrm{RF}}+\omega_{\mathrm{LO}})t_n-i\phi_0})],\\ \end{aligned}

where $$\omega_{\mathrm{LO}}= 2\pi\times 2$$ GHz. After a low pass filter, the signal is then expressed as

\tag{4} \begin{aligned} E_{\mathrm{IF}, \ n} & = \frac{g}{2}[A_{\mathrm{in, \ I}}(t_n)e^{i(\omega_{\mathrm{RF}}-\omega_{\mathrm{LO}})t_n+i\phi_0}) +i A_{\mathrm{in, \ Q}}(t_n)e^{i(\omega_{\mathrm{RF}}-\omega_{\mathrm{LO}})t_n+i\phi_0}]\\ & = \frac{g}{2}[A_{\mathrm{in, \ I}}(t_n)+i A_{\mathrm{in, \ Q}}(t_n)]e^{i(\omega_{\mathrm{RF}}-\omega_{\mathrm{LO}})t_n+i\phi_0}].\\ & = \frac{g}{2}[A_{\mathrm{in, \ I}}(t_n)+i A_{\mathrm{in, \ Q}}(t_n)]e^{i(\omega_{\mathrm{IF}}t_n+\phi_0)}]\\ & = \frac{g}{2}A_{\mathrm{in}}(t_n)e^{i(\omega_{\mathrm{IF}}t_n+\phi_0 + \phi_n)}],\\ \end{aligned}

where $$A_{\mathrm{in}}(t_n)=\sqrt{A^2_{\mathrm{in, \ I}}(t_n)+ A^2_{\mathrm{in, \ Q}}(t_n)}$$, $$\phi_n=\arctan(\frac{A_{\mathrm{in, \ Q}}(t_n)}{A_{\mathrm{in, \ I}}(t_n)})$$. By the SHFQA Scope, this signal can be displayed in LabOne GUI and downloaded after a data type conversion from integer to double. The conversion factor is $$k = \sqrt{2}/g$$, therefore the result is $$E_{\mathrm{scope, \ n}}=\frac{A_{\mathrm{in}}(t_n)}{\sqrt{2}}e^{i(\omega_{\mathrm{IF}}t+\phi_0 + \phi_1)}$$ in units of RMS volt.

$$E_{\mathrm{IF}, \ n}$$ is then integrated with a weight waveform $$E_{\mathrm{weights,\ n}}=A_ne^{-i(\omega_{\mathrm{IF}}t_n + \phi')}$$,

\tag{5} \begin{aligned} E_{\mathrm{int}} & = \frac{g}{2}\sum_{n=1}^{n=N}A_{\mathrm{in}}(t_n)e^{i(\omega_{\mathrm{IF}}t_n+\phi_0 + \phi_n)}A_ne^{-i(\omega_{\mathrm{IF}}t_n + \phi')}\\ & = \frac{g}{2}\sum_{n=1}^{n=N}A_{\mathrm{in}}(t_n)A_ne^{i(\phi_0 + \phi_n+ \phi')}.\\ \end{aligned}

In Spectroscopy mode, $$A_n = 1, \ \phi' = 0$$. In Readout mode, $$E_{\mathrm{weights,\ n}}$$ is fully configurable. If $$\phi'=-\phi_0$$,

\tag{6} \begin{aligned} E_{\mathrm{int}} & = \frac{g}{2}\sum_{n=1}^{n=N}A_{\mathrm{in}}(t_n)A_ne^{i\phi_n}.\\ \end{aligned}

After a data conversion with conversion factor $$k=\sqrt{2}/g$$, the data after integration shown in the QA result logger or downloaded is

\tag{7} \begin{aligned} E_{\mathrm{int,\ QA \ result}} & = \frac{1}{\sqrt{2}}\sum_{n=1}^{n=N}A_{\mathrm{in}}(t_n)A_ne^{i\phi_n}.\\ \end{aligned}

If thresholding is selected as the result source, then

\tag{8} \begin{aligned} E_{\mathrm{threshold, \ QA \ result}} \begin{cases} 0 & \mathrm{Re}[E_{\mathrm{int,\ QA \ result}}] \le E_{\mathrm{threshold}}\\ 1 & \mathrm{Re}[E_{\mathrm{int,\ QA \ result}}] \gt E_{\mathrm{threshold}}\\ \end{cases} \end{aligned}

where $$E_{\mathrm{threshold}}$$ is the threshold, and it is a real value. For number of averages > 1, averaging is done after weighted integration when the result source is Integration and after thresholding when the result source is thresholding.

## Functional Elements

Table 2. QA result settings.
Control/Tool Option/Range Description

Run/Stop

Run/Stop the Result Logger.

Source

Integration

Data or averaged data after weighted integration.

Threshold

Data or averaged data after thresholding.

Length

$$2^0$$ to $$2^{19}$$

Number of data points to record. One data point corresponds to a single averaged output of the selected source. The granularity is 1.

Averages

$$2^0$$ to $$2^{16}$$

Number of averages per recorded data point. The granularity is 1.

Acquired

Length x Averages

Indicate the index of the data point that will be recorded next.

Mode

Cyclic

Set Cyclic averaging of the Result Logger. The first point of the Result vector is the average of the results number 1, M+1, 2M+1, and so forth, where M is equal to the Length setting. The second point is the average of the results number 2, M+2, 2M+2, and so forth.

Sequential

Set Sequential averaging of the Result Logger. The first point of the Result vector is the average of the first N results, where N is equal to the Averages setting. The second point of the Result vector is the average of the following N results, and so forth.