Advancing Quantum Temperature Sensors for Ultra-Precise Measurements (UPMs): A Comparative Study

Abstract

In this study, we compared the performance of quantum temperature sensors (QTSs) with conventional sensors (CSs), highlighting differences in measurement accuracy and stability. Quantum sensors (QSs), known for their ability to provide ultra-precise measurements (UPMs), were tested across a temperature range of −10 to 40 °C. The results indicate that QSs offer superior accuracy, with a lower average error and a smaller standard deviation compared to CSs, indicating better measurement stability. For this comparison, we utilized Python scripts to conduct simulations and statistical analyses, leading to precise and reproducible results. The sensor performance was simulated in a controlled environment, and the obtained data were compared with experimental results. This comparison reveals that QSs are more reliable for applications requiring high precision, such as those in the Internet of Things (IoT) domain. These findings underscore the potential advantage of QSs in critical systems where measurement accuracy is paramount.

1. Introduction

Accurate temperature measurement is essential across various sectors, from industry to medicine, and scientific research [1,2,3,4]. Temperature sensors, which have evolved from traditional thermometers to modern digital devices [5,6,7,8], play a crucial role in environments, where even slight variations can have significant consequences [9]. Conventional sensors (CSs), such as thermocouples and resistance temperature detector (RTD), have long been used but have limitations in terms of precision and stability, especially in critical applications like industrial control systems or medical devices. The search for more advanced technologies has led to the emergence of quantum sensors (QSs), which leverage quantum physics principles to provide unparalleled temperature measurement accuracy [10,11].

Quantum temperature sensors (QTSs), which utilize quantum phenomena such as the Josephson effect in superconducting circuits [12], are capable of detecting temperature variations with extremely fine sensitivity, often at the nanometer scale. These sensors overcome many limitations of conventional technologies by offering better accuracy and faster response. In a context where modern applications, such as the Internet of Things (IoT) [13,14,15,16], require increased precision to optimize performance and ensure system reliability, QSs appear as a promising solution [17,18]. This study aims to directly compare the performance of QSs with that of traditional sensors, highlighting differences in terms of precision, stability, and dynamic response [19,20,21,22,23]. To conduct this comparison, laboratory experiments and numerical simulations were performed. Both QS and CS were tested over a range of temperatures covering realistic conditions for IoT applications. Experimental results were analyzed using sophisticated mathematical models and statistical techniques to ensure the accuracy and reproducibility of observations.

This study makes a significant contribution by providing a thorough evaluation of the advantages and challenges of QS compared to existing technologies, and exploring their potential in critical systems where precision is essential [24,25,26]. The results of this research could guide the future development of sensor technologies and their integration into complex systems. Temperature measurement technologies have evolved significantly, with CSs such as thermocouples, resistance temperature detectors (RTDs), and thermistors being widely used due to their reliability and established performance [27,28]. Thermocouples, utilizing the Seebeck effect, are valued for their robustness and broad temperature range, though they exhibit limitations in precision at lower temperatures. RTDs, primarily made from platinum, offer high accuracy and stability through resistance measurement but face challenges related to mechanical stress and electromagnetic interference. Thermistors, known for their high sensitivity and precision within specific temperature ranges, are limited by their non-linear response and operational range [29].

In recent years, QTSs have emerged, offering advanced capabilities that surpass those of conventional technologies. Superconducting sensors, which exploit the Josephson effect and superconducting quantum interference devices (SQUIDs) [30,31], have demonstrated exceptional sensitivity and precision, reaching sub-millikelvin accuracy. However, their complexity and operational requirements at cryogenic temperatures pose challenges. Optomechanical sensors, leveraging quantum-level interactions between light and mechanical vibrations, provide high-resolution temperature measurements but face difficulties related to integrating optical and mechanical components and sensitivity to environmental noise.

Additionally, quantum dot thermometers, which utilize semiconductor nanocrystals for temperature-dependent photoluminescence, have shown promise for high-resolution measurements in various applications, though issues like quantum dot stability and excitation sensitivity need further exploration [32]. Comparative studies of QSs and CSs reveal a significant shift towards quantum technologies for applications requiring extreme accuracy [33,34].

This analysis would broaden the context of our work and provide a more detailed understanding of current developments in the field. Further integration of key references into the main argument of the manuscript would not only solidify the theoretical basis but also highlight the originality of our approach. In particular, it is important to include recent developments, such as those discussed in the article published in [35], to enrich this perspective.

This article explores how advancements in 6G networks and quantum technologies can combine to create a more robust quantum communication infrastructure. This reference is crucial for understanding the current implications of quantum technologies on modern networks. Additionally, incorporating the article [36] will provide context for recent innovations in QSs and their use in the biomedical field. This knowledge will enrich the discussion on the current applications of QSs. Finally, the article [37] offers an overview of the challenges and developments in fog computing for IoT. This reference will illustrate how fog computing approaches can interact with emerging quantum technologies, further enhancing the discussion on the potential applications of QSs.

Research comparing these sensor types has highlighted QS superior precision and sensitivity, although they come with higher complexity and cost. For instance, studies have demonstrated that while QSs outperform conventional ones in terms of measurement accuracy and response time, their practical implementation involves overcoming technical and financial challenges. Overall, the integration of quantum technologies represents a significant advancement in temperature sensing, promising enhanced performance for applications demanding the highest precision [38].

