A Quantum Approach for Exploring the Numerical Results of the Heat Equation

Abstract

This paper presents a quantum algorithm for solving the one-dimensional heat equation with Dirichlet boundary conditions. The algorithm utilizes discretization techniques and employs quantum gates to emulate the heat propagation operator. Central to the algorithm is the Trotter–Suzuki decomposition, enabling the simulation of the time evolution of the temperature distribution. The initial temperature distribution is encoded into quantum states, and the evolution of these states is driven by quantum gates tailored to mimic the heat propagation process. As per the literature, quantum algorithms exhibit an exponential computational speedup with increasing qubit counts, albeit facing challenges such as exponential growth in relative error and cost functions. This study addresses these challenges by assessing the potential impact of quantum simulations on heat conduction modeling. Simulation outcomes across various quantum devices, including simulators and real quantum computers, demonstrate a decrease in the relative error with an increasing number of qubits. Notably, simulators like the simulator_statevector exhibit lower relative errors compared to the ibmq_qasm_simulator and ibm_osaka. The proposed approach underscores the broader applicability of quantum computing in physical systems modeling, particularly in advancing heat conductivity analysis methods. Through its innovative approach, this study contributes to enhancing modeling accuracy and efficiency in heat conduction simulations across diverse domains.

1. Introduction

In the last decade, the development of quantum computing has opened up new horizons for solving complex scientific problems that were inaccessible to classical computational methods. Quantum computing has significant potential in solving differential equations, including the heat conduction equation, due to its ability to model quantum systems efficiently and process large volumes of data in parallel. This is due to the ability of quantum computers to process information in the form of quantum states, allowing for the analysis of complex systems, such as in the field of quantum mechanics and materials science. Recent studies have demonstrated the superiority of quantum algorithms in efficiently modeling physical systems, which can lead to the development of new materials, the improvement of manufacturing processes, and the development of new methods for information processing. In addition, quantum computing reduces computational time when solving complex differential equations, which has potential applications in areas requiring precise and rapid data analysis, such as climate modeling, financial analytics, and medical diagnostics.

However, accuracy remains a critical factor. Errors in quantum computing can arise from physical limitations on quantum systems, such as decoherence, algorithmic errors, and hardware implementation issues. Thus, it is essential to analyze and evaluate the accuracy of quantum methods when solving differential equations.

This study hypothesizes that a quantum algorithm for solving the one-dimensional heat equation with Dirichlet boundary conditions, utilizing discretization techniques and quantum gates, will achieve significant computational speedup and improved accuracy in modeling heat conduction processes compared to classical computational methods.

Various methods have been developed to enhance the efficiency and accuracy of solving linear and nonlinear partial differential equations (PDEs) using quantum computing [1]. These methods can be classified into fully quantum and hybrid methods, combining the advantages of classical and quantum computing. However, it is important to understand that the accuracy of any technology, including quantum computing, may be limited. Errors in quantum computing can arise due to the physical limitations on quantum systems, such as decoherence, algorithmic errors, and hardware implementation. Therefore, it is important to analyze and evaluate the accuracy of the results when using quantum methods to solve differential equations and to consider possible errors when interpreting the results.

One method for solving PDE problems is the numerical finite difference method (FDM), which approximates derivatives to find an approximate solution [2]. Also worth noting are linearization methods, which transform nonlinear equations into a system of linear equations, which can then be solved using quantum algorithms. For example, in [3], the Carleman linearization method was used, which transforms a system of nonlinear differential equations into an infinite-dimensional system of linear differential equations. Moreover, in [4,5,6], the Schrödingerization method was developed to solve various PDEs. In practice, the use of hybrid methods is often more effective. In [7], the authors put forward a solution to an instance of the Navier–Stokes equations using quantum feature maps and differentiable quantum circuits (DQCs). Variational quantum algorithms (VQA), which adjust the parameters of quantum circuits in a manner akin to machine learning techniques, have also been employed to tackle differential equations, such as the heat equation and Poisson’s equation, as seen in references [8,9,10].

The versatility of quantum computing in solving PDEs is further highlighted by research exploring a diverse array of additional methods [11,12].

