SQ-CARS: A Scalable Quantum Control and Readout System

The Python based framework allows us to create an abstraction layer, which masks the underlying details of the hardware implementation and offers clean and user friendly configuration interface to the physics user community. The whole architecture of the system is divided between Programmable Logic (PL) and the Processing System (PS) as shown in Figure 1. ZCU111 is based on XCR28DR SoC, which has multicore ARM processor (PS) and Programmable FPGA Logic (PL). The PS runs Python productivity for Zynq (PYNQ) framework on Linux OS and can configure and control the PL design using the Overlays.

We utilized the available RFdc python code[23, 33] and made modifications to incorporate Multi-Tile Synchronization (MTS). Python classes were created for other hardware IPs on the PL to facilitate communication between the framework and hardware design modules. A hierarchical design approach is adopted to ensure a well-structured system which would be beneficial as the number of qubits and their corresponding IPs scale up.

Figure 2 illustrates the block diagram of the framework running on the PS and its logical connection with the SQ-CARS hardware design running on the PL. The communication between the PS and PL can be divided into two parts: the Configuration & Sync channel, and the data channel. Both channels are implemented using AXI interfaces available between the PS and PL. The Configuration & Sync channel employs lightweight communication using Memory Mapped Input Output (MMIO) PYNQ library for read/write operations. The data channel utilizes the AXI Direct Memory Access (DMA) driver to collect data from the PL. The modular framework can be easily adapted for other versions of the RFSoC board and can be initialized and used as a Jupyter notebook or a Python script. The Experiment Orchestrator (Exp Orch) shown in Figure 2 receives the experimental parameters and controls the overall flow of the experiment. The mainConfig class holds the system configuration, such as the bitstream file, frequency settings, amplitude settings, remote host and port information, trigger settings, and more. The rfdcConfig class is responsible for configuring the RF data converters. It creates handles for the DAC and ADC channels, assigns Block RAM (BRAM) addresses, and sets up the RF data converter for operation. This class also includes methods for initializing and running the Multi-Tile Synchronization (MTS) process. The DAC and Readout classes handle the Signal Generation (AWG) and Readout pipeline respectively. They set NCO frequencies and phases, configure the DMA, and manage the data streaming to a remote host. The Experiment Orchestrator initiates the experiment by loading the waveform into the corresponding DAC’s BRAM. Once the experiment is started, no further communication between the framework and PL is required, which reduces the experiment duty cycle.

Listing 1 shows snippet of the code used for taking experiment parameter input from the user. Snippet shown in Listing 2 shows the initialization process by providing the handle to user for modifying various parameters on the fly according to the requirement of the experiment.

The control side of superconducting quantum system involves generation of microwave pulses of various shapes and duration depending on the experiment at hand. DACs of this board are packaged into 2 tiles (Tile-0 and Tile-1), each tile containing 4 DACs, thus providing total 8 channels. Tile-0 Channels are used to generate control microwave pulses, while Tile-1 DACs are used to generate corresponding readout pulses. Tile-1 DACs can also be used independently to generate control pulses, extending the design’s capability to control upto 8-qubits.

Each of the DACs can be controlled independently by the Arbitrary Wave Generator (AWG) block which is shown in Figure 3. The parameters that control the shape, amplitude, duration etc. of the microwave signals are controlled by this block. It can generate arbitrary waveform by reading continuous samples from local memory of PL (BRAM). This allows for real-time play of the samples and provides control to the user for changing the shape and duration of the waveform. These samples are written from Python framework along with other experimental parameters like mode, loop and time delays to the BRAM. The available local memory in PL is $4.75$MB of BRAM. Each sample needs two bytes of storage. When all the eight DAC channels are used, a maximum storage of $311$K samples can be allotted for each channel. The current design supports $64$K samples per channel, which is equivalent to $80~{}\mu$s of playtime at the current FPGA clock frequency of $192$ MHz, which is a sufficient number for most experimental purposes. The signal generator has the capability to play the flat part or time between pulses of the waveform utilizing an internal counter, without actually storing the respective portions of it in BRAM. This effectively allows us to play pulses of longer duration more than 80 $\mu$s in experiments like Time Rabi and Ramsey experiment.