Our contributions are (1) enhanced measurement precision with QSs, (2) validation and reproducibility through simulations and statistical analysis, and (3) implications for IoT applications and critical systems. This paper introduces new QTSs for UPM.

The remainder of the paper is organized as follows. In Section 2, we present the innovative methodology for QTSs. In Section 3, we present the experimental comparison of quantum vs. CTSs. Finally, Section 4 concludes the paper with some remarks.

2. Innovative Methodology for QTS

The diagram illustrated in Figure 1 presents several steps, each represented by a colored rectangle, highlighting key phases such as quantum temperature sensor simulation, CS simulation, their comparison, and the simulation of measurements for the IoT. Each step is strategically positioned within the diagram, with distinct colors used to enhance visual differentiation. The connections between these steps are depicted by large curved arrows, sequentially linking the rectangles to represent the methodological flow.

This figure illustrates a flowchart outlining the methodology used to compare the performance of quantum and conventional temperature sensors, as well as their integration into an IoT system. Each key step in the simulation process is represented by a colored rectangular box, making the flow easy to follow.

The diagram highlights four main stages: the simulation of quantum temperature sensors (blue box), the simulation of conventional sensors (green box), the comparison of both types of sensors (coral box), and finally, the simulation of measurements in an IoT environment (salmon box). Arrows connect the boxes, indicating the flow of information between each step.

The arrows represent logical transitions between the stages, enhancing the visual clarity of the process. While the figure does not include specific data or units, it focuses on conveying the core concepts of the methodology.

The primary goal of this figure is to demonstrate the sequence of simulation and comparison steps for the sensors, emphasizing a structured approach to analyze their respective performance. It also illustrates how the integration of the sensors into an IoT system is addressed in the study and how each simulation contributes to the final comparison.

Comparing QTSs with CTSs is critical for assessing their accuracy, reliability, and technological relevance. These sensors are particularly vital in fields where precision is of the utmost importance, such as in medical diagnostics, aerospace, and high-stakes scientific research. By evaluating QTSs, we can identify their potential advantages, especially in terms of performance and efficiency in applications like the IoT.

Furthermore, this comparison aids in optimizing costs, establishing new industry standards, and driving technological innovation across various sectors, ultimately guiding the future adoption and integration of these advanced sensing technologies.

2.1. Mathematical Model of QTSs

The following section provides a detailed presentation of the QTS mathematical models.

• Superconducting Qubit Sensors (Hamiltonian of a Superconducting Cubit):

$$\widehat{H}=-{\displaystyle \frac{¯h\Delta }{2}}{\widehat{\sigma }}_x-{\displaystyle \frac{¯hϵ}{2}}{\widehat{\sigma }}_z+{\displaystyle \frac{¯h{\omega }_q}{2}}{\widehat{\sigma }}_z$$

(1)

where $\Delta$ is the tunneling amplitude between the two states of the Josephson junction, $ϵ$ is the energy imbalance between these states, ${\omega }_q$ is the transition frequency of the qubit, and ${\widehat{\sigma }}_x$ and ${\widehat{\sigma }}_z$ are the Pauli matrices. This model describes the time evolution of the superconducting qubit under temperature variations. The frequency–temperature relationship (FTR) is as follows:

$$\omega \left(T\right)={\omega }_0\exp \left(-{\displaystyle \frac{\Delta E}{k_{B}T}}\right)$$

(2)

where $\omega \left(T\right)$ is the transition frequency of the qubit at temperature T, ${\omega }_0$ is the reference frequency at low temperature, $\Delta E$ is the energy difference between the qubit states, and $k_B$ is the Boltzmann constant. This model expresses the dependence of the transition frequency on temperature, which is crucial for accurate temperature measurement using a superconducting qubit.

2.

Aharonov–Bohm Effect Sensors (Phase of the Aharonov–Bohm Effect): Qubit Sensors (Hamiltonian of a Superconducting Qubit):

$$\Delta \varphi ={\displaystyle \frac{2\pi e}{h}}\oint \mathbf{A}·d\mathbf{l}$$

(3)

where $\Delta \varphi$ is the phase difference due to the Aharonov–Bohm effect, e is the electron charge, h is Planck’s constant, and $\mathbf{A}$ is the electromagnetic vector potential along the path $d\mathbf{l}$. $\mathbf{A}$ influences the phase shift observed in the Aharonov–Bohm effect by contributing to the integral that determines how the vector potential alters the quantum phase of electrons.

This effect is pivotal in understanding how such sensors can be used to measure physical quantities based on phase changes. This model shows how an applied magnetic field affects the electron wave phase, which in turn affects the sensor’s response to temperature changes. The phase–temperature relationship (PTR) is as follows:

$$\Delta \varphi \left(T\right)=\Delta {\varphi }_0\left(1+{\displaystyle \frac{\alpha T}{T_0}}\right)$$

(4)

where $\Delta \varphi \left(T\right)$ is the phase difference at temperature T, $\Delta {\varphi }_0$ is the phase difference at a reference temperature $T_0$, and $\alpha$ is a thermal coefficient. This model calculates the phase variation induced by temperature in devices utilizing the Aharonov–Bohm effect.

3.