Quantum approaches for solving the heat equation are based on discretizing the PDE and reducing the problem to solving a system of linear equations, which are solved mainly using approaches based on the Harrow, Hasidim, and Lloyd (HHL) algorithm [13,14]. As a rule, a variational quantum linear solver (VQLS) is used for linear systems, based on minimizing a nonlinear cost function, as in [15,16,17,18,19]. All of these implementations demonstrated an exponential increase in computational speed as the number of qubits increased, which varied from 3 to 11. In [16], three implementations of the above approach were implemented: the direct variational method, the Hadamard testing approach, and the most efficient approach using a tree Ansatz. The VQLS is well analyzed in [17], where the authors propose a hybrid quantum-classical algorithm called a VQLS for solving linear systems on quantum computers of the near future. The authors present efficient quantum schemes for estimating $C$, which is the cost function in the VQLS, while providing evidence of the time complexity involved in its estimation. Typically, the time complexity of a quantum algorithm depends on the number of qubits in the quantum circuit, but using the Seidel method to divide the complete system of equations into subsystems and their iterative solution allows one to bypass the restrictions on the number of qubits and their connections [20]. In [21], a quantum algorithm was presented for the heat conduction problem by discretizing the PDE.

In addition to the above methods, recent advances in the quantum algorithms for dissipative nonlinear differential equations have significantly improved computational efficiency, as in [3], where a quantum algorithm for these equations assuming strong dissipation relative to nonlinear and forcing terms was developed. Utilizing Carleman linearization, it maps nonlinear differential equations to a system of linear equations solved by the quantum linear system algorithm, rendering it suitable for strongly dissipative systems such as certain fluid dynamics and epidemiological models.

A review of the literature showed that quantum algorithms for solving partial differential equation problems show an exponential increase in the computation speed with an increasing number of qubits. However, at the same time, there is an exponential increase in the relative error and cost function. Quantum modeling is used to address this problem, evaluating its potential impact on the understanding and methodology for modeling the one-dimensional heat equation.

Many previous studies on the heat equation have used quantum algorithms and cost functions [22]. To evaluate the effectiveness of this approach, the probability distribution of the outcomes was calculated and presented as a function of the standard deviation. Thus, this study aims to identify opportunities for using quantum computing in various industries to achieve more accurate and efficient results in thermal conductivity modeling. The quantum algorithm has the potential to provide higher accuracy compared to classical numerical simulation methods and may also be more efficient in solving problems with a large number of parameters.

In the context of this study, a quantum approach to the simulation modeling of temperature distribution is presented. This study applies the Qiskit quantum programming toolkit to solve heat conduction problems using quantum gates and an iterative process of heat propagation over time. The purpose of this study is to evaluate the effectiveness and potential of using quantum computing to simulate temperature distribution.

2. Methodology

The principles of developing a quantum algorithm for solving the heat equation are based on the mathematical modeling of the process of heat propagation and the use of quantum computing to approximate it. This section will discuss the detailed steps of the implementation of developing a quantum algorithm for solving the heat equation. Figure 1 shows the architecture of the proposed quantum algorithm.

The architecture consists of several key components that sequentially perform transformations and calculations to obtain a solution to the heat equation. It begins with the discretization of the one-dimensional heat equation by the central difference method. Then, the initialization of quantum states encoding the initial temperature distribution in the qubits occurs. Next, quantum gates are used to model the time evolution of the system based on the Trotter–Suzuki decomposition. These operations are implemented in the form of a quantum circuit. After the circuit is run on a quantum computer or simulator, the states of the qubits are measured, yielding a probability distribution of the final states. The final stage is the calculation of the relative error between the obtained distribution and the expected temperature distribution, which allows us to evaluate the effectiveness of the quantum solution for the thermal conductivity problem.

2.1. Basic Equations and Discretization

The heat conduction equation usually describes the mathematical modeling of the process of heat propagation in a medium. The mathematical models that describe one-dimensional thermal boundary value problems with Dirichlet boundary conditions are presented below. The heat equation is used to determine the change in temperature over time $t$. For the one-dimensional case, the heat equation can be approximated as in Equation (1):

$${\displaystyle \frac{\partial T\left(x,t\right)}{\partial t}}=k{\displaystyle \frac{{\partial }^{2}T\left(x,t\right)}{{\partial x}^2}},$$

(1)

where $T(x,t)$ is the temperature at point $x$ at time $t$, and $k$ is the thermal conductivity coefficient. The thermal conductivity coefficient ($k$) is a parameter that describes the ability of a material to conduct heat. To address the impact of the thermal conductivity parameter $k$ on the accuracy of quantum simulations, we chose a low value of 0.01. Higher values of $k$ can lead to decreased accuracy and smoothness in the temperature distribution [21]. The Dirichlet boundary conditions are applied at the initial point, $x=0$, and at the end point, $x=1$, as given by the formula:

$$T\left(0,t\right)=0, t\ge 0 \mathrm{and} T\left(1,t\right)=1, t\ge 0.$$

(2)

The initial conditions at t = 0 is

$$T\left(x,0\right)=x, 0\le x\le 1.$$

(3)

These mathematical models of Equations (1)–(3) describe the heat conduction in a one-dimensional uniform rod of length one unit with no internal heat sources, thermal diffusivity one, perfect lateral insulation, and initial condition $x$ when $0\le x\le 1$. Both the left end and the right end are insulated and kept at 0.