The mode allows to play continuous waveform or a fixed number of pulses depending on the experiment requirement, thus catering to a wide spectrum of quantum experiments. The loop parameters help to run an experiment multiple times repeatedly to collect as many measurements as required. The Controller also controls the generation of a trigger which facilitates for the timely capture on the readout side. The timing parameters decide the delay of the trigger and width of the arbitrary wave pulses which are fully programmable. The sample data read from BRAM is sent to Data Splitter and the amplitude of each sample is scaled by the Scaler block. These scaled samples are further merged and the interleaved $I$ and $Q$ samples are fed to the RF-DAC. The Channel Select allows us to select whether control or readout pulses be played through DACs.

The RF-DAC consists of a First In First Out (FIFO), an interpolation filter, a mixer, and a DAC as shown in Figure 1. The scaled sample values from the Arbitrary Wave Generator are fed into AXI Stream (AXIS) FIFO of the corresponding DAC channel. At higher sampling rates of Data Converters, the data streaming clock of the DAC cannot be pushed in the orders of their sampling rates. So a digital upsampling is necessary. This function on the DAC side is realized by the Interpolation Filters. The interpolation rate can be chosen among 1x, 2x, 4x, and 8x.

Algorithm 1 shows the pseudo code of the control and readout that suits the needs of the user in characterizing a superconducting qubit. This algorithm is implemented as the entirety of the hardware architecture.

Using this algorithm, basic characterizing experiments such as the measurement of energy relaxation time, dephasing, Rabi-oscillations in time-domain or power domain etc. can be configured easily. It takes arguments such as Ramsey, Time-Rabi, T₁, Power-Rabi, $\epsilon_{A}$, $\epsilon_{T}$, $\epsilon_{t}$, $Experiments$ and $N_{iter}$ as the inputs. $N_{iter}$ indicates the the number of RF pulses that needs to be sent to the dilution refrigerator. The parameters $\epsilon_{A}$, $\epsilon_{T}$, and $\epsilon_{t}$ represent the increments in power, trigger delay and time between the pulses respectively. Depending on the kind of experiment that needs to be performed, these parameters are initialised and updated accordingly as shown in Algorithm 1.

According to Nyquist criteria, the sampling rate limits the frequency that can be faithfully reconstructed to be less than half the sampling rate. However, in practice, when a signal is sampled, its images appear at higher frequencies. Each Band of the spectrum with width $\frac{F_{S}}{2}$ is termed as Nyquist zone (NZ). For example, the range from DC to $\frac{F_{S}}{2}$ is termed as first-Nyquist zone, $\frac{F_{S}}{2}$ to $F_{S}$ is called second-Nyquist zone and so on [34]. The images above the first-Nyquist zone can be utilized according to the need. The output voltage of DAC can be represented [35] as,

where $x\left(t\right)$ is the desired waveform whose samples are being generated, $r(t)$ is the reconstruction waveform, and $T=\frac{1}{F_{S}}$. Taking Fourier transform on both sides of the above equation, we get

where $V\left(\omega\right)$, $X\left(\omega\right)$ and $R\left(\omega\right)$ represent the Fourier transform of $v\left(t\right)$, $x\left(t\right)$ and $r\left(t\right)$, respectively. From equation 1 and 2, it becomes evident that the sampled signal is passed through a system having transfer function $R\left(\omega\right)$ and when a signal is sampled we get its copies in higher frequency ranges. Therefore, the output signal strength of the copies in different range of frequencies gets affected according to the response of $R\left(\omega\right)$.

The RFSoC supports two modes of operation: Normal mode or Non-Return-to-Zero (NRZ) mode and Mix mode or Return-to-Complement (RTC) mode which determines $R\left(\omega\right)$. In the NRZ mode of operation the DAC uses a fixed level reconstruction waveform during one clock cycle. It has high output power in the first Nyquist zone, but low output power in the second Nyquist and beyond. RTC mode or the mixed mode, it outputs the sample for the first half of the clock period and then inverts the sample for the second half of the clock period. The resulting frequency response shows high power in the second Nyquist zone and attenuation in the first Nyquist zone. The output power goes to zero at DC, and 2$F_{S}$. This mode provides highest power for the second Nyquist zone applications. Mathematically, the reconstruction waveforms can be written as,

The maximum sampling rate of DACs in RFSoC is $6.554$ GSPS. Since the frequencies necessary for the control side of qubit are typically in the range of $4-8$ GHz, the desired RF band for signal synthesis fall in the second Nyquist zone. It can be accessed by operating in mix mode while maintaining highest power. The major advantage of this approach is that RF signal is generated using onboard NCOs and a digital IQ-mixer. Therefore, it allows a full vector control of amplitude, frequency and phase on the fly. To suppress the images in other Nyquist zones and signal conditioning, we use standard coaxial band-pass filters. In principle, these filters can be incorporated on a custom daughter board. A schematic of the flow and different Nyquist zones are shown in Figure 4.

On the readout side, the signals from the dilution refrigerator are fed to RF-ADCs. The board has four tiles, each tile containing two ADCs. Among the eight ADCs in total, four ADCs are differential ended and four ADCs are single ended. The design supports four independent readout lines. The user has the flexibility to chose between a single-ended or differential ended ADC for each readout channel which is facilitated by Channel Select. It also allows us to lock ADC to any particular DAC channel for capturing the readout based on the internal trigger. The ADCs are designed to directly measure the signals in higher Nyquist zones. The zone of operation is indicated to the digital calibration engine of RFSoC to ensure optimal performance of the ADC. The input samples are provided by the parallel digital interface of the high-speed ADC. The incoming signal is down-mixed using the internal NCO of RF-ADC. Upon down-mixing and decimation, the signal is low-pass filtered (LPF) and smoothed using a moving-average (MA) filter. The NCO frequency, cut-off frequency of the LPF can be configured using the Python framework based on experiment. The in-phase $(I)$ and quadrature $(Q)$ samples received after the filter are fed to Rotation block which performs,

Such a rotation block, equivalent to the auto-alignment of phase in a lock-in measurement, is a helpful feature while performing qubit readout using single quadrature. The angle of rotation can be changed by the user using the Python framework. This estimation can be used to determine the qubit state or to provide an active feedback to the control side to further fine-tune the superconducting system.

The filter and rotation blocks can also be bypassed to capture the raw data. The resultant samples are fed into AXI Stream FIFO for transfer to PS using AXI-DMA block. The AXI Stream handshake signals like tvalid and tlast are generated using Readout Control Block, depending on the internal trigger received from Signal Generation Block. Stream to Memory Mapped (S2MM) port of AXI-DMA engine is connected to the PS via AXI Slave interface on ZYNQ SoC. The DMA engine is configured in Scatter Gather (SG) mode to facilitate use of cyclic buffer and eliminate need for PS to continuously provide descriptors to the DMA engine. To fetch the descriptors, a separate SG port of DMA is used. A custom wrapper is created in PYNQ to support DMA in Scatter Gather mode. Both descriptors and data buffers for DMA are kept in DDR memory attached to PS. Each readout channel is equipped with its own separate DMA channel, providing fine grained control on data capture on readout side. The readout data received on PS is transferred to remote PC using the PS GEM-3 Ethernet which is either saved in file or played using python scripts for initial startup of the experiment.

To control and measure multiple qubits, the output waveforms of multiple ADC and DAC channels requires to be synchronised both in time and phase. This can be achieved by an external 10 MHz clock which is fed to LMK04208 to synchronize the timebase of the FPGA board with other instruments and Multi Tile Synchronisation (MTS). The RF-ADC and RF-DAC constitute dual clock FIFOs. The data converters in a single tile share the clocking and data infrastructure, therefore the FIFO latency within a tile remains the same. However, the latency of the FIFOs can vary from one tile to another. MTS enables to achieve the relative multi-tile alignment. A single master clock generates all clock signals required for RF data converters and the programmable logic. A custom python wrapper for C-drivers is developed to configure and invoke MTS.

To measure the power dependency in NRZ and RTC modes, the DAC output response is being characterized by enabling the NCO whose frequency is changed on the fly and the corresponding magnitude is recorded. To compensate for the loss in magnitude, an inverse sinc filter is applied in both modes of operation. The recorded DAC output response is shown in Figure 8(a). The DAC output response in NRZ mode follows a sinc function. It is evident from the Figure 8 that the signal power is maximum in the second and third Nyquist zone when operating in the RTC mode.

The phase synchronisation between the control and readout channels is critical in a multi-qubit system. QICK[28] has reported achieving inter-channel phase synchronisation using the tprocessor. However, their approach does not consider the variable delay introduced by the RF-SoC pipeline, due to NCOs and FIFOs. To address this limitation, our work SQ-CARS, utilises MTS with on-board NCOs enabled, to generate and capture phase synchronised signals, demonstrating inter-channel synchronisation from generation by DACs to capture on ADCs.

