Time-Symmetric Quantum Method Efficiently Calculates Molecular and Material Energy States.

A novel time-symmetric approach efficiently solves the Hamiltonian eigenvalue problem, simultaneously determining both ground and highest excited states without prior eigenstate computation. This non-variational method avoids error accumulation and barren plateaus, utilising linear combinations of unitaries and Monte Carlo techniques to compute energy bandwidths and identify topological states in molecular and condensed matter systems.

The determination of a system’s energy eigenstates, a fundamental task in quantum mechanics, typically proceeds via iterative methods that build up solutions sequentially, potentially accumulating error and facing limitations in accessing higher energy states. Researchers are now presenting a novel approach that exploits the inherent time symmetry present in quantum systems, allowing for the simultaneous determination of all eigenstates without reliance on prior calculations. This new algorithm, detailed in their paper, leverages the coherence between forward and backward time evolution to directly identify any eigenstate and its corresponding energy. The work is the result of a collaboration between Shijie Wei, Bozhi Wang and Guilu Long from the Beijing Academy of Quantum Information Sciences, Jingwei Wen from China Mobile (Suzhou) Software Technology Company Limited, Xiaogang Li from Peking University, Peijie Chang from Tsinghua University, and Franco Nori from RIKEN, and is titled “A Time-Symmetric Quantum Algorithm for Direct Eigenstate Determination”.

Researchers present a novel approach to solving the Hamiltonian eigenvalue problem, leveraging time symmetry to achieve computational efficiencies beyond those of conventional methods. The Hamiltonian operator, central to quantum mechanics, describes the total energy of a system, and its eigenvalues represent the possible energy levels. Existing computational techniques typically rely on a fixed direction of time evolution, requiring iterative calculations to determine each energy level sequentially. This new methodology, however, exploits the inherent coherence existing between forward and backward time evolution, allowing simultaneous determination of both the ground state – the lowest energy level – and the highest excited states. It also facilitates direct identification of arbitrary eigenstates, bypassing the need for iterative computation and the associated accumulation of errors.

The algorithm’s non-variational nature distinguishes it from many existing techniques. Variational methods refine an approximate solution iteratively, seeking to minimise an energy functional. This new approach, instead, directly targets specific eigenstates, ensuring convergence and mitigating the ‘barren plateau’ problem. This problem, a significant obstacle in quantum computation, arises when the gradients used to optimise parameters become vanishingly small, hindering the learning process. Researchers implement the required non-unitary evolution – a process that does not preserve probability, essential for exploring time-symmetric dynamics – using both linear combination of unitaries (LCU), a standard technique for constructing quantum circuits from simpler operations, and Monte Carlo methods, demonstrating flexibility in implementation.

Applications of this method extend across diverse areas of physics and materials science. The team successfully computes energy bandwidths and spectra for various molecular systems, accurately characterising their properties. Furthermore, the algorithm identifies topological states in condensed matter systems, specifically within the Kane-Mele and Su-Schrieffer-Heeger models. The Kane-Mele model describes a two-dimensional topological insulator, a material that conducts electricity on its surface but acts as an insulator in its interior, exhibiting a quantum spin Hall effect where electrons with opposite spins travel in opposite directions. The Su-Schrieffer-Heeger model, a one-dimensional system, exhibits topological edge states, localised electronic states appearing at the boundaries of the material.

Current research focuses on enhancing the algorithm’s scalability to larger and more complex systems. Investigations into the performance of different implementations, particularly concerning resource requirements and computational cost, are ongoing. Exploration of hybrid quantum-classical approaches, combining the strengths of both computational paradigms, presents a promising avenue for further development.

This time-symmetric method offers a significant advancement in computational physics and materials science, efficiently calculating quantum system eigenstates and spectra. Researchers are now investigating its application to dynamical properties and time-dependent phenomena in both molecular and condensed matter systems, suggesting a broad scope for future research.