# A Quantum Approach for Exploring the Numerical Results of the Heat Equation
## 3 Results  
This section presents the results and analysis of the probability distributions obtained by applying the quantum scheme. First, a test instance is solved on 3, 5, and 7 qubits. Moving on to the analysis of the results of the thermal conductivity modeling, it should be noted that this study involved assessing the correspondence between the results of the quantum modeling and the analytical solution. Figure 3 shows these results. Here is what each line on the graph represents:  
* Classical state: this is the assumed analytical solution to the heat conduction problem;
* Quantum solutions: these are the results of thermal conductivity simulations using a quantum approach. Each point on the graph presented in Figure 3 corresponds to the state of the qubits (or other elements of the system), and the values on the vertical axis represent the probabilities of these states. The graph helps us see how well the quantum simulation results agree with the analytical solution or other simulation methods. Next, Figure 4 visualizes the probability of a quantum state with 3000 dimensions through a histogram, displaying the number of quantum states on the x-axis and their probabilities on the y-axis. Figure 3: Comparison of thermal conductivity simulation results. The visualization described in Figure 4 aids in comprehending the probability distribution of states within a quantum system following a series of measurements. Each state of a quantum system possesses a certain probability of being detected after measurement, and the histogram facilitates the visualization of these probabilities for each state. In quantum thermal conduction simulations, statistical data processing plays a key role. This is because the measurement results may be subject to random fluctuations or noise. The more measurements are performed, the higher the statistical accuracy of the data obtained. Figure 5 shows the dependence of accuracy on the number of measurements on a logarithmic scale. \"Shots\" refers to an array of logarithmic values representing the number of dimensions, while precision is determined as the inverse square root of the measurement count. The graph in Figure 5 shows how measurement accuracy changes with an increasing number of shots. A logarithmic scale on the x-axis is used to more clearly show the changes in accuracy over a wide range of values, revealing how accuracy increases quickly with a small increase in the number of measurements at the beginning, followed by a slower rate of increase. The relative error value is 0.0029. This means that the standard deviation between the actual probability distribution of qubit states obtained from the simulation and the desired probability distribution is 0.0029. A lower error value indicates a closer match between the actual and expected state probabilities. If we strive for the most accurate simulation of the temperature distribution in the system, then the value of 0.0029 can be considered quite low,  
Figure 4: Probabilities of classical states with 5000 shots. Figure 5: The dependence of accuracy on the number of measurements on a logarithmic scale. which indicates a good quality of the simulation. However, if the accuracy requirements are higher, further optimization or refinement of the model may be required. As part of this study, tests were carried out on various quantum devices, including the ibmq_qasm_simulator and simulator_statevector simulators, as well as the ibm_osaka cloud quantum computer. The simulation results, as shown in Table 1 and Figure 6, demonstrate that as the number of qubits increases, the relative error decreases for all devices considered. Quantum devices such as the simulator_statevector provide a lower relative error compared to the ibmq_qasm_simulator and ibm_osaka. This may be due to differences in the types of simulations and algorithms used. A lower relative error indicates more accurate simulation and a closer match between the actual and expected qubit state probabilities.