To test the multi channel phase synchronisation on control side, we generate a $1$ MHz sinusoid by two DACs on ZCU111. These signals are being observed on an oscilloscope for jitter measurement. The channel to channel jitter of two DACs is measured by repeatedly sampling the jitter $4000$ times with a time interval of $100~{}\mu s$ . The histogram of jitter is shown in Figure 6. The standard deviation of channel to channel jitter is $\sim 0.6~{}ps$ which is better than $\sim 5~{}ps$ and $<1~{}ps$ as reported in Ref [29] and Ref [31], respectively. Low standard deviation of jitter shows stable channel synchronisation to support high-precision synchronized qubit operations.

In order to demonstrate the multi channel phase synchronised capture on readout side, a $1$ MHz sinusoid generated by DACs as earlier (which are already phase synchronised) is fed to ADCs. The resultant signal at the output of the ADC, the $I$ and $Q$ channels are captured. and found to be in phase synchronisation. Figure 7 shows the output of $I$-channels of two such ADCs.

Latency is an important metric for qubit readout, active feedback, and error-correction protocols. The RFSoC ADC and DAC pipelines have a series of modules of FIFO, Interpolation/Decimation filters, Mixers as shown in Figure 1. These modules can be used or bypassed as per the requirement. The latency changes as we bypass or utilise these modules. Latency measurement was done using loopback between DAC and ADC and observing the markers in Integrated Logic analyzer (ILA) running at $192$ MHz clock. The interpolation rate of DAC is 8x and decimation rate of ADC is set to 4x, the mixer is enabled and the DAC and ADC channels are synchronised using MTS. The round-trip latency including DAC, ADC and the digital AXI interfaces associated with the data converters is measured to be 48 cycles equalling $250$ ns with the design operating at $192$ MHz. This number is comparable to the latency measurement reported without MTS by Quantum Instrumentation Control Kit (QICK) [28]. Each half of the measured latency is contributed by RF-DAC and RF-ADC pipelines.

Another important benchmarking metric for continuous mode of operation is the single-sideband phase noise. A frequency tone (carrier) is generated, and at various offset frequencies from the carrier, the power is measured in a specified bandwidth of 1 Hz.

We use a signal analyzer (Rohde and Schwartz FSV-13) to perform the phase noise measurements. Figure 8(b) shows the measurement of SSB phase noise plotted for different offset frequencies around the carrier frequency of $4.5$ GHz generated from the XCZU28DR device. The measured phase noise of $-102$ dBc/Hz at $4.5$ GHz carrier frequency at offset 10 kHz is comparable to the phase noise performance of standard test and measurement RF Signal generators [36, 37, 38, 39, 40].

While generation of signal using multi-zone Nyquist technique expands the scope of frequency domain capabilities of a DAC, the images generated in other zones need to be carefully suppressed to achieve a practically useful spurious-free dynamic range (SFDR) [41]. Figure 8(c) shows the various spurs for a carrier tone of 4.5 GHz while using a standard bandpass-filter minicircuit VBFZ-3590-S+(15542). With this general purpose filter, we achieve a SFDR of nearly 45 dB. Our focus in this study has been on the generation of control signals for the superconducting qubits in the frequency range of 4-4.5 GHz.

It is important to mention here that this value is currently limited by the choice of the filter, and can be further improved by using tunable cavity or switchable filter banks [42, 43].

The ADCs are designed to directly measure the signals in higher Nyquist zones. The zone of operation is indicated to the digital calibration engine of RFSoC to ensure optimal performance of the ADC. A 6 GHz signal at -23 dBm is fed to ADC in mix-mode operating at a sampling rate of 4.096 GSPS. The resultant image would appear at $6000-4096=1904$ MHz. The frequency response of the signal captured by ADC is shown in Figure 9.

To measure the linearity of ADC response in the Mix-mode, a signal is fed with varying input power for frequencies between $6-7$ GHz and the resultant amplitude (dBFs) is recorded. The measured amplitude at the ADC shows a good linear behaviour with respect to the input power as shown in Figure 10.

