Generalized Quantum Functional Expansion (QFE) Solver for Stochastic Differential Equations- Paper Implementation Project

[limitless-100]

A) Introduction

The research paper to be implemented is "Generic and Scalable Differential Equation Solver for Quantum Scientific Computing" by Jinhwan Sul and Yan Wang (Georgia Institute of Technology). It proposes a Quantum Functional Expansion (QFE) framework, which encodes the functional expansion of a solution into a quantum state and solves its time evolution using Variational Quantum Simulation (VQS). The paper introduces a parallel Pauli operation strategy, significantly improving the scalability of VQS by reducing the number of required quantum circuits from O(M² + 4ⁿM) to O(M²).

The QFE framework enables the solution of a wide range of differential equations, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). My main priority is SDEs, and other types of equations will be added in the future.

B) Technical Approach

• Quantum Functional Encoding: Encode differential equation solutions into quantum states using functional expansion.
• Variational Quantum Simulation (VQS): Solve time evolution using parameterized ansatz circuits and McLachlan’s variational principle.
• Parallel Pauli Operation Strategy: Reduce circuit count by implementing parallel execution of Pauli operations.
• Equation Implementation: Solve ODEs, PDEs, and SDEs using Galerkin and collocation methods, validating accuracy and scalability.

C) A High-Level Example Demonstrating Key Concepts

To illustrate the Quantum Functional Expansion (QFE) framework, consider solving a simple ordinary differential equation (ODE):

$$ \frac{du(t)}{dt} = -\lambda u(t) $$

where ( \lambda ) is a constant.

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Classical Approach

The analytical solution is:

$$ u(t) = u_0 e^{-\lambda t} $$

If solving numerically, discretization methods like Euler’s method or Runge-Kutta can be used.

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Quantum Functional Expansion Approach

1. Quantum Functional Encoding:

Expand ( u(t) ) in terms of basis functions:

$$ u(t) \approx \sum_i \mu_i(t) \phi_i $$

Encode expansion coefficients ( \mu_i(t) ) as a quantum state:

$$ | \psi(t) \rangle = \sum_i \mu_i(t) | i \rangle $$

2. Variational Quantum Simulation (VQS):

• Map the time evolution equation to ansatz parameters ( \theta(t) ).
• Use McLachlan’s variational principle to update parameters dynamically.

3. Parallel Pauli Operations for Efficient Computation:

• Instead of running ( O(M^2 + 4^n M) ) circuits, use a single quantum circuit to compute multiple Pauli operations in parallel.

4. Extracting the Solution:

• Use quantum amplitude estimation to compute expectation values.
• Convert back to classical form by measuring qubit states.

[TomerGoldfriend]

Sounds interesting @limitless-100 !

Please note that we accept high-quality implementations to our repository and will be glad to accept a contribution that meets our standards.

Feel free to reach out to the community for any questions.

Good luck!