Quantum Algorithms Achieve Ground State Preparation with Steps Scaling Linearly with Spins

Quantum computers promise to revolutionise fields from medicine to materials science, but preparing the correct initial state for a calculation remains a significant challenge. Mahum Pervez, Ariq Haqq, and Nathan A. McMahon, alongside Christian Arenz and colleagues at Arizona State University and Leiden University, now demonstrate a new approach to this problem using Riemannian gradient descent. Their work establishes a clear relationship between the structure of a quantum system’s Hamiltonian and the number of computational steps needed to accurately prepare its ground state, offering a pathway to more efficient quantum algorithms. By developing and testing approximations of this method, alongside practical device implementations, the team reveals a scalable strategy for achieving guaranteed convergence, bringing fault-tolerant quantum computation a step closer to reality.

Quantum Optimization Using Manifold Geometry This research focuses on improving variational quantum algorithms (VQAs) through optimization techniques inspired by differential geometry and manifold learning. The core idea involves representing quantum states and parameters as points on a manifold, a curved space, allowing the use of specialized optimization algorithms. Key concepts include Riemannian geometry, tangent spaces, and geodesics, which define movement and optimization directions within this curved space. The goal is to enhance the performance of VQAs, such as VQE and QAOA, for solving problems in quantum chemistry, materials science, and machine learning. Specific areas of improvement include more accurate gradient estimation, increased robustness against noise, scalability to larger problems, and better initialization of the optimization process. The research also explores the application of these techniques to quantum machine learning, including quantum neural networks and variational quantum circuits, and the development of quantum kernels to improve classical machine learning algorithms. Furthermore, the work addresses challenges in representing classical data in quantum states. The theoretical foundation relies heavily on differential geometry, Riemannian manifolds, optimization theory, graph theory, and linear algebra. Recognizing the limitations of current quantum hardware, the research also considers the impact of noise and errors, and explores error mitigation techniques and hardware-efficient algorithms. Emerging trends include adaptive optimization, second-order optimization, symmetry exploitation, data re-uploading, and analyzing the geometric structure of parameter spaces. The study establishes a relationship between the number of steps required for Riemannian gradient descent (RGD) to achieve a desired precision, the Hamiltonian’s spectral gap, the overlap between initial and ground states, and the target precision. Numerical analysis of RGD for Ising Hamiltonians reveals that the scaling depends on the Hamiltonian structure, with linear scaling for a one-dimensional chain with nearest-neighbor interactions and quadratic scaling for complete graphs. To accelerate convergence, the team engineered a stochastic variant of RGD that projects the Riemannian gradient into polynomial-sized subspaces, building upon a previously introduced stochastic protocol. This technique allows for efficient implementation on quantum hardware while maintaining convergence guarantees. Experiments compared the performance of this stochastic RGD with the exact RGD method across varying subspace dimensions, demonstrating the impact of subspace size on convergence speed. Rigorous analysis establishes an upper bound for the number of steps needed to prepare the ground state to a given precision. The resulting algorithms were implemented on quantum devices, utilizing both Trotterization and a stochastic drift-inspired protocol to realize the randomized RGD approach. This device implementation enables the exploration of ground state problems on small-scale quantum hardware, providing data for validating the theoretical findings and assessing practical feasibility. The research demonstrates that projecting the Riemannian gradient into polynomial-sized subspaces significantly speeds up convergence, offering a pathway to overcome challenges associated with exponential circuit depth or width requirements in adaptive quantum circuits.

Ground State Preparation Scales with Hamiltonian Structure Scientists have developed and analyzed randomized Riemannian gradient descent algorithms for preparing ground states on near-term quantum devices, achieving significant insights into the convergence behavior of these methods. The work demonstrates that the number of steps required to prepare a ground state depends on the Hamiltonian’s structure; for a one-dimensional Ising Hamiltonian with nearest-neighbor interactions, the algorithm scales linearly with the number of spins, while for all-to-all couplings, it scales quadratically. These findings suggest the algorithm can reflect the inherent difficulty of the ground-state problem itself. To enable efficient implementation, researchers approximated the Riemannian gradient through random projections into polynomial-sized subspaces spanned by Pauli operators, preserving convergence guarantees. Experiments revealed that increasing the size of these random subspaces accelerates convergence, closely tracking the full Riemannian gradient descent, though at the cost of increased circuit complexity, establishing a tunable trade-off between speed and resources. Benchmarking two quantum device implementations, first-order Trotterization and a quantum stochastic drift-inspired protocol, showed comparable asymptotic scaling, with the randomized implementation achieving lower circuit depth. Data from quantum device implementations on a two-qubit system closely matched numerical simulations, demonstrating the potential of the approach. However, for a three-qubit system, accumulated noise prevented convergence to a ground state, even with transpiler-level circuit optimization, highlighting the limitations of current hardware. Measurements confirm that the randomized implementation generates fewer unitaries and achieves a lower upper bound for circuit depth compared to the Trotter-based approach, as demonstrated for systems up to six qubits. Specifically, the randomized method requires approximately 10 5 unitaries for six qubits, while the Trotter method requires over 10 6 . Researchers demonstrated that the number of steps required for Riemannian gradient descent to achieve a desired precision depends on the Hamiltonian’s structure, specifically scaling linearly for a one-dimensional Ising model with nearest-neighbor interactions and quadratically for an Ising model with all-to-all coupling. This suggests the algorithm can reflect the inherent difficulty of a given ground-state problem. To enable efficient implementation on quantum computers, the team approximated the Riemannian gradient through random projections into polynomial-sized subspaces. Results indicate that increasing the size of these subspaces accelerates convergence, though at the cost of increased circuit complexity, establishing a tunable balance between speed and resource requirements. Two quantum device implementations, based on Trotterization and a quantum stochastic drift-inspired protocol, were developed and benchmarked, exhibiting comparable scaling but with the randomized implementation achieving lower circuit depth. While simulations of a two-qubit system closely matched numerical predictions, experiments with three qubits revealed that accumulated noise prevented convergence to the ground state, even with circuit optimization. This highlights both the potential and current limitations of implementing this approach on existing quantum hardware. The research establishes a foundation for Riemannian gradient-based quantum optimization, offering a pathway to implement each adaptive step with polynomial resources while preserving convergence guarantees, and suggests that continued development of randomized manifold-based techniques, alongside advances in quantum devices and compilation strategies, may further expand the feasibility of ground-state preparation. 👉 More information 🗞 Riemannian gradient descent-based quantum algorithms for ground state preparation with guarantees 🧠 ArXiv: https://arxiv.org/abs/2512.13401 Tags: