#hdesalgorithm

ColibriTD

The  IBM Qiskit Functions Catalogue now includes ColibriTD's quantum-classical hybrid partial differential equation (PDE) solution QUICK-PDE. Based on IBM's H-DES (Hybrid Differential Equation Solver) technique, QUICK-PDE lets researchers and developers solve domain-specific multiphysics PDEs via cloud access to utility-scale quantum devices.

QUICK-PDE

QUICK-PDE was developed by quantum-powered multiphysics simulation company ColibriTD. IBM Qiskit Functions Catalogue lists it as an application function. QUICK-PDE is part of ColibriTD's QUICK platform.

The function lets researchers, developers, and simulation engineers solve multiphysics partial differential equations using IBM quantum computers in the cloud. For domain-specific partial differential equation solutions, it simplifies and makes development easier.

It works

ColibriTD's unique H-DES algorithm underpins QUICK-PDE. To solve differential equations, trial solutions are encoded as linear combinations of orthogonal functions, commonly Chebyshev polynomials. The function is encoded using $2^n$ Chebyshev polynomials, where $n$ is the number of qubits.

Variable Quantum Circuit (VQC) angles parametrise orthogonal functions.

The function is embedded in an ansatz-created state and evaluated by observable combinations that allow its assessment at any time.

Loss functions encode differential equations. By altering the VQC angles in a hybrid loop, trial solutions are brought closer to real solutions until a good result is achieved.

A solution can use many optimisers. You can chain optimisers to follow a gradient by using a global optimiser like “CMAES” (from the cma Python package) and then a fine-tuned optimiser like “SLSQP” from Scipy for the Material Deformation scenario.

Noise reduction is built into the algorithm. The noise learner strategy can mitigate noise during CMA optimisation by stacking identical circuits and assessing identical observables on various qubits within a larger circuit, reducing the number of shots needed.

Different qubits can encode each variable's function. The function may choose appropriate default values, but users can change them. The ansatz depth per function can also be changed. Adjustable variables include the number of shots needed per circuit. Since there are several optimisation processes, the shots parameter is a list whose length matches the number of optimisers used. Computational Fluid Dynamics and Material Deformation have preset shot values.

Users can choose “RANDOM” or“PHYSICALLY_INFORMED” for VQC angle initialisation. “PHYSICALLY_INFORMED” is the default and often works, but “RANDOM” can be used in other cases.

Use cases and multiphysics capabilities

QUICK-PDE solves complex multiphysics problems. We cover two key use cases:

Computational fluid dynamics

The inviscid Burgers' equation and fundamental CFD model are issues. This equation simulates non-viscous fluid flow and shockwave propagation for automotive and aerospace applications.

The Navier-Stokes equations for fluid motion have an inviscid Burgers' equation at zero viscosity ($\nu = 0$). $fracpartial upartial t + ufracpartial upartial x = 0$1117, where $u(x,t)$ is the fluid speed field

When $a$ and $b$ are random constants and $u(t=0, x) = axe + b$, the current implementation only allows linear functions as initial conditions. Change these constants to see how they affect the solution.

The CFD differential equation arguments are on a fixed grid: space ($x$) between 0 and 0.95 with 0.2375 step size and time ($t$) with 30 sample points. The dynamics of new reactive fluids for heat transfer in tiny modular reactors can be studied using QUICK-PDE.

MD: Material Deformation

Second is Material Deformation (MD), which studies 1D mechanical deformation in a hypoelastic regime like a tensile test. Simulation of material stress is crucial for manufacturing and materials research.

Problem: a bar with one end dragged and one fixed. This system of equations includes a strain function ($\sigma$) and a stress function ($u$).

A surface stress boundary condition ($t$) represents the labour needed to stretch the bar in this use case.

MD differential equation arguments use a fixed grid ($x$) between 0 and 1 with a 0.04 step size.

Future versions of QUICK-PDE will include the H-DES algorithm to handle higher-dimensional problems and additional physics domains like electromagnetic simulations and heat transport.

Usability, Accessibility:

IBM Quantum Premium, Dedicated Service, and Flex Plan users can use QUICK-PDE.

The function must be requested by users.

The quantum workflow is simplified by application functions like QUICK-PDE. They use classical inputs (such physical parameters) and return domain-familiar classical outputs to make quantum approaches easier to integrate into present operations without needing to build a quantum pipeline.

This allows domain experts, data scientists, and business developers to study challenges that require HPC resources or are difficult to solve.

The function supports “job,” “session,” and “batch” execution modes. The default mode is “job”. A dictionary contains input parameters.

Use_case (“CFD” or “MD”) and physical_parameters specific to the use case (e.g., a, b for CFD; t, K, n, b, epsilon_0, sigma_0 for MD) are crucial. Users can adjust nb_qubits, depth, optimiser, shots, optimizer_options, initialisation, backend_name, and mode using optional arguments.

The function's output is a dictionary of sample points for each variable and its computed values. For instance, the CFD scenario provides u(t,x) function values and t and x samples. In MD, x samples and function values for u(x) and sigma(x) are presented. The resulting array's structure matches the variables' alphabetic sample points.

Benchmarks for Inviscid Burgers' equation and Hypoelastic 1D tensile test show statistics like qubit usage, initialisation method, error ($\approx 10^{-2}$), duration, and runtime utilisation.

A tutorial on modelling a flowing non-viscous fluid with QUICK-PDE covers setting up starting conditions, adjusting quantum hardware parameters, performing the function, and applying the results. The manual provides MD and CFD examples.