The proposed architecture has been implemented using Vivado 2020.2 and the configuration parameters of the experiment like the name of the experiment, pulse duration, time between pulses, etc., are being passed using SQ-CARS Python framework running on PYNQ v2.7. Table I show a comparison of the proposed SQ-CARS with the state-of-the-art Quantum control platforms.

The major challenges in the design of an integrated control and readout system are the scalability, direct synthesis and capture of microwave signals, on-board data processing and an unified user interface. The existing approaches, such as QubiC [27], QICK [28], and Yang et al.[29], do not provide a comprehensive end-to-end solution for addressing all the challenges aforementioned. These approaches rely on discrete components and custom clock distribution modules at the baseband level, resulting in increased complexity and challenges in unified control. QICK does not make use of the on-board NCOs for generation and capture of microwave frequencies. These works use external analog mixers to up-convert the baseband signals. Their additional utilization of external analog mixers for RF conversion introduces problems such as leakage, periodic calibration etc. Although QICK achieves inter-channel phase synchronization using a custom state machine, it fails to address the variable delay caused by the RF-SoC pipeline due to NCOs and FIFOs and does not support MTS. ICARUS-Q[30] performs MTS but again like QICK, it does not utilize the on-board NCOs. This limits on-the-fly synchronized control of frequency and phase across different channels and leaves the variable delay caused by the RFSoC pipeline unaddressed.

To overcome the limitations of the previous works, our work, SQ-CARS combines MTS with on-board NCOs to directly synthesize, control and capture phase-synchronized signals, demonstrating inter-channel synchronization among all data converters. We also developed a Python-based framework, by creating an abstraction layer that conceals the complexities of hardware implementation. It offers a user-friendly configuration interface to the physics user community, simplifying their interaction with the underlying hardware. Leveraging the existing RFdc Python code, we enhanced it by incorporating Multi-Tile Synchronization (MTS) functionality which enables on-the-fly synchronized control of frequency and phase of different channels. In addition, we developed Python classes for other hardware IPs within the programmable logic (PL), enabling seamless communication between the framework and hardware design modules. This streamlined approach enhances flexibility and ease of use for researchers and engineers working with the framework. To enable low-latency active feedback, on-board processing of acquired readout signals is essential. SQ-CARS incorporates on-board tunable filtering, rotation, and averaging blocks, which are absent in ICARUS-Q, Yang et al., and QubiC.

In Table II, we compare the resource utilization of SQ-CARS with existing platforms. Except ICARUS-Q, no other work has reported resource utilization. Hence, it is written ‘Not Reported’ indicating the absence of the report of the corresponding parameter in the literature. SQ-CARS achieves waveform playtime of 80us, a crucial experimental parameter, by leveraging on-board NCOs. This is significantly higher than ICARUS-Q, which only offers a playtime of 10us while utilizing similar BRAM resources per channel. Another main contribution lies in the on-board information processing pipeline employed by SQ-CARS, which is absent in ICARUS-Q. SQ-CARS consumes 46.38% of available LUTs and 42.96% of available BRAM resources which leaves ample room for expanding the system’s capacity to accommodate a larger number of qubits, ensuring scalability.

The Power consumption of SQ-CARS has been generated using Vivado 2020.2 enabling the power optimisation feature. SQ-CARS consumes consumes 1.456 W of Static power and 20.353 W of Dynamic power, out of which only 15% is due to Logic. The rest of the power is majorly due to hard IPs like RF-DACs, RF-ADCs etc. These power numbers are unreported by any of the previous works.

In conclusion, SQ-CARS offers a comprehensive end-to-end solution, with the set of features including direct microwave synthesis and capture, long-term inter-channel phase synchronization, a scalable and flexible architecture, a digital pipeline for information processing, and a user-friendly unified interface, all on a single platform. These features, except for those found in Presto [31], enable SQ-CARS stand apart from the previous approaches.

To generate the microwave control pulses for the qubit, we use RFSoC DAC. As the qubit frequencies fall in the second-Nyquist zone, we use a wide-band Pasternack (PE 15A1002) amplifier to boost the signal amplitude to overcome the attenuation before sending it into the fridge. The baseband readout signal was generated using UHF lock-in amplifier from Zurich Instruments with AWG option. We used a home-built frequency up-converter and down-converter setup to generate the microwave readout pulse and to demodulate the readout signal coming from the fridge. We then determine the in-phase and quadrature components of the readout pulse and the subsequent qubit state determination. To address the issue of cross-platform triggering, we generated a trigger from ZCU111 analog output channel and used it to arm the sequencer on UHF-AWG. This straightforward handshaking allows us to correct any trigger latency by advancing the trigger generation on ZCU111. Moreover, it allows defining simple “loops” on ZCU111 so that the entire experiment can be controlled from the board. It is important to mention that, in principle, the fast ADC channels available on ZCU111 can be used for qubit readout and the entire control and readout part can be done on the same board.

