One-Step Run Example Command

python ./scripts/algorithm.py

Skill: Advection Equation Solver via Schrödingerization

Skill Objective

Enable the agent to solve the 1D Advection Equation using a Schrödingerization-based framework, supporting both:

• Classical simulation
• Quantum (Trotterized) simulation

---

Mathematical Model

Advection Equation

$$
\frac{\partial u}{\partial t}

a \frac{\partial u}{\partial x}
$$

• $a$: convection velocity
• Models transport phenomena (fluid flow, waves, etc.)

---

Core Capabilities

1. PDE Parsing

Extract problem parameters:

• Domain $[0, L]$
• Final time $T$
• Grid size $N = 2^{n_x}$
• Boundary condition (Periodic / Dirichlet / others)
• Initial condition $u(x,0)$
• Velocity $a$

---

2. Spatial Discretization

Construct grid:
$$
x_i = i \Delta x, \quad \Delta x = \frac{L}{N}
$$
Build first-order derivative operator:

A0, b0 = first_order_derivative(
    N=Nx,
    dx=dx,
    boundary_condition=bd,
    scheme=scheme
)

Construct system:
$$
A = a A_0, \quad b = a b_0
$$

---

3. Unitary Structure Detection

Agent must determine whether Schrödingerization is needed.

Case A: Already Unitary

If:

• Central difference scheme
• Periodic boundary

Then:
$$
H = -a \hat{p} \approx \frac{a D^{\pm}}{i}
$$
Properties:

• $A$ is skew-Hermitian
• Evolution is already unitary
• Skip Schrödingerization

---

Case B: Non-unitary System

If:

• Upwind scheme
• Dirichlet boundary
• Source term $b \neq 0$

Then:

• Apply full Schrödingerization procedure

---

4. Schrödingerization Transformation

Transform:
$$
\frac{du}{dt} = Au + b
\quad \Rightarrow \quad
\frac{d\psi}{dt} = -iH\psi
$$

If $A$ is Hermitian-compatible:

$$
H = iA
$$

Otherwise:

• Use auxiliary variable lifting
• Apply Fourier transform
• Construct Hamiltonian:

$$
H = \eta H_1 + H_2
$$

---

5. Hamiltonian Construction

Split operator:
$$
A = H_1 + iH_2
$$
Construct:

• $H_1 = \frac{A + A^\dagger}{2}$
• $H_2 = \frac{A - A^\dagger}{2i}$

Final Hamiltonian:
$$
H = D \otimes H_1 + I \otimes H_2
$$
The Schrödingerization framework can be referred to in './Schr_skills.markdown'.

---

6. Time Evolution Methods

(A) Classical Schrödinger Solver

from unitarylab.library import schro_classical
u = schro_classical(
    A,
    u0,
    T=T,
    na=na,
    R=R,
    order=order,
    point=point,
    b=b
)

---

(B) Quantum Trotter Evolution

Hamiltonian splitting:
$$
H = H_1 + H_2
$$
Trotter approximation:
$$
e^{-iHt}
\approx
\left(e^{-iH_1 \Delta t} e^{-iH_2 \Delta t}\right)^{N_t}
$$

from unitarylab.library import schro_trotter
u, qc = schro(
    u0=u0,
    H1=H1,
    H2=H2,
    Nt=Nt,
    na=na,
    R=R,
    order=order,
    point=point,
    b=b,
    theta=theta
)

---

7. Output Generation

Agent must return:

• Numerical solution $u(x,T)$
• Grid information $(x, \Delta x)$
• Hamiltonian structure
• (Optional) quantum circuit $qc$
• Visualization-ready data

---

Execution Workflow (Agent Pipeline)

1. Parse input parameters
2. Construct spatial grid
3. Build derivative operator $A_0$
4. Form system $A = aA_0$
5. Detect unitary structure
   • If unitary → direct evolution
   • Else → Schrödingerization
6. Construct Hamiltonian
7. Select solver:
   • Classical / Quantum
8. Perform time evolution
9. Recover solution
10. Output results

---

Accuracy & Stability Considerations

• Grid size $n_x$ controls spatial accuracy
• Time step $\Delta t$ affects Trotter error
• Scheme choice:
  • Central → high accuracy, unitary
  • Upwind → stable but dissipative
• Schrödingerization introduces:
  • auxiliary dimension cost
  • smoother but higher-dimensional system

---

Usage Interface

algo = AdvectionEquationAlgorithm()
result = algo.run(params)

---

Skill Trigger Conditions

Activate when:

• PDE contains:
  • first-order spatial derivative
  • transport/advection structure
• Keywords:
  • “advection equation”
  • “transport equation”
  • “hyperbolic PDE”
  • “Schrödingerization + PDE”

---

Summary

This skill implements a hybrid computational framework:
$$
\text{Advection PDE}
;\rightarrow;
\text{Discretization}
;\rightarrow;
\text{(Optional) Schrödingerization}
;\rightarrow;
\text{Unitary Evolution}
$$
It bridges:

• Classical numerical PDE methods
• Quantum Hamiltonian simulation