Variational ansatz-based quantum simulation of imaginary time evolution

Abstract: Imaginary time evolution is a powerful tool for studying quantum systems. While it is possible to simulate with a classical computer, the time and memory requirements generally scale exponentially with the system size. Conversely, quantum computers can efficiently simulate quantum systems, but not non-unitary imaginary time evolution. We propose a variational algorithm for simulating imaginary time evolution on a hybrid quantum computer. We use this algorithm to find the ground-state energy of many-particle systems; specifically molecular hydrogen and lithium hydride, finding the ground state with high probability. Our method can also be applied to general optimisation problems and quantum machine learning. As our algorithm is hybrid, suitable for error mitigation and can exploit shallow quantum circuits, it can be implemented with current quantum computers.

• The paper introduces a hybrid quantum-classical algorithm leveraging a variational ansatz to simulate non-unitary imaginary time evolution.
• It adapts McLachlan's variational principle to update ansatz parameters, enabling shallow circuit implementations suitable for current quantum hardware.
• Numerical results on molecules like H₂ and LiH demonstrate accurate ground state energy estimation, paving the way for broader quantum applications.

Overview of Variational Ansatz-Based Quantum Simulation of Imaginary Time Evolution

Imaginary time evolution is a mathematical technique frequently employed in exploring quantum systems, especially for determining ground state energies and simulating thermal properties. While computationally daunting on classical systems due to exponential scaling with system size, quantum computers offer a promising alternative. However, they traditionally struggle with non-unitary processes such as imaginary time evolution. The paper discusses a novel hybrid quantum-classical variational algorithm designed to address these challenges by simulating imaginary time evolution efficiently.

Algorithm Description

The proposed algorithm leverages a variational principle to approximate the non-unitary imaginary time evolution in quantum systems. It reformulates the problem such that it can be addressed using a combination of quantum computations for state representations and classical computations for parameter optimization. Utilizing variational ansatz states, the algorithm adapts McLachlan's variational principle to update parameters, ensuring that the system approximates the imaginary time evolution while remaining operable within the constraints of unitary operations. This hybrid approach allows for mitigating errors and using shallow quantum circuits, making it compatible with current quantum hardware capabilities.

Numerical Performance

The authors demonstrate the application of their algorithm by estimating the ground state energies of molecular systems such as Hydrogen (H₂) and Lithium Hydride (LiH). The numerical simulations show that the approach accurately identifies ground states, illustrating its potential to handle quantum chemistry problems that are intractable by classical means. The reported results suggest the variational imaginary time approach not only circumvents some limitations of classical optimizations like gradient descent—which is prone to getting trapped in local minima—but also offers robustness in finding ground states efficiently.

Implications and Future Directions

The presented algorithm holds promise for addressing a range of complex optimization problems beyond quantum chemistry, including general optimization and quantum machine learning applications. Using quantum computers to simulate imaginary time evolution could significantly enhance the efficiency of solving optimization problems, especially when integrated with the quantum approximate optimization algorithm (QAOA). Moreover, the algorithm facilitates the preparation of thermal states, which has further potential applications in simulating statistical mechanics and applying machine learning techniques conditioned on Gibbs distributions.

Practically, this method could underpin quantum solutions in chemistry, materials science, and beyond, particularly where insight into ground states or thermal properties is crucial. The scalability of the approach to more significant problems hinges on further developments in quantum hardware and the refinement of ans\"atze suited to various application domains.

Theoretically, designing more powerful and expansive ans\"atze to represent a broader class of states within the imaginary time evolution spectrum remains an open research challenge. Solutions may involve exploring new classes of trial states that provide more flexibility or approximating nontrivial Hamiltonians with reduced computational resources.

Conclusion

This work constitutes a step forward in using quantum computing to simulate processes that are hard to tackle with classical systems. While demonstrating encouraging results with current quantum technology, the continued development of this field holds the potential to unlock new capabilities across various scientific and computational domains. Future research should aim to refine ansatz design and expansion, improve noise resilience, and generalize the algorithm's applicability to a more extensive array of quantum and optimization challenges.