It involves measuring the excited state qubit population while varying the amplitude/length of the Gaussian pulse resonant with the qubit frequency, the measurement colloquially known as power/time Rabi measurement. We implemented this experiment on device D1, by inserting a rectangular pulse with a variable number of samples at the center of the $X_{\pi}$ Gaussian pulse. We use 260 ns long Gaussian pulses for these measurements with a standard deviation of 65 ns. The duration of the qubit control pulse can be adjusted by adjusting the number of sample points in the rectangular pulse. The top part of Fig. 12(a) panel shows the pulse sequence. Any trigger latency between ZCU11 and the readout setup (UHF) is adjusted by adjusting the trigger delay. The bottom panel shows the coherent time oscillations as the duration of the rectangular portion of the control pulse is increased. The measurement of Rabi oscillations in time and power domain allows to calibrate the microwave pulses necessary for qubit operation.

After getting the calibrated $X_{\pi}$ from the above measurement, it is straightforward to carry out energy relaxation time measurement. The setup is same as the Rabi oscillation measurement setup with a minor change in the pulse sequence. Here we first send a calibrated $X_{\pi}$ pulse and then wait for time $t$, before performing a readout as shown in Fig. 12(b). Due to on-the-fly configuration of the delay time, the entire measurement can be controlled by ZCU111 unit. For a given wait time, we generate 50,000 $X_{\pi}$ pulses at a repetition rate of 5 kHz and determine the excited state population. Fig. 12(c) shows the result from $T_{1}$ measurements of D1 device. The value of $T_{1}$ was extracted by fitting the following equation $A+Be^{-t/T_{1}}$. The results from the $T_{1}$ experiment along with the fits, yielded a $T_{1}$ of approximately 30.5 $\mu$s.

Next, we carry out the Ramsey experiment, a measurement of dephasing rate. The protocol requires the generation of two $X_{\pi/2}$ pulses with a variable time-delay between them, and it is followed by the qubit measurement. A schematic of the pulse sequence and result from the Ramsey fringe experiment is shown in Fig. 12(c). As the experiment can be completely controlled by the ZCU111, the procedure remains similar to the $T_{1}$ measurements. Here the trigger for the readout unit is aligned with the end of the second $X_{\pi/2}$ pulse and a delay between the two $X_{\pi/2}$ pulses is varied. By fitting the measured curve of the Ramsey fringes using $A+B\cos(2\pi\Delta t+\phi)e^{-t/T_{2}^{R}}$, one can obtain the values for $T_{2}^{R}$ and detuning $\Delta$, respectively. In this context, the detuning represents the deviation of the microwave pulse frequency from the qubit frequency. For the device D1, we measured $T_{2}^{R}$ of 6.25 $\mu$s and detuning of 0.8 MHz. It is important to point out here that since both the $X_{\pi/2}$ pulses are generated by the internal digital IQ-mixer and common NCO, a well-defined phase relation holds between the two pulses. Such functionality makes it very easy to implement where X- and Y- qubit rotations can be made by controlling the signals on I and Q inputs of the digital mixer.

To compare the performance of RFSoC and standard test and measurement instruments, we compare the results of qubit characterization, $T_{1}$ and $T_{2}^{R}$, on the same device during the same cooldown run. The conventional measurements were carried out by generating the drive pulses using the single-sideband modulation technique and a vector signal generator. The baseband drive signals were generated using a commercially available AWG. In the Fig. 13, we show the comparative results from another device (D2) by measuring it using general test and measurement equipment and RFSOC. We do not observe any statistically significant difference between the results. Given the ease of the measurements, as no periodic calibration for sideband suppression is necessary, we also repeated the $T_{1}$ measurements of device D1 over 24 hours and recorded statistical variations in the $T_{1}$ measurements. Fig. 12 (d) shows a histogram of 47 $T_{1}$ measurements from the same device.