1-D Seismic Inversion Using Quantum Computing Simulation with Sources and Loss Functions

A quantum-classical hybrid simulation for 1-D elastic wave propagation in heterogeneous media. The code combines a classical finite-difference (leapfrog) PDE solver with a Hamiltonian-based quantum time-evolution circuit built using Qiskit 2.x, following the framework of Schade et al.

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Acknowledgements & Literature

This work is built upon and references the following repositories and publications:

Reference	Link
Schade, M. et al. (2024) — Quantum Wave Equation Solver	github.com/malteschade/Quantum-Wave-Equation-Solver
Schade, M. et al. (2024) — Quantum Wave Simulation with Sources and Loss Functions	github.com/malteschade/Quantum-Wave-Simulation-with-Sources-and-Loss-Functions
Schade, M. et al. (2023) — arXiv preprint	arXiv:2312.14747

Author

Najlah Rupaidah (NIM 1227030025) Geophysics Specialization, Department of Physics, Faculty of Science and Technology Universitas Islam Negeri Sunan Gunung Djati, Bandung, Indonesia

Co-author: bex

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Features

• Classical 1-D wave solver — finite-difference leapfrog scheme supporting heterogeneous density and elastic modulus (Dirichlet / Neumann BCs).
• Hamiltonian-based quantum circuit — matches the structure of Schade et al. Fig. A1:
  • State Preparation (R_Y, X, Z gates)
  • Time Evolution (exp(-iHt) unitary block)
  • Observable (Pauli basis rotation via H gates)
  • Measurement (all qubits)
• Quantum reconstruction — simulates exact (statevector), shot-noise, and hardware-noise quantum reconstructions.
• Publication-quality plots — forward simulation multiplot, energy, overlap, error, and circuit diagram styled to match the reference.
• Data export — JSON config, pickle results, and Excel workbook with multiple sheets.

Requirements

Package	Version
Python	>= 3.10
Qiskit	>= 2.0
qiskit-aer	>= 0.15
numpy	>= 1.23
scipy	>= 1.9
matplotlib	>= 3.6
openpyxl	>= 3.1
pylatexenc	>= 2.10

Install all dependencies:

pip install qiskit qiskit-aer numpy scipy matplotlib openpyxl pylatexenc

Project Structure

101225_malte_schade_integration/
├── main.py                  # Main simulation script
├── README.md                # This file
├── penjelasan_kode.txt      # Simple code explanation (Bahasa Indonesia)
├── figures/
│   ├── forward_sim.png      # 2x3 multiplot: medium properties + wave snapshots
│   ├── energy.png           # Total energy vs time
│   ├── overlap.png          # Quantum state overlap with initial condition
│   ├── error.png            # Relative L2 reconstruction error
│   └── circuit.png          # Quantum circuit diagram (Fig. A1 style)
└── data/
    └── <timestamp>/
        ├── configs.json     # Experiment parameters
        ├── data.pkl         # Simulation results (pickle)
        └── results.xlsx     # Simulation results (Excel, 6 sheets)

Usage

Run the full simulation pipeline:

python main.py

This executes 7 steps:

1. Classical wave simulation (leapfrog finite-difference)
2. Build quantum circuit (Group 0, Index 10)
3. Save experiment data (JSON + pickle)
4. Save Excel workbook (6 sheets)
5. Generate forward simulation multiplot
6. Generate analysis plots (energy, overlap, error)
7. Generate circuit diagram

Quantum Circuit Structure

The circuit follows the Hamiltonian simulation approach from Schade et al.:

         ┌──────────┐ ░ ┌──────────────┐ ░ ┌───┐ ░ ┌─┐
  q_0: ──┤ R_Y(-θ)  ├─░─┤              ├─░─┤ H ├─░─┤M├───
         └──────────┘ ░ │              │ ░ └───┘ ░ └╥┘┌─┐
  q_1: ───────────────░─┤  exp(-iHt)   ├─░───────░──╫─┤M├─
         ┌───┐┌───┐  ░ │              │ ░ ┌───┐ ░  ║ └╥┘┌─┐
  q_2: ──┤ X ├┤ Z ├──░─┤              ├─░─┤ H ├─░──╫──╫─┤M├
         └───┘└───┘  ░ │              │ ░ └───┘ ░  ║  ║ └╥┘┌─┐
  q_3: ───────────────░─┤              ├─░───────░──╫──╫──╫─┤M├
                      ░ └──────────────┘ ░       ░  ║  ║  ║ └╥┘
  c: 4/═════════════════════════════════════════════╩══╩══╩══╩═

Section	Purpose
State Preparation	Encode initial conditions (displacement + velocity) into qubit amplitudes via R_Y, X, Z gates
Time Evolution	Apply Hamiltonian exp(-iHt) as a unitary block, where H is derived from the 1-D elastic wave equation
Observable	Rotate into Pauli measurement basis (XZXZ) using Hadamard gates
Measurement	Measure all qubits in computational basis

Excel Output

The results.xlsx workbook contains 6 sheets:

Sheet	Contents
Configuration	All experiment parameters
Medium	Grid position, density (rho), elastic modulus (mu)
TimeSeries	Time step, time, energy
Overlaps	Time, squared quantum overlap
WaveFields	Full displacement field at every time step
CircuitParams	Qubit count, Hilbert dimension, evolution time, observable

License

This project is for academic and research purposes. Please cite the original references when using this code.