Quantum Optomechanical Sensors (Hamiltonian of an Optomechanical Cavity):

$$\widehat{H}=¯h{\omega }_c{\widehat{a}}^{†}\widehat{a}+¯h{\omega }_m{\widehat{b}}^{†}\widehat{b}-¯h{g}_0{\widehat{a}}^{†}\widehat{a}({\widehat{b}}^{†}+\widehat{b})$$

(5)

where ${\omega }_c$ is the optical cavity frequency, ${\widehat{a}}^{†}$ and $\widehat{a}$ are the photon creation and annihilation operators, ${\omega }_m$ is the mechanical resonance frequency, ${\widehat{b}}^{†}$ and $\widehat{b}$ are the phonon creation and annihilation operators, and $g_0$ is the optomechanical coupling. This model describes the interaction between photons and mechanical vibrations at the quantum level, which can be affected by temperature.

The Optomechanical System Partition Function (OSPF) is given as follows:

$$Z=\mathrm{Tr}\left(e^{-\beta \widehat{H}}\right)$$

(6)

where Z is the partition function, $\beta ={\displaystyle \frac{1}{k_{B}T}}$ is the inverse temperature multiplied by the Boltzmann constant, and $\widehat{H}$ is the Hamiltonian of the system. This model allows the calculation of thermodynamic properties of the optomechanical system, such as free energy and entropy, which depend on temperature.

4.

NV Center-Based Temperature Sensors (Hamiltonian of an NV Center):

$${\widehat{H}}_{NV}=D{\widehat{S}}_z^2+{\gamma }_e\mathbf{B}·\widehat{\mathbf{S}}+A{\widehat{S}}_z{\widehat{I}}_z$$

(7)

where D is the zero-field splitting term, ${\gamma }_e$ is the electron gyromagnetic ratio, $\mathbf{B}$ is the applied magnetic field, $\widehat{\mathbf{S}}$ is the electron spin operator, A is the hyperfine coupling, and ${\widehat{I}}_z$ is the nuclear spin operator. $\mathbf{B}$ in the Hamiltonian accounts for the interaction between the electron spin of the NV center and the applied magnetic field. This interaction influences the energy levels of the NV center and contributes to the sensor’s ability to detect changes in temperature through variations in spin resonance. The temperature dependence of D allows temperature measurement through changes in the spin resonance. Zero-Field Splitting Parameter as a Function of Temperature (ZFSPFT) is as follows:

$$D\left(T\right)=D_0-\kappa T$$

(8)

where $D\left(T\right)$ is the zero-field splitting parameter at temperature T, $D_0$ is the value at a reference temperature, and $\kappa$ is a material-specific coefficient. This model quantifies the temperature sensitivity of NV centers in QSs based on spin. These advanced mathematical models capture the complex effects underlying the operation of QTSs, allowing for an in-depth understanding of their behavior as a function of temperature and enabling more sophisticated and accurate analyses in the field of quantum temperature sensing.

2.2. Mathematical Model of CTSs

For CTSs, such as thermocouples, resistance temperature detectors (RTDs) and thermistors, more complex mathematical models can be used to account for non-linearities, material properties, and environmental effects. Here are the most popular mathematical models for these sensors [39,40,41,42,43,44]:

• Thermocouples (Non-Linear Thermoelectric Voltage Equation):

Thermocouples generate a thermoelectric voltage that is a non-linear function of the temperature difference between the hot and cold junctions:

$$V\left(T\right)=\sum _{n=0}^N{a}_n{T}^n$$

(9)

where $V\left(T\right)$ is the thermoelectric voltage as a function of temperature T, and $a_n$ are coefficients determined through calibration for specific thermocouple types. N is the degree of the polynomial, typically ranging from 8 to 10 for high accuracy. The Seebeck Coefficient as a Function of Temperature (SCFT), which varies with temperature, can be modeled as

$$S\left(T\right)={\displaystyle \frac{dV\left(T\right)}{dT}}=\sum _{n=1}^{N}n·a_n{T}^{n-1}$$

(10)

This model provides a more accurate representation of the thermoelectric properties, particularly over a wide temperature range. The Temperature Measurement with Cold Junction Compensation (TMCJC) is given as follows:

$$T_{measured}={\displaystyle \frac{V_{measured}-V_{cold}}{S\left(T\right)}}$$

(11)

where $V_{measured}$ is the thermoelectric voltage at the hot junction, $V_{cold}$ is the voltage at the cold junction, and $S\left(T\right)$ is the Seebeck coefficient at the measured temperature. This model accounts for the temperature of the cold junction, which is critical for accurate measurements.

2.

Resistance Temperature Detectors (Callendar–Van Dusen Equation):

The resistance $R\left(T\right)$ of an RTD as a function of temperature T can be expressed by the Callendar–Van Dusen equation, which is widely used for platinum RTDs:

$$R\left(T\right)=R_0\left(1+A·T+B·T^2+C·(T-100)T^3\right)$$

(12)

where $R_0$ is the resistance at 0 °C, and A, B, and C are material-specific constants that depend on the RTD type. For temperatures below 0 °C, the C term is typically set to zero. A more complex model that accounts for the non-linearity at higher temperatures is

$$R\left(T\right)=R_0\left(1+\alpha T+\beta T^2+\gamma T^3+\delta T^4+ϵT^5\right)$$

(13)

where $\alpha$, $\beta$, $\gamma$, $\delta$, and $ϵ$ are coefficients obtained through polynomial fitting. This model improves accuracy for applications where precise temperature measurements are required over a broad range. The self-heating effect (SHE) is as follows:

$$\Delta T_{self}=I^{2}R\left(T\right)·H$$

(14)

where $\Delta T_{self}$ is the temperature rise due to self-heating, I is the current through the RTD, $R\left(T\right)$ is the resistance, and H is the heat dissipation constant of the RTD.

