#hhlalgorithm

The Quantum IBM IBM scientists are using quantum computing to solve the chaotic, shifting differential equations, one of science and engineering's hardest problems. These advances are significant for computational science. From river rapids to stock market fluctuations, these principles underpin practically every field that deals with change over time.

HHL Algorithm

A new quantum method for solving Ax=b linear equations is Harrow-Hassidim-Lloyd (HHL). Quantum computing can transform processor performance by using quantum information and entanglement. Sometimes the HHL algorithm is exponentially faster than standard methods.

HHL Algorithm Function

HHL transforms the linear system into a quantum state. The following steps are necessary:

The approach first determines matrix A's eigenvalues using Quantum Phase Estimation (QPE).

Quantum gates calculate the inverse of these eigenvalues.

Finally, solution reconstruction provides a quantum version of the solution vector ‘x’. This approach efficiently manipulates complex quantum linear systems.

HHL iteratively for non-Hermitian problems

The researchers apply HHL to non-Hermitian systems with a smart modification. The non-Hermitian complex scaled Hamiltonian matrix (H) is extended to form a bigger Hermitian matrix (A). The HHL algorithm, designed for Hermitian operators, can be used. Additionally, an iterative strategy is described to solve the eigenvalue problem for resonant states by reformulating the Schrödinger equation to make the final eigenvector a fixed point.

The self-consistent iterative HHL quantum algorithm uses an initial trial wave function and energy, improves them, and solves until a desired accuracy is obtained. Real and imaginary components of complex wave functions are handled independently. A projection method ensures orthogonality and prevents convergence to previously established eigenstates when finding consecutive eigenvectors.

Eigenvector Continuation and Complex Scaling

Eigenvector continuation (EC) and complex scaling (CSM) for nuclear resonances are needed by the HHL algorithm:

Complex Scaling (CSM): This mathematical method allows resonances, naturally unstable quantum states, to be treated and precisely computed as quasi-bound states. Resonant states and continuum spectrum are separated by complex scaling transformation, which rotates coordinates.

EC is a powerful method that reduces the dimension of the eigenvalue problem, enhancing computational performance. To do this, “training eigenvectors” shrink the subspace. After deriving training eigenvectors from bound states, EC with complex scaling estimates resonant states. This method dramatically reduces the computational dimension and qubit count for quantum computation, improving numerical stability and efficiency for many-body systems and parameter-dependent problems.

These “emulators,” which always use eigenvector continuation, mark a major step towards faster and more computationally feasible reaction result prediction methods. This is crucial for reactor physics, nuclear astrophysics, and basic nuclear theory.

Meaning of Nuclear Resonances

The interaction of two nuclei, such as the decay of two alpha (α) particles, causes nuclear resonances. Safety-critical reactor models, nuclear structural physics, and nuclear astrophysics require quasi-stationary states. Due to their intrinsic fading (non-Hermitian) nature, quantum algorithms and linear algebraic approaches struggle with their computing challenges.

Bound states are stable and Hermitian, but resonances require more intricate treatment. This is because their energy spectrum has an imaginary decay chance element. Traditional computational methods like diagonalisation and complex scaling are resource-intensive and don't scale well as the system size expands.

Alpha-Alpha System validation

To test their new technique, researchers utilised quantum simulations to establish the resonant state of the alpha-alpha ($\alpha-\alpha$) system. Simulations of the two alpha particles' collision employed a simple Gaussian potential. A G-wave resonant state with an energy between 11.8079 and 1.8085i MeV was estimated for this system using the R-matrix technique.

Iterative HHL, EC, and CSM correctly estimated the resonant energy with a complex scaling angle of 20 degrees. The quantum simulation used 1-bit precision and yielded results that matched traditional approaches. After five rounds, the second eigenvalue, which represents the resonant energy, converged to 11.8211 - 1.8107i MeV, demonstrating the algorithm's computing efficiency and reliability. This proves the new non-Hermitian quantum eigensolver's reliability.

Significance of This Development

Quantum Computation for Non-Hermitian Systems

Quantum algorithms usually assume probability-conserving Hermitian operators. IHHL, CSM, and EC have helped researchers solve complex fading quantum states problems that classical quantum algorithms struggle with.

Efficiency of Resources

Eigenvector continuation reduces dimensionality and processing resources. Since it requires fewer qubits and shallower circuits, the approach is more compatible with near-term quantum hardware despite noise and qubit restrictions.

Interdisciplinary Relevance