Penalty-free Quantum Algorithm Finds Energy Eigenstates, Leveraging Exponential Hilbert Space Growth

Determining the energy states of a system, a fundamental challenge in physics and chemistry, often demands immense computational power. Nannan Ma, Heng Dai from the National University of Singapore, and Jiangbin Gong, alongside their colleagues, now present a new quantum algorithm that efficiently finds these energy states without relying on approximations or classical computing assistance. The team’s fully quantum approach avoids the penalties typically associated with such calculations, offering a significant advantage over existing methods. This development expands the computational tools available to tackle problems currently beyond the reach of even the most powerful classical computers, potentially accelerating progress in materials science, drug discovery, and fundamental physics.

Quantum Simulation and Ground State Finding

This collection of research papers explores methods for simulating quantum systems and finding their ground states using quantum computers. A central challenge lies in the exponential growth of computational complexity as the system size increases, demanding innovative approaches. Researchers are particularly focused on variational quantum algorithms (VQAs), which combine classical optimization with quantum computation, making them suitable for near-term, noisy quantum devices. A key technique involves imaginary time evolution (ITE), a process analogous to running the system forward in time with an imaginary time parameter.

This effectively filters out excited states, leaving only the ground state. Overcoming limitations of current quantum hardware, such as limited qubit counts and coherence times, is a significant focus, with researchers developing techniques to compress quantum circuits, decompose complex gates, and design effective parameterized quantum circuits (ansatze) to accurately represent ground states. Several papers demonstrate successful implementations of quantum imaginary time evolution (QITE) for finding ground states, extending the technique to simulate finite-temperature properties and open quantum systems. Machine learning techniques are also being integrated to improve the efficiency of quantum simulations and design better ansatze.

A promising new direction involves state-based quantum simulation, which represents the quantum state using classical parameters, potentially reducing the complexity of quantum circuits and improving scalability. The work by Alipour and Ojanen presents a state-based quantum simulation of ITE, offering a potentially scalable approach by shifting computational burden from the quantum computer to a classical computer. This reduces the need for complex quantum circuits, making it more compatible with near-term devices, although it introduces challenges related to classical memory requirements and maintaining accuracy. Overall, this research represents a vibrant area aimed at harnessing quantum computers to solve complex problems in physics, chemistry, and materials science, with a strong emphasis on practical algorithms for current and near-future hardware.

Eigenstate Preparation via Imaginary Time Evolution

Researchers have developed a novel quantum algorithm for determining both ground and excited states of many-body Hamiltonians, avoiding the need for penalty functions or variational steps. The algorithm begins by decomposing the target Hamiltonian into two components, each with readily solvable eigenstates. Imaginary time evolution (ITE) is then implemented by rescaling and shifting the eigenvalues of these components, allowing for the construction of a density matrix through stochastic sampling of projectors, directly preparing the ground state on the quantum circuit. To isolate excited states, the team employs a state-based simulation protocol, building upon the established ground state projector.

This technique enables the sequential determination of excited states through stochastic sampling and ITE, without requiring complex ansatz structures. Crucially, expectation values of the Hamiltonian components can be directly measured from the quantum circuits, providing a means to determine the corresponding eigenvalues. The method effectively represents the Hamiltonian as an equivalent mixed-state density matrix, transforming the eigenvalue problem into a density matrix manipulation task. By skillfully combining stochastic sampling with state-based simulation, this study delivers a powerful new approach to finding multiple eigenstates of complex Hamiltonians, circumventing limitations inherent in existing quantum algorithms. The algorithm demonstrates the ability to accurately determine both ground and excited states, offering a significant advancement in the field of quantum simulation.

Efficient Excited State Discovery via Iterative Evolution

This research presents a novel algorithm for finding both ground and excited states of quantum systems, achieving progress without relying on computationally expensive techniques like penalty functions or hybrid classical-quantum approaches. The core of the method involves an iterative process, termed imaginary time evolution (ITE), which efficiently progresses towards the system’s lowest energy states. The accuracy of the discretization of time steps is determined by a value proportional to the square of the total evolution time. To find excited states, the team employs a strategy of “lifting” the ground state by adding a projector to the system’s Hamiltonian.

This allows the ITE process to converge on the first excited state, and the process can be repeated to find subsequent excited states. A key innovation is the implementation of this lifting process using a state-based simulation, which introduces an ancillary system and a control system to realize the evolution within the system’s Hilbert space. The algorithm, initialized with a specific state, iteratively samples projectors and updates the system’s state, achieving convergence towards the desired eigenstates. The team confirms that the probability of obtaining the correct state in each implementation step is consistently high, demonstrating the algorithm’s effectiveness and reliability.

Eigenstate Discovery Beyond Ground State Methods

This research demonstrates a fully quantum algorithm capable of finding both ground and excited states of a Hamiltonian, moving beyond methods limited to finding only the ground state. The algorithm achieves this through a combination of stochastic sampling and quantum circuit implementation, avoiding the need for complex unitary operations or assumptions about energy gaps. Results show the successful identification of low-lying eigenstates with high fidelity, and the calculated energy gaps closely match known values, indicating the algorithm’s accuracy. The authors acknowledge that the overall success rate of the algorithm currently relies on multiple post-selection steps, and improving this rate for larger-scale problems requires further investigation.

Future research directions include exploring the optimal balance between accuracy and success rate, and fully exploiting symmetries present in physical systems to reduce computational costs. Additionally, the team highlights the need to assess the algorithm’s performance on real quantum hardware and to develop strategies for mitigating errors arising from both quantum and classical noise. This work provides a concrete example of using quantum circuits to address classically intractable problems and has broad applicability to a range of quantum tasks, demonstrating the potential of quantum algorithms to advance scientific discovery.