To solve this equation numerically, various discretization methods are applied [23]. One approach is to use central differences to approximate the second derivative with respect to the spatial variable $x$, as in Equation (4):

$${\displaystyle \frac{T_i^{n+1}-T_i^n}{\Delta t}}=k{\displaystyle \frac{T_{i+1}^n-2T_i^n+T_{i-1}^n}{{\left(\Delta x\right)}^2}},$$

(4)

where $T_i^n$ represents the temperature at node $i$ at time level $n.$ Here, $\Delta t$ is the time step, and $\Delta x$ is the spatial step.

In its original form, the heat conduction equation represents a classical model that describes the propagation of heat in a medium. However, a direct transition to quantum circuits from this model is impossible. Instead, by using discretization methods, a difference equation was obtained that could already be adapted for quantum computing. This difference equation was used for each time step, using the temperature values at the previous time level to obtain the evolution of temperature in space and time.

2.2. Converting Heat Equations to Quantum Form

The conversion of the heat equation into the quantum form is performed using quantum gates and quantum states. For the one-dimensional case, the following algorithm is used:

• The initialization of quantum states that represent the temperature distribution in space;
• The application of quantum gates that emulate the evolution operator corresponding to the propagation of heat in space and time;
• The measurement of states to obtain information about temperature distribution.

2.2.1. Initialization of Quantum States

Let us represent the temperature distribution at time level $t$ at point $x$ using a quantum state on qubit $i$, where $i$ corresponds to the index of the point $x_i=i\Delta x$, and $\Delta x$ is the spatial step. The initial temperature distribution in the system can be encoded into qubit states. For example, if we have $N$ points in the system, we can use $N$ qubits, where each qubit represents one point, and the initial state is determined by their values.

Let us denote the initial temperature distribution as $T_i^0$, where $T_i^0$ represents the temperature at point $x_i=i\Delta x$ at time $t=0$. We can encode this initial distribution into the quantum states as follows:

$$\begin{gathered} If T_i^0=1: \mid 1⟩i, \\ If T_i^0=0: \mid 0⟩i. \end{gathered}$$

where $\mid 1⟩i$ represents that the temperature at point $x_i=i\Delta x$ is 1 and $\mid 0⟩i$ represents that the temperature at point $x_i=i\Delta x$ is 0.

Therefore, the initial quantum state representing the temperature distribution at time level t can be written as the tensor product of the individual qubit states:

$$\mid T\left(0\right)⟩=\mid T_1^0⟩\otimes \mid T_2^0⟩\otimes \dots \otimes \mid T_N^0⟩,$$

(5)

$\mid T\left(0\right)⟩$ is the initial quantum state representing the temperature distribution at time level $t=0$, and $\mid T_i^0⟩$ is the quantum state representing the temperature at point $x_i=i\Delta x$ at time $t=0.$

To encode the $x$ co-ordinate in this example, qubits are used, where each qubit represents a specific position $x_i$ on the spatial axis. Thus, the qubit $\mid T\left(xi\right)⟩$ encodes information about the initial temperature at point $x_i$ based on the function $T\left(x\right).$ Each qubit contains information about the temperature at the corresponding point $x_i$, which allows us to represent the temperature distribution in the system in the form of a quantum state.

2.2.2. The Use of Quantum Gates for Propagation

To propagate the temperature distribution in quantum form, we needed to design quantum gates that simulate the evolution of the system governed by the Hamiltonian operator $H$. The Hamiltonian for the heat conduction problem in discrete form models the differential operator from the heat equation.

To implement the time evolution of a quantum state corresponding to the heat equation, we used the Trotter–Suzuki expansion. This method allows one to approximate the time evolution operator, which greatly simplifies its implementation on a quantum computer. To evolve the quantum state $\mid \psi i\left(t\right)⟩$ over a small time step $\Delta t$, we used the time evolution operator $U$, which is related to the Hamiltonian $H$ as follows.

$$U=e^{-iH\Delta t},$$

(6)

where $\Delta t$ is a small time step. The Hamiltonian H in discrete form for the heat equation can be written as

$$\mathrm{H}=-\mathrm{k}{\sum }_{\mathrm{i}=1}^{\mathrm{N}-1}({\mathsf{\sigma }}_{\mathrm{i}}^{+}{\mathsf{\sigma }}_{\mathrm{i}+1}^{-}+{\mathsf{\sigma }}_{\mathrm{i}}^{-}{\mathsf{\sigma }}_{\mathrm{i}+1}^{+}-2{\mathsf{\sigma }}_{\mathrm{i}}^{\mathrm{z}}).$$

(7)

Here, ${\mathsf{\sigma }}_{\mathrm{i}}^{+}$+ and raise and lower operators ${\mathsf{\sigma }}_{\mathrm{i}}^{-}$−—raise and lower operators, a ${\mathsf{\sigma }}_{\mathrm{i}}^{\mathrm{z}}$—Pauli-Z operator acting on the ith qubit.

For a small time step $\Delta t$, the time evolution operator can be approximated using the Trotter–Suzuki expansion. The Trotter–Suzuki decomposition allows us to represent the evolution operator of a complex system as a product of the evolution operators for simpler subsystems. For the Hamiltonian of thermal conductivity $H$, we can identify components that can be conveniently implemented on a quantum computer. Let us consider the decomposition into two Hamiltonians $H_1$ and $H_2$, corresponding to the interaction of neighboring qubits and the self-interaction of each qubit, respectively.

Interaction of neighboring qubits:

$$H_1=-\mathrm{k}{\sum }_{\mathrm{i}=1}^{\mathrm{N}-1}({\mathsf{\sigma }}_{\mathrm{i}}^{+}{\mathsf{\sigma }}_{\mathrm{i}+1}^{-}+{\mathsf{\sigma }}_{\mathrm{i}}^{-}{\mathsf{\sigma }}_{\mathrm{i}+1}^{+}).$$

(8)

Self-interaction of each qubit:

$$H_2=\mathrm{k}\sum _{\mathrm{i}=1}^{\mathrm{N}-1}\left(2{\mathsf{\sigma }}_{\mathrm{i}}^{\mathrm{z}}\right).$$

(9)

Then, the time evolution operator can be approximated as follows:

$$\mathrm{U}{=e}^{-iH\Delta t}\approx {({\mathrm{e}}^{-{\displaystyle \frac{\mathrm{i}{\mathrm{H}}_1\mathsf{\Delta }\mathrm{t}}{\mathrm{m}}}}{\mathrm{e}}^{-{\displaystyle \frac{\mathrm{i}{\mathrm{H}}_2\mathsf{\Delta }\mathrm{t}}{\mathrm{m}}}}\dots {\mathrm{e}}^{-{\displaystyle \frac{\mathrm{i}{\mathrm{H}}_{\mathrm{p}}\mathsf{\Delta }\mathrm{t}}{\mathrm{m}}}})}^{\mathrm{m}},$$

(10)

where $H=H_1+H_2+\dots +H_p$—decomposition of the Hamiltonian into parts that are easier to implement. Here, $m$ is the number of times used to repeat the product of the exponentials to improve the accuracy of the approximation. The larger $m$, the more accurate the approximation of the Trotter–Suzuki expansion. In the limit at $m\to \infty$, the approximation becomes accurate.

2.3. Quantum Circuit

To implement a quantum circuit that models temperature propagation, one begins by initializing a quantum state $\mid T \left(0\right) ⟩,$ representing the initial temperature distribution. Next, the Trotter–Suzuki expansion is used to approximate the time evolution operator $U=e^{-iH\Delta t}$ for a small time step $\Delta t$. The Hamiltonian $H$ is decomposed into parts $H_i$, which are easier to implement using quantum gates. The gates for interaction between adjacent qubits are implemented through CNOT gates and single-qubit rotations, while self-interaction terms are implemented through Z-rotations $Rz\left(\theta \right)$ on each qubit. For each time step, the quantum circuit involves applying these gates sequentially to all decomposed components of the Hamiltonian. Finally, measuring the final state of the system gives the temperature distribution at the next point in time. Repeating these steps allows us to simulate the evolution of the thermal field over time. A quantum circuit for heat propagation is illustrated in Figure 2.

The level of obtaining a quantum circuit opens up the possibility of its execution both on a quantum computer and on a simulator. The result of this process is a probability distribution of outcomes. To evaluate the performance of a given circuit, an evaluation function is used, which allows us to quantify how well the circuit solves a given problem. The formula for the scoring function can be written as the standard deviation between the actual probability distribution of the qubit states obtained from the simulation and the desired probability distribution, which is the expected temperature distribution in the system. Let $p_i$ be the probability of state $i$ and $N$ be the total number of states; then, the formula will look like this:

$$Relative error=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(p_i-{Desired Probability}_i \right)}^2},$$

(11)

where the desired probability is the expected probability of state $\mathrm{i}$ in the desired distribution.

