Statistical complexity and quantum simulation of stochastic processes

Overview

This project explores the simulation and analysis of stochastic processes, focusing on classical and quantum complexity. It includes implementations for the Upset-Gambler process, a non-Markovian system, and demonstrates how quantum systems can efficiently model such processes. Two distinct simulation methods are implemented for running the Upset-Gambler process:

1. Encoding outputs onto four separate registers and measuring all at once.
2. Using a single register with repeated measurement and resetting after each step.

Key objectives:

• Compare classical complexity ($C_\mu$) and quantum complexity ($C_q$) for stochastic processes.
• Simulate processes using Qiskit and analyze results from both simulators and quantum hardware.
• Study hardware calibration data to understand its impact on quantum performance.

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Features

• Classical and quantum complexity calculation for stochastic processes.
• Two simulation methods for the Upset-Gambler process:
  • Method 1: Outputs encoded onto 4 separate registers.
  • Method 2: Outputs generated via single register resetting.
• Hardware execution of quantum circuits on IBM Quantum devices.
• Calibration data analysis, including coherence times, gate errors, and measurement errors.

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Directory Structure

stochastic-process-simulation/
├── complexity_analysis.py             # Classical and quantum complexity analysis
├── device_analysis.py                 # Analysis of IBM Quantum hardware calibration data
├── ibm_brisbane_calibrations_2025...  # Calibration data (CSV file)
├── m1_upset_gambler_simulator.py      # Method 1: Upset-Gambler simulation (Simulator)
├── m1_upset_gambler_QPU.py            # Method 1: Upset-Gambler simulation (Quantum Hardware)
├── m2_upset_gambler_simulator.py      # Method 2: Upset-Gambler simulation (Simulator)
├── m2_upset_gambler_QPU.py            # Method 2: Upset-Gambler simulation (Quantum Hardware)
├── requirements.txt                   # Python dependencies
└── README.md                          # Project documentation

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Simulation Methods

Method 1: Multiple Registers

• Outputs are encoded into four separate registers, representing four output symbols of the process.
• All registers are measured at the end of the circuit.
• Advantage: Provides a clean mapping of all symbols simultaneously.
• Example: Circuit creates and measures outputs like $[x_1, x_2, x_3, x_4]$.

Method 2: Single Register

• A single register is measured and reset repeatedly to produce each output symbol sequentially.
• Advantage: Reduces hardware qubit usage, making it efficient for noisy or limited devices.
• Example: Sequential generation of $[x_1], [x_2], [x_3], [x_4]$ over time.

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Getting Started

Prerequisites

• Python 3.9+
• Libraries:
  • qiskit, qiskit_aer, qiskit_ibm_runtime
  • matplotlib, numpy, pandas, seaborn
• Access to IBM Quantum:
  • Create an account at IBM Quantum.
  • Retrieve your API token and set it up in Qiskit.

Installation

Clone the repository and install the required dependencies:

git clone https://github.com/Gaurang-Belekar/Quantum-simulation-of-stochastic-processes.git
cd  Quantum-simulation-of-stochastic-processes
pip install -r requirements.txt

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Usage

1. Classical and Quantum Complexity Analysis

Run the script to compute and plot classical and quantum complexities for the Upset-Gambler process:

python complexity_analysis.py

2. Simulating the Upset-Gambler Process

Method 1 (Four Registers - Simulator):

python m1_upset_gambler_simulator.py

Method 2 (Single Register - Simulator):

python m2_upset_gambler_simulator.py

3. Running on IBM Quantum Hardware

Method 1 (Four Registers - Hardware):

python m1_upset_gambler_QPU.py

Method 2 (Single Register - Hardware):

python m2_upset_gambler_QPU.py

4. Hardware Calibration Data Analysis

Analyze IBM Quantum hardware calibration data for coherence times, gate errors, and other metrics:

python device_analysis.py

Note: Please add your IBM ACCESS TOKEN in the m1 and m2 QPU files to run it on actual hardware

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Results

Simulation Insights:

• Both methods successfully simulate the process, but Method 2 (single register) is more hardware-efficient.
• Classical vs Quantum Complexity: Quantum systems ($C_q$) are more memory-efficient compared to classical systems ($C_\mu$).

Hardware Observations:

• Noise and decoherence affect the fidelity of the quantum hardware results.
• Hardware calibration data highlights coherence times and error rates that influence performance.

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Contributing

Contributions are welcome! Please follow these steps:

1. Fork the repository.
2. Create a feature branch:

git checkout -b feature-name

3. Commit changes and open a pull request.

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License

This project is licensed under the MIT License. See the LICENSE file for details.

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Reference

[1] Felix C. Binder, Jayne Thompson and Mile Gu, Phys. Rev. Lett. 120, 240502 (2018)
[2] Farrokh Vatan and Colin Williams, Phys. Rev. A 69, 032315 (2004)
[3] Thomas J. Elliott, Chengran Yang, Felix C. Binder, Andrew J. P. Garner, Jayne Thompson and Mile Gu, Phys. Rev. Lett. 125, 260501 (2020).