Qutrit Harrow-Hassidim-Lloyd Algorithm Achieves Reduced Qudit Count for Hydrogen Molecule Potential Energy Calculations

The quest for quantum advantage in solving complex calculations drives innovation in quantum algorithms, and researchers continually explore how different quantum systems can enhance computational power. Tushti Patel from IIT Tirupati and V. S. Prasannaa now present a significant advance in this field, extending the well-known Harrow-Hassidim-Lloyd (HHL) algorithm to utilise qutrits, quantum bits with three levels instead of the standard two. This work demonstrates that a qutrit-based HHL implementation requires fewer quantum resources, specifically qudits, than its qubit counterpart while maintaining a comparable number of computational gates, potentially paving the way for more efficient quantum simulations and calculations, and offering a promising step towards practical quantum computation. The team validates their approach with simple matrices and applies it to a challenging problem in computational chemistry, calculating the potential energy curve of a hydrogen molecule, showcasing the algorithm’s potential for real-world applications.

Qutrit HHL Algorithm Solves Quantum Chemistry Problem

Scientists have expanded the capabilities of the Harrow-Hassidim-Lloyd (HHL) algorithm by adapting it to utilize qutrits, quantum systems possessing three distinct levels instead of two. This involved designing a complete circuit and developing a program for implementing the qutrit HHL algorithm, successfully demonstrating its feasibility and opening the door to potential advantages over qubit-based approaches. Initial tests using simple matrices confirmed the algorithm’s accuracy. The team then applied the qutrit HHL algorithm to a challenging problem in quantum chemistry, specifically calculating the potential energy curve for a hydrogen molecule using a split valence basis, demonstrating its ability to tackle complex molecular systems.

A key aspect of the research involved a direct comparison of the resources required for both qubit and qutrit HHL implementations. Results demonstrate that, for a fixed level of precision, the qutrit HHL circuit requires fewer qudits than its qubit counterpart. Furthermore, the number of two-qudit gates remains comparable between the two implementations, suggesting that the reduction in qudit count does not come at the expense of increased circuit complexity. This finding indicates that qutrit HHL could potentially offer a more efficient way to solve linear systems relevant to quantum chemistry and other fields, establishing a foundation for exploring the benefits of higher-dimensional qudits in quantum algorithms.

Qutrit HHL Algorithm For Molecular Simulation

This work extends the Harrow-Hassidim-Lloyd (HHL) algorithm to utilize qutrits, quantum units with three possible states. Researchers designed a circuit for this qutrit-based HHL algorithm and implemented it in a quantum computing framework, successfully demonstrating its functionality through calculations involving simple matrices and quantum chemical calculations on a model hydrogen molecule. These calculations generated a potential energy curve with high precision, achieving correlation energies within 0. 02 percent of those obtained using established classical methods like CISD and LCCSD. Results demonstrate that, for a given level of precision, a qutrit HHL implementation requires fewer qudits than its qubit counterpart, specifically by a factor of log2(3). While the number of two-qudit gates required remains comparable, the qutrit version shows reductions in gate requirements for certain modules, such as the Quantum Fourier Transform, by as much as 50 percent. The team acknowledges that realizing the benefits of this approach depends on the availability of quantum hardware capable of reliably creating and manipulating high-quality qutrits, and future work will likely focus on optimizing the algorithm and exploring its application to more complex quantum chemical problems.

Ternary Qudits Enhance Linear System Solutions

Scientists are exploring the use of qudits, quantum digits with more than two levels, as an alternative to traditional qubits to improve the efficiency and accuracy of quantum algorithms. This research focuses on ternary qudits, possessing three levels, and their application to the Harrow-Hassidim-Lloyd (HHL) algorithm for solving linear systems of equations, a crucial task in many scientific fields. The work centers on applying these concepts to solving linear systems arising in quantum chemistry, specifically in calculating molecular energies. The Harrow-Hassidim-Lloyd algorithm offers a potential exponential speedup over classical algorithms for solving linear systems under certain conditions.

This research investigates how representing numbers and matrices in base-3, the ternary system, can be leveraged for implementing algorithms using ternary qudits. The team applies this approach to solving the linear systems that arise in quantum chemistry calculations, such as finding the ground state energy of molecules, using the Linearized Coupled Cluster (LCC) method. Statevector simulation is employed to test and compare the performance of the algorithms. The research presents evidence suggesting that ternary qudits can offer advantages over qubits in terms of resource requirements for solving linear systems. Ternary representations can lead to more compact representations of data and potentially reduce the number of quantum gates needed, improving the accuracy of the HHL algorithm, particularly when dealing with challenging mathematical problems. The authors demonstrate how to apply their ternary qudit-based algorithms to solve linear systems arising in quantum chemistry calculations, specifically in the context of LCC, positioning the quantum algorithms as potential alternatives to classical methods.