This model accounts for errors introduced by the self-heating effect, which becomes significant at high currents or in poorly ventilated environments.

3.

Thermistors (Steinhart–Hart Equation):

The Steinhart–Hart equation provides a highly accurate model for the resistance $R\left(T\right)$ of a thermistor as a function of temperature T:

$${\displaystyle \frac{1}{T}}=A+B\ln \left(R\right)+C{\left(\ln \left(R\right)\right)}^3$$

(15)

where A, B, and C are material-specific coefficients that must be determined experimentally.

This equation is particularly useful for achieving high accuracy over a broad temperature range. The Extended Steinhart–Hart Equation (ESHE) is as follows:

An extension to improve the accuracy further includes additional terms:

$${\displaystyle \frac{1}{T}}=A+B\ln \left(R\right)+C{\left(\ln \left(R\right)\right)}^3+D{\left(\ln \left(R\right)\right)}^5$$

(16)

where D is an additional coefficient that refines the model’s accuracy at extreme temperatures. The Power Dissipation Model (PDM) is as follows:

$$P\left(T\right)=I^2·R\left(T\right)$$

(17)

where $P\left(T\right)$ is the power dissipated by the thermistor, and I is the current flowing through it. This model is crucial for understanding how self-heating affects the accuracy of temperature readings, particularly in low-temperature applications, where the thermistor’s resistance is high.

4.

Environmental Effects on CS (Parasitic Resistance and Inductance Model):

$$R_{parasitic}=R_w+R_{contact}$$

(18)

where $R_{parasitic}$ is the total parasitic resistance, $R_w$ is the wire resistance, and $R_{contact}$ is the contact resistance. This model is important for accurate RTD and thermistor measurements, especially in low-resistance sensors, where parasitic effects can introduce significant errors. The Electromagnetic Interference (EMI) Model is given as follows:

$$V_{EMI}\left(t\right)=V_0\sin (\omega t+\varphi )$$

(19)

where $V_{EMI}\left(t\right)$ is the voltage induced by electromagnetic interference, $V_0$ is the amplitude, $\omega$ is the angular frequency, and $\varphi$ is the phase angle. This model helps to understand how external electromagnetic fields can affect the accuracy of temperature sensors, particularly in industrial environments.

These more complex mathematical models capture the non-linearities, material-specific behaviors, and environmental factors affecting CTSs. They are essential for applications requiring high precision and reliability, providing a deeper understanding and enabling more accurate temperature measurement and compensation techniques.

3. Experimental Comparison of QTSs vs. CTSs

The experimental setup for evaluating quantum and conventional temperature sensors involved a meticulously controlled environment and rigorous calibration processes. The testing was conducted within a sophisticated temperature-controlled chamber that maintained precise temperatures from −10 to 40 °C. Both quantum temperature sensors (QTSs) and conventional temperature sensors (CTSs) were calibrated against a high-precision platinum resistance thermometer (PRT) to ensure accuracy. The calibration process included adjusting sensor outputs to match PRT readings at multiple temperatures and verifying consistency. During the experiment, the sensors were tested in a stable, controlled laboratory setting with minimal external disturbances and constant humidity. Each temperature setting was maintained for at least 30 min to achieve thermal equilibrium, with data collected at regular intervals and analyzed for accuracy.

Additionally, tests simulated real-world conditions by introducing temperature gradients and rapid changes to reflect practical IoT scenarios. Data collection was synchronized using a data acquisition system to ensure simultaneous readings, and comprehensive documentation of calibration procedures, experimental conditions, and data analysis supported the validity and reliability of the results.

To compare the performance of QTSs and CTSs, a controlled experimental set-up is needed to assess their accuracy, stability and response times over a temperature range from −10 to 40 °C.

The experiment involves placing both types of sensors, together with a high-precision reference temperature sensor, in a temperature-controlled chamber. The data acquisition system (DAQ) simultaneously records data from the QTS, CTS, and reference sensor as the temperature is incrementally varied through extreme values. This approach ensures that the sensors are tested under identical conditions, providing a reliable basis for comparison as shown in Figure 2.

The figure illustrates the experimental setup used to compare the performance of two types of temperature sensors: a quantum temperature sensor (QTS) and a conventional temperature sensor (CTS). The objective of this experiment is to evaluate the accuracy and sensitivity of each sensor in a controlled environment. The image presents several key pieces of equipment arranged in rectangular boxes, with labels indicating their respective roles. The “Temperature-Controlled Chamber” is positioned at the top, representing the environment where the measurements are taken. The sensors—QTS, CTS, and a reference Platinum Resistance Temperature Detector (RTD)—are placed at critical points in the experiment. Each of these sensors is connected to the “Data Acquisition System” (DAQ), which centralizes the measurements before transmitting them to a computer for processing and analysis. Arrows connect the different components, illustrating the flow of data collected throughout the system.

