# Weighted Integration and Result Logging

 This tutorial is applicable to all UHFQA Instruments.

## Goals and Requirements

This tutorial explains how to analyze a pulsed signal using the weighted integration units and record the result with the Result Logger in the Quantum Analyzer Result tab.

The measurements in this tutorial can be performed using simple loop back connections.

## Preparation

Connect the cables as illustrated below. Make sure that the UHF unit is powered on and connected by USB to your host computer or by Ethernet to your local area network (LAN) where the host computer resides. After starting LabOne the default web browser opens with the LabOne graphical user interface.

Figure 1. UHF connections for the Result Logger tutorial

The tutorial can be started with the default instrument configuration (e.g. after a power cycle) and the default user interface settings (e.g. as is after pressing F5 in the browser).

## Test Signal Generation

First, we enable both outputs of the UHF instrument.

Table 1. Settings: observe the pulse waveform
Tab Sub-tab Section # Label Setting / Value / State

In / Out

Signal Outputs

1

On

ON

In / Out

Signal Outputs

2

On

ON

Copy the following code into the Sequence Editor in the AWG tab.

const length = 4096;
const f_s = 1.8e9;
wave w_I = gauss(length, length/2, length/8)
wave w_Q = gauss(length, length/2, length/8)

setTrigger(AWG_INTEGRATION_ARM);
var result_length_by_2 = getUserReg(0)>>1; // We divide by two using the
// bit shift operator because
// each loop contains two pulses
var result_averages = getUserReg(1);

repeat (result_averages) {
repeat (result_length_by_2) {
playWave(w_I, w_Q);
setTrigger(AWG_INTEGRATION_ARM + AWG_INTEGRATION_TRIGGER + AWG_MONITOR_TRIGGER);
setTrigger(AWG_INTEGRATION_ARM);
waitWave();
wait(1024);

playWave(0.5*w_I, 0.5*w_Q);
setTrigger(AWG_INTEGRATION_ARM + AWG_INTEGRATION_TRIGGER + AWG_MONITOR_TRIGGER);
setTrigger(AWG_INTEGRATION_ARM);
waitWave();
wait(1024);
}
}
setTrigger(0);

Upload this program to the AWG by clicking on "Save" or "To Device". The program generates a series of modulated Gaussian pulses with two alternating amplitudes. There are two nested loops: the number of pulses in the inner loop is determined by User Register 1 and would correspond to the number of points in a pulsed measurement (e.g., a Rabi flopping). The outer loop corresponds to the number of identical repetitions to be performed in order to allow averaging and is determined by User Register 2. Apply the settings in the following table to configure the AWG output as well as the User Registers.

Table 2. Settings: configure the AWG output and User Registers
Tab Sub-tab Section # Label Setting / Value / State

AWG

Control

Rerun

OFF

AWG

Control

Output 1

Amplitude (FS)

1.0

AWG

Control

Output 1

Mode

Plain

AWG

Control

Output 2

Amplitude (FS)

1.0

AWG

Control

Output 2

Mode

Plain

AWG

Control

User Registers

Register 1

100 (Length)

AWG

Control

User Registers

Register 2

1 (Averages)

## Configure the Integration Units

The AWG generates pulses with cosine and sine carrier waves at 10 MHz. In the QA Input tab, we have to define integration weights that enable demodulation of these signals at the right frequency. We choose channel 1 for this measurement.

Table 3. Settings: define the integration weights
Tab Sub-tab Section # Label Setting / Value / State

QA Input

Control

Weights / Generate

Amplitude

1

QA Input

Control

Weights / Generate

Frequency

10 M

QA Input

Control

Weights / Generate

Window Length

4096

QA Input

Control

Weights / Generate

Channel

1

QA Input

Control

Weights / Generate

Set

click

QA Setup

Integration

Mode

Standard

QA Setup

Integration

Length

4096

Upon clicking on Set in the QA Input tab, the sinusoid integration weights are displayed in the plot area of this tab as shown in the figure below. We denote these dimensionless weights as WI,q(t) and WQ,q(t), where q = 1,…,10 denotes the measurement channel, and I and Q correspond to the two signal inputs.

Figure 2. Integration weights as displayed in the Quantum Analyzer Input tab

We can check if the frequency of the sinusoids agree with the signal programmed in the AWG. The measurement is performed similarly as described in the previous tutorial. In the QA Input tab, set the Length to 4096 and the Averages to 64, and click on Run/Stop. Then, start the AWG by clicking on Run/Stop in the AWG tab. The peak amplitude of about 530 mV corresponds to the average between the high and low amplitude pulses generated by the AWG. We can see that the frequencies of the weights and the signal agree. An undesired delay between the signal and integration period can be compensated by using the Delay setting in the Deskew section of the QA Setup tab. Phase offsets can be compensated by directly changing the phase of the generated signal or the integration weights. Alternatively, the Rotation coefficients in the QA Setup tab allow you to rotate the signal in the complex plane after the integration. This can also be used to compensate a phase offset between signal and integration weight, but it doesn’t have an effect on the raw signal displayed in the Input Monitor.

