One-Step Run Example Command

python ./scripts/algorithm.py

Skill: 1D Backward Heat Equation Solver via Schrödingerization

Skill Objective

Enable the agent to stably solve ill-posed PDEs, especially the 1D backward heat equation, by:

• Transforming exponential growth into controlled evolution
• Applying Schrödingerization
• Supporting both classical and quantum-inspired solvers

---

Mathematical Model

Backward Heat Equation

$$
\partial_t u = H u, \quad u(0) = u_0
$$

• Typically $H = -\Delta$（or discrete Laplacian）
• Ill-posedness:
  • Exponential increase in the frequency mode index
  • Extremely sensitive to initial values

---

Core Capabilities

1. Ill-posedness Handling

Agent must:

• Detect exponential growth instability
• Apply warped phase transformation

Introduce:
$$
w(t,p) = e^{-p} u(t)
$$
Transform equation:
$$
\frac{d}{dt} w = -H \partial_p w
$$

---

2. Spatial Discretization

Construct grid:
$$
x_i = i\Delta x, \quad \Delta x = \frac{L}{N}
$$
Construct Laplacian operator (via finite difference):

A = -a * CDiff(
    N=Nx,
    dx=dx,
    order=2,
    scheme=scheme,
    boundary=bd
).get_matrix()

---

3. Schrödingerization Mapping

Transform:
$$
\frac{du}{dt} = A u
\quad \Rightarrow \quad
\frac{d\psi}{dt} = -iH\psi
$$
Steps:

1. Auxiliary lifting (introduce $p$)
2. Fourier transform in $p$
3. Construct Hamiltonian

$$
H = D \otimes H_1 + I \otimes H_2
$$

---

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

4. Classical Solver (Matrix Exponentiation)

Use Schrödingerization-based classical solver:

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

Characteristics:

• Stable compared to direct backward evolution
• Avoids catastrophic amplification

---

5. Quantum-Inspired Solver (Trotter Method)

Hamiltonian splitting:

$$
H = H_1 + H_2
$$

Lie-Trotter decomposition:

$$
e^{-iHt}
\approx
\left(e^{-iH_1 \Delta t} e^{-iH_2 \Delta t}\right)^{N_t}
$$

func1, func2 = (a * TDiff(nx, dx, 2, scheme=scheme, boundary=bd)).data()

H1 = func1(dt / R)
H2 = func2(dt)

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

---

6. Block Encoding (Extension Capability)

• Reserved for quantum linear algebra methods
• Currently fallback to classical solver

u = backHeatEquationAlgorithm._solve_block(eq)

---

7. Boundary Condition Handling

Agent supports:

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

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

• Neumann:
  $$
  \partial_x u = 0
  $$

Implementation:

if bd == "periodic":
    x = np.arange(0, L, dx)
elif bd == "neumann":
    x = np.arange(0, L+dx, dx)

---

8. Solution Recovery

After evolution in transformed space:
$$
u(t) = e^{p} w(t,p)
$$

w0 = np.exp(-p) * u0
u_t = np.exp(p) * w_t

---

9. Visualization Capability

Solution Plot

fig, ax = plt.subplots()
ax.plot(x, u)
ax.set_title("Backward Heat Solution")
fig.savefig("solution.svg")

Quantum Circuit Visualization

qc.draw(filename="circuit_full.svg")
H1.decompose().draw(filename="circuit_H1.svg")
H2.decompose().draw(filename="circuit_H2.svg")

---

Execution Workflow (Agent Pipeline)

1. Parse input parameters
2. Identify ill-posed structure
3. Apply warped transformation
4. Construct spatial operator $A$
5. Perform Schrödingerization
6. Choose solver:
   • Classical (default)
   • Trotter (quantum-inspired)
7. Run time evolution
8. Recover physical solution
9. Generate visualization
10. Output results

---

Output Specification

Agent must return:

• Solution $u(x,t)$
• Grid data $(x, \Delta x)$
• Stability-related parameters
• (Optional) quantum circuit $qc$
• Visualization files

---

Stability & Numerical Considerations

• Backward heat equation is severely ill-posed
• Schrödingerization acts as:
  • implicit regularization
  • structure-preserving transformation

Accuracy depends on:

• grid size $N$
• auxiliary dimension $n_a$
• Trotter step $N_t$
• smoothness of lifting function

---

Skill Trigger Conditions

Activate when:

• PDE contains:
  • diffusion operator with reversed time
• Keywords:
  • “backward heat equation”
  • “ill-posed PDE”
  • “unstable evolution”
  • “regularization via Schrödingerization”

---

Summary

This skill provides a robust computational framework for ill-posed PDEs:
$$
\text{Backward Heat}
;\rightarrow;
\text{Warped Transformation}
;\rightarrow;
\text{Schrödingerization}
;\rightarrow;
\text{Unitary Evolution}
$$
It enables:

• Stable numerical simulation
• Quantum-compatible formulation
• Integration of classical and quantum methods