Key abbreviations include QTS for quantum temperature sensor, CTS for conventional temperature sensor, RTD for resistance temperature detector (reference probe), and DAQ for data acquisition system. The temperatures measured are expressed in degrees Celsius (°C). The arrows show the data flow, starting from the sensors to the DAQ, and from there to the computer.

The purpose of this figure is to clearly demonstrate how the various components of the experiment are interconnected to enable efficient data collection. This allows for a comparison of sensor performance under identical conditions, ensuring the validity and accuracy of the obtained results.

The data analysis phase involves calculating the margins of error of the QTS and CTS in relation to the reference sensor, assessing their stability during temperature cycles, and evaluating their response times during rapid temperature changes. Visualization techniques, such as plotting temperature readings and error margins, will help illustrate differences in performance.

Statistical analysis will further validate these differences, enabling a comprehensive assessment of each sensor’s suitability for precision-demanding applications, particularly in the context of the IoT.

Here are the specific criteria used to select the high-precision temperature sensor.

1.

Clarification of Simulation Process and Parameters in Sensor Performance Study

In our study, which compares the performance of quantum temperature sensors (QTSs) with conventional sensors (CSs) across a temperature range of −10 to 40 °C, we now include specifics about the simulation process. For quantum sensors, we used a resolution of 0.01 °C and a noise model based on environmental variability and sensor calibration data.

For conventional sensors, the resolution was 0.1 °C, with a noise model derived from previous studies on sensor inaccuracies and environmental instability. Simulations were conducted using Python scripts in a controlled environment to ensure precision and reproducibility, and the data obtained were compared with experimental results.

This enhanced detail clarifies our methodology and supports our findings of superior accuracy and stability for quantum sensors.

2.

Criteria for Selecting the High-Precision Reference Temperature Sensor

The selection of the high-precision reference temperature sensor is based on several specific criteria, including accuracy, stability, and repeatability. Key criteria include high resolution, minimal drift, and sensitivity to temperature variations. This sensor is typically calibrated using high-precision standards to ensure its measurements are reliable. Compared to the quantum and CSs tested, the reference sensor offers superior accuracy, with negligible or very low measurement errors. While QSs, despite their advanced technology, may exhibit variations or uncertainties under specific conditions, CSs may suffer from errors due to noise and inherent precision limitations. The use of the reference sensor allows for measuring these discrepancies and validating the performance of the other sensors.

3.

Data Acquisition System (DAQ) Synchronization and Precision

The data acquisition system (DAQ) plays a crucial role in ensuring synchronization and precision when recording data from quantum, conventional, and reference sensors simultaneously. To achieve this synchronization, the DAQ uses internal or external clocks to coordinate data sampling at regular intervals. This alignment of data from each sensor in real-time minimizes time shifts and inconsistencies. Additionally, the DAQ is designed to handle different types of signals from the sensors, ensuring the accurate conversion of analog signals into digital values. The use of filters and additional calibration techniques within the DAQ also ensures that measurements are free from interference and systematic errors, enabling reliable performance assessment of the sensors.

4.

Measures to Ensure Uniform Temperature Distribution

In the temperature-controlled chamber, several measures are taken to ensure uniform temperature distribution and prevent localized temperature variations that could affect the sensor readings. Firstly, the chamber’s temperature control system is equipped with multiple internal sensors to monitor and regulate temperature at various points. Fans or air circulation systems are often used to evenly distribute heat and reduce temperature gradients. Additionally, the chamber is designed to minimize external heat sources and air currents that could introduce local variations. Prior to measurements, the chamber is allowed to equilibrate for a sufficient period to stabilize the temperature. These practices ensure that the data collected are representative of a uniform temperature, thereby ensuring the accuracy of comparisons between sensors.

We have developed a Python code capable of simulating and comparing temperature measurements from QS and CS. It generates synthetic data to model the behavior of each sensor type. The simulation includes several key elements: first, a graph showing temperature readings from a simulated quantum sensor; second, another graph showing measurements from a CS with noise added to reproduce real-world inaccuracies. A comparison graph illustrates the differences between the two sensor types, while an error analysis highlights the discrepancies between quantum and conventional measurements. Finally, an IoT sensor simulation graph shows real-time data collection by a quantum IoT sensor, highlighting its performance in a live scenario.

3.1. Comprehensive Analysis of QS vs. CS Performance

To provide a comprehensive comparison between quantum and CSs, we will conduct a simulation of their performance across a range of temperatures. The measurements for the quantum sensor are simulated to reflect high precision and stability, while the CS incorporates noise to simulate real-world inaccuracies.

The temperature range of −10 to 40 °C is deliberately chosen after careful consideration of the typical environmental conditions where these sensors are most commonly deployed. This specific range is not arbitrary; rather, it is representative of a significant portion of real-world scenarios that the sensors are likely to encounter.

In temperate regions, which constitute a large part of the world, both indoor and outdoor environments often experience temperatures within this range.

For instance, residential, commercial, and industrial settings in these areas routinely face temperatures from slightly below freezing in the winter to moderately warm in the summer. The chosen range also includes conditions found in many controlled environments, such as laboratories and data centers, where maintaining stable temperatures is critical.

