University of Bordeaux builds stable quantum Krylov method

Researchers from the University of Bordeaux, CNRS, and LOMA have developed orthogonal quantum Krylov diagonalization (OQKD), a new quantum computing method that reformulates the classical Lanczos recursion directly at the operator level. Unlike existing quantum Krylov diagonalization methods, OQKD eliminates the need for regularization of overlapping matrices and promises improved numerical stability and accuracy in quantum computations. The researchers report that they achieved this by representing the Lanczos vector as a polynomial transformation of the Hamiltonian. This stability increase comes without any computational penalty. OQKD uses block encoding and generalized quantum signal processing to achieve the same asymptotic query complexity as established Chebyshev-based QKD methods. Numerical simulations confirm the classical Lanczos convergence and numerical stability of the proposed method. Based on the OQKD framework, researchers introduced a restarted state preparation protocol that replaces a single high-order polynomial transform with a set of fixed low-order transforms to maintain affordable block encoding success probability while maintaining comparable convergence. These results demonstrate that the restarted protocol is a promising state preparation strategy for quantum phase estimation.

New quantum computing techniques directly mirror the stability of classical algorithms and may enable more reliable computations. This replication is also extended to convergence properties, providing a more predictable and stable path to finding the low-energy spectrum of quantum many-body Hamiltonians. Numerical simulations of the J1-J2 Heisenberg model confirm the classical Lanczos convergence and numerical stability of the proposed method, and the scaling of the measurement complexity is established analytically. This suggests that its applications in quantum computing may be further explored.

Quantum computing has increasingly focused on subspace diagonalization techniques, particularly quantum Krylov diagonalization (QKD), in the pursuit of efficient ways to determine the energy levels of complex quantum systems. These approaches aim to circumvent the exponential scaling challenges of classical computation by leveraging the principles of quantum mechanics to generate and manipulate the relevant computational space. Based on the OQKD framework, the team introduced a restarted state preparation protocol that replaces a single high-order polynomial transform with a set of fixed low-order transforms, maintaining comparable convergence and affordable block encoding success probabilities. These advances establish OQKD as a robust quantum analog of the classic Lanczos algorithm and provide a promising path toward more reliable and scalable quantum simulations.

Existing approaches are promising for computing quantum many-body Hamiltonians, but require regularization of the overlap matrix, which limits numerical stability and accuracy. The team aims to solve this problem using the newly developed orthogonal quantum Krylov diagonalization (OQKD) framework. The central problem is that many QKD methods approximate Krylov vectors without guaranteeing orthogonality, leading to instability and reduced accuracy. Kirby et al. proposed a quantum Krylov construction based on block encoding and quantization of Chebyshev polynomials, but the researchers point out that “orthogonality between Krylov vectors is still not maintained.” This lack of orthogonality reflects the challenges faced by classical numerical linear algebra, where ill-conditioned conditions necessitate orthogonalization techniques such as the Lanczos algorithm. This maintains comparable convergence and affordable block encoding success probability.

This means that even though this method handles numerical errors better, it still requires the same amount of quantum operations. Based on the OQKD framework, researchers introduced a rebooted state preparation protocol that replaces complex high-order polynomial transforms with a sequence of simpler fixed-order transforms to maintain comparable convergence while maintaining affordable block encoding success probabilities. These results establish OQKD as an orthogonal quantum analog of the classical Lanczos algorithm and identify the restarted protocol as a promising state preparation strategy for quantum phase estimation.

This validation confirms the classical Lanczos convergence and numerical stability of the proposed method, and analytically establishes the measurement complexity scaling. Based on the OQKD framework, researchers introduced a restarted state preparation protocol that replaces a single high-order polynomial transform with a set of fixed low-order transforms, maintaining comparable convergence and affordable block encoding success probability. This important improvement eliminates the need for regularization of overlapping matrices, potentially streamlining computation and accelerating convergence.

OQKD directly reformulates the classical Lanczos recursion to improve stability, but implementing the single advanced polynomial transformation required to prepare the state can be resource-intensive. To avoid this, the team introduced a restarted state preparation protocol based on the OQKD framework. It replaces a single high-order polynomial transform with a set of fixed low-order transforms, maintaining comparable convergence and affordable block encoding success probabilities. This relaunched protocol focuses on maintaining affordable block encoding success probabilities, a key element in translating theoretical quantum algorithms to real-world applications. This approach allows for efficient Krylov state preparation without sacrificing the accuracy obtained through OQKD orthogonalization techniques and eliminates the need for regularization of overlapping matrices. The development of this protocol is important because it addresses a key challenge in quantum computing: balancing algorithmic efficiency with the constraints of available quantum resources.

Quantum phase estimation (QPE) aims for accurate eigenvalue estimation through controlled time evolution, but is often hampered by what researchers call orthogonality catastrophes, where overlapping initial states affect the probability of success. Variational quantum algorithms (VQA) designed for short-term processors suffer from optimization challenges such as sterile plateaus, noise sensitivity, and dependence on selected Ansatz representability. However, OQKD takes a different approach and focuses on building orthogonal Krylov subspaces to improve numerical stability. This eliminates the need for regularization of overlapping matrices and is an important step towards more reliable quantum computation. Based on the OQKD framework, the researchers introduced a restarted state preparation protocol by replacing complex polynomial transformations with low-order sequences. This maintains an affordable block encoding success probability while maintaining comparable convergence. These results establish OQKD as an orthogonal quantum analog of the classical Lanczos algorithm and identify the restarted protocol as a promising state preparation strategy for quantum phase estimation.