Figure 3. Integration weights and signal as displayed in the Quantum Analyzer Input tab

In the Quantum Analyzer Input tab, apply the settings in the table below in order to conduct a measurement without averaging.

Table 4. Settings: configure the Result Logger
Tab Sub-tab Section # Label Setting / Value / State

QA Result

Control

Result Wave

Source

Rotation

QA Result

Control

Result Wave

Length

100

QA Result

Control

Result Wave

Averages

1

QA Result

Control

Result Wave

Run / Stop

ON

Click on "Start/Stop" in the AWG tab in order to run the AWG. The lines setTrigger(AWG_INTEGRATION_TRIGGER + AWG_INTEGRATION_ARM); and setTrigger(AWG_INTEGRATION_ARM); in the AWG sequence program generate a rising edge trigger to start the weighted integration. 100 consecutive dual-channel pulses are acquired, integrated, and displayed. The figure below shows the signal as displayed in the QA result tab.

Figure 4. Measurement result as displayed in the Quantum Analyzer Result tab. The numerical values depends on the phase relation between generation and demodulation, which in turn depends on the cable length and on the Delay setting in the QA Setup tab.

These values in the graph are the integrated in-phase voltages V'I,q after application of the Rotation coefficient of channel q. Since we left the Rotation coefficient at its default, which is 1+0i, the voltage V'I,q is equal to the voltage VI,q before rotation. This voltage VI,q, along with its quadrature counterpart VQ,q, is obtained as

$$$\tag{1} V_{I,q} = f_{s}\int_{0}^{T}W_{I,q}(t)V\prime_{1,q}(t)dt$$$
$$$\tag{2} V_{Q,q} = f_{s}\int_{0}^{T}W_{Q,q}(t)V\prime_{2,q}(t)dt$$$

where T is the length (in units of time) of the integration weights WI,q(t) and WQ,q(t), and V'1,q(t) and V'2,q(t) are the input signals after application of the deskew matrix accessible in the QA Setup tab, and fs is the input sampling rate 1.8 GSa/s. Since in our case we didn’t change the deskew matrix from its default which is the unit matrix, these two signals are identical to the raw input signals on Signal Inputs 1 and 2, V1,q(t) and V2,q(t).

Expressed in terms of discrete-time samples n, the formula above become

$$$\tag{3} V_{I,q} = f_{s}\sum_{n=0}^{N-1}W_{I,q}(t)V\prime_{1,q}(t)$$$
$$$\tag{4} V_{Q,q} = f_{s}\sum_{n=1}^{N-1}W_{Q,q}(t)V\prime_{2,q}(t)$$$

where N is equal to the length (in units of samples) of the integration weights.

 The integrity of the data vector of the Result Logger depends on this unit receiving a number of AWG Integration triggers equal to the product of the Averages and Length settings of the Result Logger. If this condition is violated, the data of the Result Logger may get corrupted. Often, the first point of the Result vector may then contain an outlier value. Notably, the Result Logger may be driven into an invalid state by generating an endless series of triggers, e.g. by enabling the AWG in Rerun mode.

In the given case, the integrated voltages in Figure 4 alternate between about 240 V and 480 V. This value lies in the range [– 4096×1.5 V, + 4096×1.5 V] determined by the maximum length of the integration window and the maximum input voltages. Note that the measured values depend on the phase relation between generated signal demodulation weights, which in turn depend on the cable length and on the Delay setting in the QA Setup tab.

In the next step, we can apply a voltage threshold in order to discriminate the analog signal. In practice, this allows us to convert the analog readout signal in a discrete qubit state readout result. In the Thresholds section of the QA Setup tab in line 1, enter a value of 360, which lies in between the high and low readout signals in the previous measurement. Change the Source signal in the QA Result tab to Threshold, click Reset, and run again the AWG. The following table summarizes these settings.

Table 5. Settings: apply a threshold
Tab Sub-tab Section # Label Setting / Value / State

QA Setup

Thresholds

1

Level

360

QA Result

Control

Result Wave

Source

Threshold

QA Result

Control

Result Wave

Reset

click

QA Result

Control

Result Wave

Run / Stop

ON

AWG

Control

Start / Stop

ON

Figure 5. Measurement result as measured in the Quantum Analyzer Result tab

The discretized readout results can be output with low latency as digital signal on the DIO connector on the back of the instrument. To this end, set the Digital I/O Mode to QA Result in the DIO tab. Please refer to the Digital Interface Specifications for a specification of the pin assignment.