One-Step Run Example Command

python ./scripts/algorithm.py

Skill: Quantum Simulation of 1D Heat Equation

1. Mathematical Formulation

1.1 1D Heat Equation

$$
\frac{\partial u}{\partial t} = a \frac{\partial^2 u}{\partial x^2} + f(x)
$$

• $a > 0$: diffusion coefficient
• $f(x)$: source term

1.2 Schrödingerized Hamiltonian

$$
\frac{d\psi}{dt} = -i H \psi
$$

with simplified Hamiltonian for periodic, source-free cases:
$$
H \approx -a \hat{\eta} \otimes \hat{p}^2 \approx a D_{\eta} \otimes D^\Delta
$$

• $\hat{\eta}$: auxiliary operator introduced by Schrödingerization
• $D_\eta$: discretized auxiliary operator
• $D^\Delta$: discrete Laplacian

Note: General source or non-periodic BC requires full Schrödingerization.

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

2. Supported Features

• PDE Type: 1D parabolic diffusion
• Boundary Conditions: Dirichlet, Periodic
• Initial Conditions: sine, Gaussian, custom
• Solvers: Classical matrix exponentiation, Trotter splitting
• Automatic finite-difference Laplacian assembly
• Visualization + quantum circuit export

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3. Full Algorithm Pipeline (Step-by-Step)

Step 1: Parse Input Parameters

L, T, nx, na, R, order, point, f0 = eq.get_common_coefficients()
bd = eq.boundary.type

Nx = 2**nx
dx = L / Nx
x = np.arange(0, L, dx)
u0 = f0(x)  # initial condition

---

Step 2: Discretization

Construct 2nd-order differential operator:

A0, b0 = second_order_derivative(
    N=Nx,
    dx=dx,
    boundary_condition=bd
)

A = a * A0
b = a * b0 + f(x)

---

Step 3: Schrödingerization

Transform non-unitary diffusion equation:
$$
\frac{du}{dt} = A u + b \quad \longrightarrow \quad \frac{d\psi}{dt} = -i H \psi
$$

• Ensures unitary evolution
• Required for general source or non-periodic BC

---

Step 4: Time Evolution

Classical Schrödinger Solver

u = schro(
    A,
    u0,
    T=T,
    na=na,
    R=R,
    order=order,
    point=point,
    b=b
)

Trotterized Quantum Evolution (Optional)

Split Hamiltonian $H = H_1 + H_2$ and apply Trotter decomposition:
$$
e^{-iHt} \approx \left(e^{-i H_1 \Delta t} e^{-i H_2 \Delta t}\right)^{N_t}
$$

u, qc = schro(
    u0=u0,
    H1=H1,
    H2=H2,
    Nt=Nt,
    na=na,
    R=R,
    order=order,
    point=point,
    b=b,
    theta=theta
)

---

Step 5: Visualization

ax.plot(x, u, "r-", linewidth=2)
ax.fill_between(x, u, alpha=0.3)

• Generates 1D solution plot
• Optional animation for time evolution

---

Step 6: Quantum Circuit Export

qc.draw(filename="heat_1d_circuit.svg")

• Full circuit for Hamiltonian simulation
• Includes Trotter decomposition if used

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4. Boundary Conditions

• Dirichlet: $u(0,t)=0, u(L,t)=0$
• Periodic: $u(0,t) = u(L,t)$

---

5. Initial Conditions

• Sine: $u(x,0) = \sin(2 \pi x / L)$
• Gaussian: $u(x,0) = \exp(-x^2)$
• Custom: user-defined $f_0(x)$

---

6. Finite-Difference Scheme

• Central Difference (2nd-order Laplacian):

$$
\Delta u_i \approx \frac{u_{i+1} - 2 u_i + u_{i-1}}{\Delta x^2} \Delta t
$$

• Automatically assembled into matrix $A$

---

7. Outputs

• Solution array $u(x,T)$
• 1D plot of solution
• Full quantum circuit diagram
• Optional Trotter decomposition diagrams

---

8. Trigger Phrases

• Quantum simulation of 1D heat equation
• Schrödingerization-based solver for diffusion PDE
• Trotter quantum evolution of parabolic PDE

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9. Use Cases

• 1D diffusion and conduction problems
• Heat transfer simulations
• Benchmarking quantum PDE algorithms
• Educational demonstrations of Schrödingerization

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Summary

• Standardized step-by-step quantum simulation pipeline for 1D Heat Equation
• Handles Dirichlet/Periodic BCs and custom initial conditions
• Supports classical and Trotterized quantum evolution
• Automates Laplacian assembly, visualization, and circuit export
• Fully consistent with Advection / Burgers / General Linear PDE / Elastic Wave skill style