This formula calculates the standard deviation between the actual and expected probabilities of all states, which allows one to evaluate how close the simulation results are to the expected values. The desired probability depends on what temperature distribution can be obtained in the final system. If the temperature is uniformly distributed throughout the system, then each state should have the same probability.

For a system with $N$ qubits, where each qubit can be in the $\mid 0⟩$ or $\mid 1⟩$ state, there will be a total of $2^N$ different states. If we want a uniform distribution, then the desired probability of each state will be ${\displaystyle \frac{1}{\left(2^N\right)}}$.

Thus, the desired probability for each state $i$ will be:

$${Desired Probability}_i={\displaystyle \frac{1}{2^N}}.$$

(12)

By using the relative error and the desired probability distribution, the effectiveness of the quantum circuit for solving the thermal conductivity problem can be estimated. The smaller the relative error value, the closer the actual probability distribution is to the desired one and the more efficient the scheme. In this case, the relative error is based on the standard deviation, and the desired probability distribution corresponds to a uniform temperature distribution.

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.

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, 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. Relative error results on various quantum devices.

Device	Characteristic	Number of Qubits	Relative Error
ibmq_qasm_simulator	General, context-aware	32	0.0029296875
simulator_statevector	Schrödinger wavefunction	32	0.0023456456
ibm_osaka	Eagle r3	127	0.0254375

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.

From the data provided, it is clear that the simulators have different performance and relative errors depending on the number of qubits. For the simulator ibmq_qasm_simulator, which is generic and context-aware, the relative error decreases as the number of qubits increases. For example, when using 3 qubits, the error is about 12.5%, and when using 7 qubits, the error is already reduced to approximately 0.29%. The simulator simulator_statevector, which simulates the Schrödinger wave function, also shows a decrease in the relative error with an increasing number of qubits. For example, when using 3 qubits, the error is about 12.5%; when using 15 qubits, the error drops to about 0.015%.

In the case of a real quantum computer from IBM, the designated ibm_osaka with the “Eagle r3” architecture, the relative error also depends on the number of qubits. For example, when using 5 qubits, the relative error is about 16.58%; when using 10 qubits, the error drops to about 0.47%.

These results show that when simulating or using quantum devices, both simulators and real quantum computers, it is important to consider the number of qubits and their characteristics, as this significantly affects the accuracy of the results.

The algorithm offers several advantages, including high accuracy compared to classical numerical simulation methods, efficiency for large systems with many parameters, and versatility for solving other problems involving partial differential equations. However, it also faces limitations such as implementation complexity due to the complex nature of quantum computing, system size limitations imposed by the limited number of qubits in quantum devices, and susceptibility to noise, which can affect the accuracy of the results.

4. Conclusions

This article demonstrates the practical application of temperature distribution modeling using quantum computing. A quantum algorithm based on the principles of quantum modeling has been developed, which makes it possible to effectively simulate the evolution of the temperature distribution considering thermal conductivity. Using the statevector and qasm simulations, including in a cloud-based quantum computer, it has been discovered that quantum algorithms can significantly improve the accuracy and efficiency of modeling physical processes such as thermal conductivity. Numerical experiments have revealed a direct relationship between the number of qubits and the accuracy of the simulation. An analysis of various parameters has shown that the time complexity of the quantum algorithm is effectively scaled with the accuracy and size of the equation. The model analysis confirmed good agreement with the analytical solutions, which is confirmed by the low value of the relative error.

In addition, successful modeling of one-dimensional thermal conductivity problems has confirmed the effectiveness of the proposed method based on quantum gates and states. The results of heuristic scaling showed an almost logarithmic dependence on 1/ε and a linear dependence on the number of qubits.

However, it should be noted that in this study, only the small dimensions of the equation were taken into account due to computational limitations. In addition, the optimization of the quantum circuit and the use of an optimizer may be the subject of further research. This opens up new prospects for further research aimed at improving and optimizing methods for solving such problems using quantum computing.

In the future, it is planned to expand the scope of quantum thermal conductivity modeling methods to larger-scale systems and more complex physical phenomena. Further research may include optimizing algorithms, using hybrid algorithms, increasing the number of qubits to improve accuracy and efficiency, and investigating the effects of various factors on simulation results. This will allow us to better understand the possibilities and limitations of quantum methods in solving problems of thermal conductivity and contribute to the development of this field of science.