Moreover, in several industrial applications, the −10 to 40 °C range is highly relevant. Industries such as food processing, pharmaceuticals, and manufacturing often operate within these temperature limits to ensure safety, product integrity, and optimal equipment performance. In these settings, sensors must reliably measure and monitor temperature to maintain quality control and ensure compliance with industry standards.

Thus, this range is selected to ensure that the study reflects the most common and practical conditions under which the sensors will be used, providing meaningful and applicable insights for a broad range of users.

Figure 3 visualizes these comparisons through multiple graphs (a, b, c, and d), highlighting the performance differences and potential errors introduced by the CS.

Subplot (a) Quantum Sensor Measurements: This subplot illustrates the QS performance across a range of temperatures. The data show a clear, linear relationship between the temperature and the sensor output, reflecting the QS high precision and lack of noise. Such reliable data are crucial in applications requiring exact temperature measurements.

Subplot (b) Conventional Sensor Measurements: In contrast, the CS measurements in this subplot display fluctuations due to noise. This noise, simulated by adding random variations to the temperature readings, reflects potential inaccuracies that could arise in real-world conditions, potentially leading to errors in sensitive applications.

Subplot (c) Comparison of Sensor Measurements: This subplot compares the outputs from both sensors. The QS data remain stable and consistent, while the CS data show noticeable variability. This comparison highlights the superior accuracy of the quantum sensor, making it more suitable for tasks requiring precise temperature monitoring.

Subplot (d) Measurement Error (Quantum–Conventional): Finally, this subplot shows the measurement error between the two sensors, calculated as the difference between the QS and CS outputs. The error plot reveals how the CS inaccuracies manifest across different temperatures, emphasizing the importance of sensor choice in critical applications.

This setup provides a clear visual comparison between the QS and CS, demonstrating the potential advantages of QS in precision measurement scenarios.

The use of a CS as a reference in this study is a strategic choice to provide a baseline for comparison, allowing for a more comprehensive evaluation of the QS performance. While it might seem counterintuitive at first, comparing QS against both CS and a high-precision reference sensor offers several key advantages.

Firstly, including a CS enables us to demonstrate the incremental improvements and specific advantages that QSs bring to the table. By comparing the QS to a technology that is already well understood and widely used, we can highlight where the QS excels, particularly in areas such as sensitivity, stability, and resistance to noise.

Secondly, this comparison allows us to establish the quantum sensor’s performance in real-world conditions. Conventional sensors are still prevalent in many applications, and understanding how QSs perform in relation to these widely used tools is crucial for industry adoption.

The CS serves as a familiar benchmark, helping to contextualize the quantum sensor’s performance for practitioners who are used to working with traditional methods.

Lastly, the inclusion of a CS underscores the potential of the QS to surpass existing technologies. By setting a CS as the reference point, we can more effectively demonstrate the superiority of QSs, not just in theory but in practice, across various conditions and environments.

3.2. Fatigue Analysis and Durability Assessment of the QS

To assess the lifespan of our quantum sensor and determine when it becomes less reliable, which is crucial for long-term applications, we conducted a comprehensive study on the fatigue and durability of the sensor. The following Figure 4 illustrates the observed behavior.

To simplify the data and clarify Figure 4, we present a table.

Table 1. Measurements of sensor performance over time at 20 °C.

Usage Time (Arbitrary Units)	Measurement at 20 °C (°C)
0.000000	20.000000
111.111111	20.012019
222.222222	20.030623
333.333333	19.456982
444.444444	20.259083
555.555556	20.170770
666.666667	19.694442
777.777778	19.830338
888.888889	18.880939
1000.000000	19.816022

displays the simulated measurements at 20 °C over time as the sensor is used, illustrating how the readings may vary due to sensor fatigue.

Table 1 presents the sensor measurements over time at a constant temperature of 20 °C. The data show the variation in sensor readings as the usage time increases. Initially, the measurements are stable around 20 °C but exhibit some fluctuations as usage time progresses. Notably, the readings deviate slightly from the baseline, reflecting the sensor’s response to prolonged use. This variation highlights the gradual impact of sensor fatigue over time, providing insight into how the sensor’s performance may change with extended operation.

3.3. Simulation of Quantum Sensor Data for IoT

Figure 5 presents data collected by a simulated quantum IoT sensor over time. The x-axis represents the actual temperature in degrees Celsius, and the y-axis shows the sensor measurement. The plot demonstrates the performance of the quantum sensor in a real-time data collection scenario.

Figure 5 depicts the simulated data collected from an Internet of Things (IoT) quantum temperature sensor. It provides a graphical representation of the sensor’s measurements against actual temperatures. The plot features data points and a line connecting them, illustrating the relationship between the actual temperature and the sensor’s recorded measurements. Key elements of the figure include the plotted data points, which represent measurements taken by the quantum IoT sensor, and the line that connects these points, providing a visual indication of the sensor’s response. The x-axis of the plot is labeled “Actual Temperature (°C)”, representing the real temperatures at which the sensor measurements were taken, while the y-axis is labeled “Sensor Measurement (°C)”, indicating the temperatures recorded by the sensor.

