Diagonalization of large many-body Hamiltonians on a quantum processor (2407.14431v3)

Abstract: The estimation of low energies of many-body systems is a cornerstone of computational quantum sciences. Variational quantum algorithms can be used to prepare ground states on pre-fault-tolerant quantum processors, but their lack of convergence guarantees and impractical number of cost function estimations prevent systematic scaling of experiments to large systems. Alternatives to variational approaches are needed for large-scale experiments on pre-fault-tolerant devices. Here, we use a superconducting quantum processor to compute eigenenergies of quantum many-body systems on two-dimensional lattices of up to 56 sites, using the Krylov quantum diagonalization algorithm, an analog of the well-known classical diagonalization technique. We construct subspaces of the many-body Hilbert space using Trotterized unitary evolutions executed on the quantum processor, and classically diagonalize many-body interacting Hamiltonians within those subspaces. These experiments show that quantum diagonalization algorithms are poised to complement their classical counterpart at the foundation of computational methods for quantum systems.

• The paper introduces the Krylov Quantum Diagonalization (KQD) method implemented on quantum processors to estimate low-energy states of large many-body systems, mitigating limitations of existing algorithms like VQE and QPE.
• Implemented on the IBM Quantum Heron R1 processor for systems up to 56 sites, the KQD method showed convergence in simulated eigenenergies and resilience to noise, overcoming limitations often seen in QPE on noisy devices.
• This research establishes KQD as a viable pathway for scalable quantum simulation of complex many-body problems, particularly relevant for condensed matter physics, and sets the stage for future advancements towards fault-tolerant capabilities.

Overview of Diagonalization of Large Many-Body Hamiltonians on a Quantum Processor

The paper "Diagonalization of large many-body Hamiltonians on a quantum processor" presents significant advancements in computational quantum sciences, particularly in estimating the low-energy states of quantum many-body systems. The research addresses inherent challenges with existing variational quantum algorithms (VQAs) like the Variational Quantum Eigensolver (VQE) and Quantum Phase Estimation (QPE), which respectively suffer from scalability and precision issues with pre-fault-tolerant quantum devices.

Methodology and Implementation

The authors propose using the Krylov Quantum Diagonalization (KQD) method on superconducting quantum processors to compute eigenenergies for two-dimensional lattice systems encompassing up to 56 sites. This method is grounded in projecting the Hamiltonian into a Krylov subspace and subsequently solving for eigenvalues through classical diagonalization.

The focus of KQD is to construct subspaces using Trotterized unitary evolution. This approach mitigates some of the errors and inefficiencies associated with parametric optimization in VQE. By leveraging inherent U(1) symmetries, the authors optimize their circuit designs, maintaining quantum coherence notably with circuits leveraging symmetry-conserving initializations and simplifying their implementations.

Empirical Results and Insights

The experiments were conducted on the IBM Quantum Heron R1 processor using configurations of $k$-particle subspaces with $k$ ranging from 1 to 5, demonstrating convergence in the simulated eigenenergies for large-scale simulations. Notably, the results revealed resilience to noise, an obstacle seen in other methods like QPE when executed on NISQ devices. The convergence of energies displayed a dependency on chosen parameters such as time step size for time evolution, critical in minimizing algorithmic error.

Contributions and Implications

The paper's methodology closes a crucial gap between small-scale demonstrations feasible with VQE and the high-circuit-depth requirements of QPE on fault-tolerant devices. The Krylov methodology addressed herein offers provable convergence guarantees, essential as quantum technologies transition from NISQ to fault-tolerant capabilities.

The application of KQD particularly in the Heisenberg model reveals its robust potential in condensed matter physics, shedding light on eigenstate estimation challenges for large systems beyond classical computational limits. This research substantiates a pathway for scalable quantum simulations, establishing groundwork for quantum computational advantages in eigenstate problems within varying scientific disciplines.

Future Directions

The research sets the stage for future innovations in subspace diagonalization across broader classes of quantum systems. Challenges such as enhancing noise resiliency and optimizing for real-world Hamiltonians in chemistry and material science are promising areas for further exploration.

Given the demonstrated viability of KQD in addressing scalability and precision gaps, future investigations might explore its integration with emerging quantum error correction strategies to further expand the scope and accuracy of quantum simulation tasks.

In conclusion, this paper represents a critical step forward in computational quantum science, offering insights applicable to quantum system analytics and advancing the practical application of quantum computers in solving complex many-body problems.