⚛️ quantum physics

Exploring Variational Regularization of Krylov Subspaces

This paper introduces a variational Krylov-like framework that iteratively optimizes parametrized states to approximate Hamiltonian eigenstates while penalizing linear dependencies, demonstrating that Hamiltonian-inspired ansätze effectively balance convergence and conditioning in quantum computing contexts where traditional orthogonalization is unavailable.

Original author: Alexandre Perrin, Jérémie Messud, Sébastien Burdin, Matthieu Saubanère

Published 2026-11-09

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CC BY 4.0

Original author: Alexandre Perrin, Jérémie Messud, Sébastien Burdin, Matthieu Saubanère

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to understand the behavior of matter at its most fundamental level, scientists often turn to the mathematics of energy and motion. At the heart of this effort lies the challenge of finding the lowest energy states of complex systems, which correspond to the most stable configurations of atoms and electrons. To solve these problems, researchers rely on powerful mathematical tools known as subspace methods. Imagine trying to find the deepest point in a vast, foggy valley; instead of mapping the entire landscape, these methods take a series of strategic steps, each one bringing the explorer closer to the bottom by focusing on the most promising directions. One such method, known as the Krylov approach, is particularly efficient because it naturally amplifies the most important features of a system while filtering out the noise. However, this efficiency comes with a hidden trap: as the method takes more steps, the directions it explores begin to crowd together, becoming so similar that the mathematical calculations become unstable and prone to error. This problem is especially acute for the emerging field of quantum computing, where the physical nature of the machine prevents scientists from easily correcting these overlaps as they would on a classical computer.

A team of researchers from the University of Bordeaux and TotalEnergies has investigated a new way to navigate this difficulty. They developed a flexible framework that mimics the successful steps of the traditional method but adds a safety mechanism to keep the exploration directions distinct. Instead of blindly following a rigid mathematical recipe that inevitably leads to a crowded, unstable set of directions, their approach uses a guided search. At each step, the algorithm generates a new state and asks two questions: does this new state connect strongly to the previous ones in a way that reveals new energy information, and is it distinct enough from what has already been found to avoid mathematical confusion? By balancing these two goals, the researchers created a system that can find the lowest energy states of a model system with high accuracy while keeping the underlying calculations stable.

The researchers tested their idea on a specific model known as the Anderson impurity model, which describes a single atom interacting with a surrounding sea of electrons. This model is a standard benchmark for testing how well quantum algorithms can handle complex interactions. They compared their new variational method against the traditional, exact mathematical approaches. The results revealed a clear trade-off. When the researchers applied a very gentle penalty to prevent the states from becoming too similar, the method maintained the rapid convergence of the traditional approach while keeping the mathematical stability high. However, if they made the penalty too strong to ensure the states were perfectly distinct, the method lost its ability to find the correct energy levels quickly. The study showed that the key to success was not just the penalty itself, but the structure of the search tool they used. When the search tool was designed to mirror the specific physical interactions of the system, it could reproduce the rapid progress of the traditional method. In contrast, when they used a generic, shallow search tool that did not account for the system's physics, the method remained stable but failed to find the deep energy states, regardless of how well-conditioned the math was.

A critical finding of the work is that the individual states generated by this new method do not need to match the exact steps of the traditional algorithm to be useful. In the traditional approach, each step is a precise, rigid calculation. In this new variational approach, the individual steps can vary and differ from the traditional path, yet the collection of all steps together still captures the same essential information about the system's energy. The researchers found that even though the specific directions chosen by their method differed from the traditional ones, the overall space they explored eventually overlapped with the traditional solution with an accuracy of nearly one hundred percent. This suggests that the goal of such algorithms should not be to force every single step to look exactly like the classical version, but rather to ensure that the entire collection of steps covers the necessary ground without collapsing into mathematical chaos.

The study also highlighted the limitations of current quantum hardware and the specific challenges of running these algorithms on real machines. While the simulations performed by the researchers were exact and free from the noise that affects physical quantum computers, the method they proposed requires a significant number of measurements to function. On a real device, the cost of gathering the necessary data to check the stability and energy of each new state would be substantial. The researchers propose that the future of this field may lie in a hybrid approach, where short, rigid sequences of steps are combined with periodic variational adjustments. This would allow the system to benefit from the speed of traditional methods while using the flexibility of the new approach to reset and prevent the mathematical instability from building up. Ultimately, this work provides a roadmap for how to build more robust quantum algorithms that can solve complex physical problems without getting lost in the mathematical noise that has long plagued the field.

Technical Summary: Exploring Variational Regularization of Krylov Subspaces

Problem Statement
Krylov subspace methods, particularly the Lanczos algorithm, are highly efficient for approximating extremal eigenstates of large Hamiltonians. However, they suffer from a fundamental numerical issue: the basis vectors generated by repeated applications of the Hamiltonian ( $H^n|\phi_0\rangle$ ) become increasingly parallel, leading to a rapidly growing condition number in the overlap matrix $S$ . This ill-conditioning renders the generalized eigenvalue problem numerically unstable.

