One-Step Run Example Command

python ./scripts/algorithm.py

Skill: 2D Heat Equation (Schrödingerization-based Solver)

Mathematical Model

2D Heat Equation

$$
\frac{\partial u}{\partial t} = a_1 \frac{\partial^2 u}{\partial x^2} + a_2 \frac{\partial^2 u}{\partial y^2} + f(x,y)
$$

• $a_1, a_2$: diffusion coefficients in x/y directions
• $u(x,y,t)$: temperature field
• $f(x,y)$: source term

Schrödingerized Hamiltonian (Periodic Form)

$$
H = - \hat{\eta} \otimes \left( a_1 \hat{p}_x^2 + a_2 \hat{p}_y^2 \right)
$$

Valid if:

1. Discrete derivative operator is Hermitian
2. Periodic BC in both x/y directions
3. Source term $f(x,y) = 0$

Otherwise, full Schrödingerization procedure is required.

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Supported Features

• 2D anisotropic heat conduction
• Boundary conditions: Dirichlet, Periodic
• Initial conditions: 2D sine wave, custom
• Quantum solvers: Classical matrix exponentiation, Trotter splitting, Block encoding (fallback)
• Tensor-product finite-difference discretization
• 3D surface / contour visualization
• Automatic quantum circuit generation

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Full Algorithm Pipeline (Step-by-Step)

Step 0: Import Libraries

Import necessary modules for parsing, solvers, operators, and circuit generation:

# import parser
from unitarylab.library import parse_equation

# import solvers
from unitarylab.library import schro_classical, schro_trotter
from unitarylab.library.differential_operator.classical_matrices import first_order_derivative, second_order_derivative
from unitarylab.library.schrodingerization.classical import circuit_classical
from scipy.integrate import cumulative_trapezoid

Step 1: Parse 2D Domain & Parameters

Extract coefficients, domain, grid, qubits, boundary type:

a1 = eq.get_parameter('a1')
a2 = eq.get_parameter('a2')
L, T, nx, na, R, order, f0 = eq.get_common_coefficients()
bd = eq.boundary.type

Nx = 2**nx
dx = L / (Nx + 1)
x = np.arange(dx, L, dx)
y = np.arange(dx, L, dx)

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Step 2: Initialize 2D Temperature Field

u0 = f0(x[:, None], y[None, :])  # 2D initial condition
u0 = u0.flatten()                # flatten for solver

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Step 3: Assemble 2D Laplacian Matrix (Kronecker Product)

A0 = CDiff(N=Nx, dx=dx, order=2, scheme=scheme, boundary=bd).get_matrix()
A = a1 * sp.kron(A0, sp.eye(Nx)) + a2 * sp.kron(sp.eye(Nx), A0)

• kron(A0, I): x-direction Laplacian
• kron(I, A0): y-direction Laplacian

---

Step 4: Build Source Term

b0 = source(x)
b = a1 * np.kron(b0, np.ones(Nx)) + a2 * np.kron(np.ones(Nx), b0)

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Step 5: Schrödingerization Solver

u = schro_classical(A, u0, T=T, na=na, R=R, order=order, b=b)
u = u.reshape((Nx, Nx))  # reshape to 2D
qc = circuit_classical(nx, na, dim=2)

The Schrödingerization framework can be referred to in './Schr_skills.markdown'.

Step 6: Trotter Quantum Circuit (Optional)

func1 = TDiff(nx, dx, 2, boundary=bd).data()[0]
D1 = lambda a: func1(a * dt/R)

H1 = Circuit(2*nx)
H1.append(D1(a1), range(nx))      # x-direction
H1.append(D1(a2), range(nx, 2*nx))# y-direction

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Step 7: Run Trotter Evolution

u, qc = schro_trotter(u0=u0, H1=H1, H2=None, Nt=Nt, na=na)
u = u.reshape((Nx, Nx))

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Step 8: Visualization

X, Y = np.meshgrid(x, y)
ax.plot_surface(X, Y, u, cmap='viridis')

---

Boundary & Initial Conditions

• Dirichlet: $u=0$ on domain boundary
• Periodic: $u(0,y)=u(L,y), u(x,0)=u(x,L)$
• Initial Condition: 2D sine wave:

$$
u(x,y,0) = \sin\left(\frac{\pi x}{L_x}\right) \sin\left(\frac{\pi y}{L_y}\right)
$$

---

Finite-Difference Scheme

2D central difference for Laplacian:
$$
\Delta u_{i,j} = \Delta t \left[ a_1 \frac{u_{i+1,j} - 2 u_{i,j} + u_{i-1,j}}{\Delta x^2} + a_2 \frac{u_{i,j+1} - 2 u_{i,j} + u_{i,j-1}}{\Delta y^2} \right]
$$

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Outputs

• 2D temperature field $u(x,y,T)$
• 3D surface plot
• Full quantum circuit diagram
• H1 decomposed Trotter circuit

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Trigger Phrases

• Solve 2D heat equation using quantum Schrödingerization
• Quantum simulation for planar heat conduction
• 2D thermal analysis
• Quantum PDE solver for heat conduction

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Use Cases

• Chip / PCB thermal simulation
• 2D material heat conduction
• Thin-plate temperature analysis
• Quantum PDE benchmarking

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Summary

This skill provides a complete quantum solution for the 2D heat equation:

• Uses Kronecker product for 2D Laplacian
• Supports anisotropic diffusion in x/y directions
• Works with classical, Trotter, and block solvers
• Automatically reshapes to 2D field
• Generates professional 3D visualizations
• Fully aligned with your implementation and mathematical framework