The figure uses the abbreviation “IoT” for Internet of Things and specifically focuses on the quantum sensor used in this context. The temperature measurements are expressed in degrees Celsius (°C). The objective of this figure is to demonstrate the performance and accuracy of the quantum IoT sensor by comparing its measurements with the actual temperatures. It illustrates how well the sensor’s recorded values align with the true temperature, providing insights into the sensor’s reliability and effectiveness in an IoT application.

Table 2. Comparison of temperature measurements across five different graphs. The table shows the temperature values in degrees Celsius recorded by various sensor types under different conditions.

Graphs	Temp (°C)	Quantum (°C)	Conv. (°C)	Error (°C)	IoT Quantum (°C)
3(a)	10	10.0	-	-	-
3(b)	15	-	14.8	-	-
3(c)	20	20.0	19.7	-	-
3(d)	25	-	-	0.5	-
5	30	30.0	-	-	29.9

above presents a comparison of temperature measurements between a quantum sensor and a CS across five distinct scenarios. Each row in the table corresponds to one of the graphs generated in the simulation. The columns display the measured temperatures, the values obtained from the QS and CS, the measurement errors between the two types of sensors, and the simulated data for a quantum IoT sensor.

Table 2 presents a comparative analysis of the performance of quantum and conventional sensors in measuring temperature at various values. The results highlight the superior accuracy of the quantum sensor, which consistently provides precise measurements closely aligned with actual temperatures. For instance, in Graphs 3(a), 3(c), and 5, the quantum sensor measures temperatures of 10 °C, 20 °C, and 30 °C, respectively, with exact accuracy. In contrast, the conventional sensor shows slight deviations from the actual temperature values. In Graph 3(b), the conventional sensor records 14.8 °C when the actual temperature is 15 °C. Similarly, in Graph 3(c), the conventional sensor records 19.7 °C compared to the actual temperature of 20 °C. These deviations, although small, highlight the relative inaccuracy of the conventional sensor compared to the quantum sensor.

The “Error” column, which appears only in Graph 3(d), indicates an error value of 0.5 °C. This suggests a slight discrepancy in the measurements, which may indicate sensor limitations under certain conditions. The IoT quantum sensor in Graph 5 also shows a highly accurate reading of 29.9 °C, further emphasizing the precision of quantum technology, even in IoT applications.

Overall, the table underscores the reliability and accuracy of quantum sensors over conventional ones, particularly in applications requiring high precision. Quantum sensors, including the IoT variant, consistently outperform conventional sensors, with minimal error in temperature measurements across a range of tested values. QSs exhibit exceptional accuracy, with measurements closely matching the actual temperature at each simulated point. The quantum sensor consistently returns values identical to the input temperature, underscoring its potential for applications requiring minimal error. In contrast, the CS shows slight deviations, with measurement errors reaching up to 0.3 °C, particularly at 20 °C. While these discrepancies are minor, they could be significant in contexts where extreme precision is essential. The difference between quantum and CS measurements as shown in the “Error” column reveals an error of 0.5 °C at 25 °C. This indicates that while the CS is relatively accurate, it is less reliable than the quantum sensor under certain conditions, possibly due to systematic errors or limitations in sensitivity.

To test the stability of the quantum sensor studied in this article, we present Figure 6, which compares the measurements of both types of sensors and analyzes the discrepancies to assess the quantum sensor’s stability. This combined view provided by Figure 6 offers a comprehensive overview of the stability and performance of the quantum sensor compared to a CS.

This Figure 6 presents a comprehensive analysis of temperature measurements comparing quantum and conventional sensors, using four distinct plots to illustrate the results. The main context of the figure is to demonstrate the stability and accuracy of a quantum temperature sensor relative to a conventional sensor, across a temperature range of −10 to 40 °C. The figure provides insights into how the two sensors perform in terms of their measurements, their error rates, and how their data are collected over time.

The first subplot (labeled a) shows the measurements taken by the quantum sensor across the specified temperature range, represented by a blue line. This plot highlights the accuracy and consistency of the quantum sensor, with minimal fluctuations observed in its recorded measurements. The second subplot (labeled b) shows the measurements from the conventional temperature sensor in red. The data appear noisier, illustrating the effect of random measurement errors inherent in conventional sensors. In both plots, the temperature is represented in degrees Celsius (°C), with the sensor measurements plotted against the actual temperatures.

The third subplot (labeled c) compares the measurements of both sensors directly, with the quantum sensor in blue and the conventional sensor in red. This comparison allows for a clear visual understanding of how the sensors behave differently under identical conditions. Finally, the fourth subplot (labeled d) focuses on the measurement error between the two sensors, represented by a green line. It demonstrates the difference between the quantum sensor’s accurate measurements and the noisy data from the conventional sensor, with the error plotted as the difference in degrees Celsius (°C).

The overall objective of the figure is to highlight the superior stability and reduced error of the quantum sensor in comparison to the conventional one. It shows that quantum sensors provide more reliable data, especially in environments where precision is crucial, such as in IoT applications where real-time monitoring is critical.

In a simulated scenario, the quantum IoT sensor also demonstrates high accuracy, recording 29.9 °C for an actual temperature of 30 °C, showcasing its effectiveness even in real-time data collection. QSs could be particularly advantageous in fields requiring UPM, such as scientific research, high-tech industries, or medical devices. CSs, though slightly less accurate, remain useful for applications where minor errors are acceptable, and factors like cost or ease of integration are more critical. This comparison highlights the superiority of QSs in precision, especially in critical or real-time scenarios, while acknowledging the continued relevance of CSs for most applications.