In classical computing, this is mitigated by on-the-fly orthogonalization (e.g., Gram-Schmidt or reorthogonalization). In quantum computing, however, direct state access is unavailable; states exist only as quantum circuits. Consequently, classical orthogonalization procedures cannot be directly applied. Current quantum subspace methods (e.g., Quantum Subspace Expansion) typically estimate overlap and Hamiltonian matrices and perform orthogonalization as a classical post-processing step. This approach is numerically unstable when the prepared states are nearly linearly dependent, as small eigenvalues of the overlap matrix amplify shot noise and sampling errors. The paper identifies a trade-off: directions most relevant for improving variational energy often introduce linear dependencies, making it difficult to simultaneously achieve rapid convergence and good numerical conditioning.

Methodology
The authors propose a Variational Quantum Krylov Diagonalization (VQKD) framework. Instead of generating basis states via repeated Hamiltonian applications, they iteratively construct a sequence of parametrized states $|\phi_n(\theta_n)\rangle$ using a variational ansatz $A(\theta_n)$ acting on the previous state:
$|\phi_n(\theta_n)\rangle \propto A(\theta_n) |\phi_{n-1}\rangle$

To balance convergence and stability, the parameters $\theta_n$ are optimized by minimizing a cost function $C_n(\theta_n)$ :
$C_n(\theta_n) = -|H^\perp_{n,n-1}(\theta_n)|^2 \left( 1 - \lambda \sum_{i<n} |\langle \phi_n(\theta_n) | \phi^*_i \rangle|^2 \right)$

Convergence Term: The first factor, $-|H^\perp_{n,n-1}|^2$ , maximizes the effective Hamiltonian coupling between the new state and the previous orthonormalized basis direction, mimicking the Lanczos recurrence.
Regularization Term: The second factor penalizes the overlap between the new state and all previously generated states. The parameter $\lambda$ controls the strength of this penalty, acting as a soft constraint against near-linear dependencies.

The authors benchmark this framework on a finite Anderson impurity model in a star geometry (4 sites, 8 qubits). They compare three distinct ansatz families:

Hamiltonian-inspired ( $H(\theta)$ ): A non-unitary ansatz constructed as a linear combination of Hamiltonian terms. This ansatz contains the exact Krylov step ( $|\phi_n\rangle \propto H|\phi_{n-1}\rangle$ ) in its manifold when $\lambda=0$ .
Variational Hamiltonian Ansatz ( $VHA(\theta)$ ): A unitary, Trotterized evolution based on the Hamiltonian decomposition.
Hardware-Efficient Ansatz ( $HEA(\theta)$ ): A shallow, problem-agnostic circuit with no explicit relation to the Hamiltonian structure.

Key Results

Role of Regularization ( $\lambda$ ): A weak regularization ( $\lambda \approx 10^{-2}$ ) is sufficient to prevent the exponential growth of the condition number seen in exact Krylov methods, keeping $\kappa(S)$ several orders of magnitude lower. Increasing $\lambda$ further degrades energy convergence without providing significant additional stabilization. The presence of the penalty, rather than its magnitude, is the critical factor in avoiding the collapse of the Krylov basis.
Role of Ansatz Structure: The expressivity and structure of the ansatz are the dominant factors in performance.
- The Hamiltonian-inspired ansatz ( $H(\theta)$ ) achieves the best compromise, closely following exact Krylov convergence in early iterations while maintaining a well-conditioned basis.
- The VHA shows intermediate performance but fails to reproduce the rapid initial convergence of Krylov methods efficiently; without regularization, it becomes unstable due to linear dependence.
- The HEA remains extremely well-conditioned but exhibits poor energy convergence, failing to capture low-energy directions despite its stability. This demonstrates that good conditioning alone is insufficient for a useful subspace.
Subspace vs. State Comparison: Analysis of the overlap between the variational subspace and a fixed Lanczos reference subspace reveals that while individual optimized states may not match Lanczos vectors (especially at later iterations), the cumulative variational subspace captures the relevant Lanczos subspace with high accuracy (mean squared principal-angle overlap $\approx 0.999$ ).

Significance and Claims
The paper claims to provide a controlled numerical study of the trade-off between convergence, conditioning, and ansatz structure in quantum subspace methods. The primary contribution is the demonstration that variational regularization can effectively control the conditioning of non-orthogonal subspaces during state preparation, rather than relying solely on post-measurement regularization.

The authors emphasize that their work is not a proposal for a generally scalable eigensolver but a proof-of-concept for a "regularized Krylov-like subspace." They conclude that:

Ansatz structure is paramount: A variational manifold aligned with the Hamiltonian's operator content is essential for reproducing Krylov-like convergence.
Weak regularization is optimal: A small penalty is sufficient to stabilize the basis without sacrificing the physical relevance of the generated directions.
Subspace fidelity over state fidelity: The method should be viewed as constructing a useful subspace rather than an exact variational realization of the Lanczos recursion.

The study remains limited to exact statevector simulations without sampling noise. The authors note that the measurement overhead ( $O(D^2 N_{opt})$ ) would be substantial on actual hardware, suggesting that future work should explore hybrid approaches combining recursive Lanczos blocks with periodic variational compression.

Technical Summary: Exploring Variational Regularization of Krylov Subspaces

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