In order to validate our methodology, we present a comprehensive comparison with the relevant literature.

Table 3. Comparison of measurement results with similar studies.

Source	Sensor Type	Measured Temp (°C)	Accuracy/Error (°C)
This Work	Quantum	10.0	0.0
This Work	Conventional	14.8	0.2
Study [17]	Quantum	9.9	0.1
Study [45]	Conventional	14.6	0.4
Study [46]	Quantum	20.1	0.1
Study [47]	Conventional	19.9	0.3

presents a comparison of similar works.

Table 3 presents a comparative analysis of the temperature measurement results from our study with those reported in previously published scientific studies [17,45,46,47]. This comparison highlights the performance advantages of the quantum sensor used in our work, as well as the relative accuracy of the CS.

QSs offer significant advantages over CSs, mainly thanks to their superior accuracy and greater consistency across different temperature ranges. This enhanced accuracy is crucial in fields such as scientific research, high-tech manufacturing and medical devices, where even minor measurement inaccuracies can have significant consequences. The implications of these findings are particularly relevant to applications requiring the highest precision, including emerging technologies and industries with stringent accuracy requirements. Our study effectively highlights the superior performance of QSs, affirming their reliability and accuracy. By comparing our results with the established scientific literature, we have demonstrated the robustness of QSs and highlighted their potential for wider application in existing and future technologies. This reinforces the significant value of QSs in scenarios where precise temperature measurement and control are essential.

The current study demonstrates the significant advantages of quantum temperature sensors (QTSs) over conventional sensors, particularly in terms of accuracy and stability. However, transitioning these advanced sensors from controlled laboratory settings to practical, everyday applications presents several challenges. These include the high cost of quantum sensors, the need for specialized calibration and maintenance, and the integration with existing systems. Addressing these challenges will require further research focused on reducing the cost of quantum sensor technology through innovation in manufacturing processes, improving the robustness of sensors to handle diverse environmental conditions, and developing user-friendly calibration techniques. Future research should also explore the scalability of quantum sensors for large-scale applications and their interoperability with other sensor technologies in IoT ecosystems. By addressing these practical considerations, the research can contribute to making quantum sensors a viable and widespread technology for various applications, from industrial monitoring to consumer electronics.

While various experimental setups and sophisticated simulation tools exist for studying sensor performance, our tool provides additional advantages by simulating a broader range of scenarios and offering a high degree of parameter control. This allows users to explore sensor behavior in conditions that may be difficult to replicate in real-world environments, such as extreme or unstable conditions. Moreover, our tool simplifies the process of running multiple simulations and performing statistical analyses, offering reproducible and accurate comparisons between quantum and conventional sensors. Its ability to model performance under varying conditions contributes to more reliable predictive insights, complementing experimental data with flexible, parameter-driven simulations.

The quantum sensors used in this study are effective within a range of −10 to 40 °C, which aligns with standard conditions for many industrial and IoT applications. However, beyond these limits, the accuracy of the sensors can be affected due to instrumentation constraints and quantum shift effects at extreme temperatures. Next, for the detection limit and quantification, while quantum sensors offer superior sensitivity, these parameters can be influenced by external factors such as environmental fluctuations and the quality of integration into more complex systems. We also highlight that their performance may be compromised by practical limitations, such as manufacturing costs and the expertise required for calibration, which are less restrictive for conventional sensors.

4. Conclusions

In this study, we have demonstrated the significant advantages of quantum temperature sensors (QTSs) over conventional sensors (CSs), particularly with respect to measurement accuracy and stability. Testing across a temperature range of −10 to 40 °C revealed that QTSs provide superior accuracy, characterized by lower average errors and smaller standard deviations compared to CSs. These results were further validated through Python simulations and statistical analyses, ensuring the reliability and reproducibility of our findings. The enhanced precision of QTSs underscores their exceptional performance and their potential for critical applications requiring high precision, such as in the Internet of Things (IoT) and other critical systems where accurate temperature control is crucial.

However, our study also reveals certain practical limitations. First, the tested temperature range of −10 to 40 °C, while suitable for many industrial and IoT applications, remains relatively narrow. Expanding this range to include extreme temperatures would provide a more comprehensive assessment of QTS performance in a wider array of environments. Secondly, while the simulations provided valuable insights, the long-term stability and durability of QTSs in real-world scenarios need to be further evaluated. Real-world conditions often involve environmental fluctuations and unforeseen operational challenges that simulations cannot fully capture. Therefore, future experimental studies should focus on assessing QTS performance over extended periods under variable and harsh conditions.

Additionally, we recognize some practical limitations in terms of the cost and complexity of QTSs. Quantum sensors, though offering unparalleled accuracy, are currently associated with higher manufacturing costs and require specialized expertise for calibration. These factors may present barriers to their large-scale adoption. Future research should explore ways to reduce these costs and simplify the integration of QTSs into existing and emerging IoT systems.

Moving forward, comparative studies with other advanced sensor technologies will be essential to further validate the broader applicability of QTSs and identify areas for improvement. Addressing these limitations will be critical for enhancing their deployment in a wide range of technological fields, allowing QTSs to fulfill their potential as key components in next-generation sensing systems.