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Green's Functions from Sample-based Krylov Quantum Diagonalization: An Impurity Solver for Dynamical Mean-Field Theory

This paper extends the sample-based Krylov quantum diagonalization method to compute single-particle Green's functions, enabling efficient impurity solvers for dynamical mean-field theory on near-term quantum hardware by reconstructing spectral functions from short-time evolutions in reduced particle-number sectors.

Original authors: Jay Patel, Chakradhar Rangi, Ka-Ming Tam

Published 2026-09-09

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CC BY 4.0

Original authors: Jay Patel, Chakradhar Rangi, Ka-Ming Tam

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 authors. For technical accuracy, refer to the original paper. Read full disclaimer

Matter that behaves in strange and stubborn ways, like the superconductors that carry electricity without resistance or the magnetic materials that power our hard drives, often defies the simple rules we use to explain ordinary solids. In these materials, electrons do not move independently; they push and pull on one another with a force so strong that the behavior of one electron is inextricably linked to the behavior of its neighbors. To understand how these materials work, scientists must calculate the "Green's function," a mathematical tool that acts like a map of how an electron moves through this crowded, chaotic environment. This map reveals the material's energy levels and how it responds to light or electricity, but calculating it for large systems is a monumental task. The number of possible arrangements for the electrons grows so explosively with the size of the system that even the most powerful supercomputers quickly run out of memory, forcing researchers to study only tiny, simplified versions of the materials they actually care about.

A team of researchers has now proposed a new way to tackle this problem using quantum computers, not by trying to simulate the entire system at once, but by taking a series of clever, small snapshots. In a study focused on a model system that mimics the behavior of electrons in a metal, the researchers adapted a technique called sample-based Krylov quantum diagonalization. Instead of asking a quantum computer to hold the entire, massive state of the system in its memory, they asked it to evolve the system for very short periods of time and then measure the results. By repeating this process many times and collecting the most common outcomes, they were able to reconstruct the full map of electron movement without ever needing to store the impossible amount of data required by traditional methods. The approach is designed to work on the noisy, imperfect quantum computers available today, which cannot yet run the long, complex circuits required by older methods.

The researchers tested their method on a specific model known as the single-impurity Anderson model, which describes a single atom interacting with a sea of electrons. They set up a simulation where this central atom was connected to a chain of twelve sites, representing the surrounding environment. In this setup, the central atom carries a strong repulsive force that prevents two electrons from occupying the same spot, a key feature that drives the transition between a metal that conducts electricity and an insulator that does not. The team used a classical computer to simulate the quantum device, allowing them to compare their new method against the exact, perfect solution for this system. They found that by sampling the quantum evolution, they could reconstruct the electron's energy map with high accuracy using only a tiny fraction of the total possible states. For a system with twelve sites, the method reduced the required computational space by three to four orders of magnitude, meaning it needed to handle only a few thousand states instead of the hundreds of millions that would be required for a full calculation.

The study revealed that the method works best when the system is in a state that is neither completely simple nor completely chaotic. When the interaction between electrons is very weak, the system behaves like a calm sea of independent particles, and when it is very strong, the electrons lock into a rigid pattern. In both of these extreme cases, the method found the answer easily. However, in the middle ground, where the material is on the verge of changing from a metal to an insulator, the electrons are in a complex, fluctuating state. Here, the researchers found that the method required a slightly larger number of samples to capture the full picture, but it still succeeded where traditional methods would have failed. The results showed that the spectral features, which are the distinct peaks and valleys in the energy map that define the material's properties, were reproduced faithfully across a wide range of interaction strengths.

One of the most significant findings was that the method did not require the quantum computer to perform complex operations that involve extra "helper" bits, known as ancillas, which are often a source of errors in current devices. Instead, the quantum device only needed to prepare the state, let it evolve for a short time, and then measure the positions of the electrons. All the heavy lifting of connecting these measurements to the final answer was done by a classical computer. This division of labor keeps the quantum circuit shallow and simple, making it a realistic candidate for the hardware available in the near future. The researchers noted that while the method is currently a simulation, the way the required number of samples grows as the system gets larger is encouraging. As the chain of sites grew from eight to twelve, the fraction of the total possible states needed to get an accurate answer actually decreased, suggesting that the method could scale to even larger systems that are currently out of reach.

The work also addressed a common question about which starting point is best for these calculations. The team tested two different ways of setting up the initial state of the system: one based on a simple, non-interacting picture and another based on a more complex, self-consistent picture. They found that the simpler starting point was just as effective as the more complex one for this specific task, which simplifies the setup for future experiments. Furthermore, they demonstrated that the method could handle the mathematical "branching" required to calculate how electrons are added to or removed from the system, a necessary step for building the full energy map. By carefully selecting which sampled states to keep and which to discard, they ensured that the final result respected the fundamental rules of physics, such as the conservation of probability.

This approach offers a promising path forward for dynamical mean-field theory, a powerful framework used to study strongly correlated materials. In this framework, a complex lattice of atoms is mapped onto a single impurity problem, which must be solved repeatedly to find the properties of the whole material. The bottleneck has always been the impurity solver, the step that calculates the electron behavior for that single site. If this solver can be run on a quantum computer with a larger number of bath sites than is currently possible, it would allow scientists to model materials with much greater precision. The researchers suggest that their method could serve as a new impurity solver, enabling the study of larger and more realistic systems than ever before. While the study was limited to a specific model and did not yet include the full self-consistent loop of the theory, the results provide a strong proof of concept that such a calculation is feasible.

The study also looked at how the method performs under different conditions, such as varying the strength of the electron repulsion. They found that the number of samples needed to reach a specific level of accuracy was not a simple straight line; it peaked at intermediate interaction strengths before dropping again at very high strengths. This behavior makes sense physically, as the intermediate region is where the system is most complex and requires the most information to describe. Despite this peak, the total number of samples required remained a small fraction of the total possible states, even for the most difficult cases. The researchers concluded that the method is robust and that the favorable scaling observed in their simulations suggests it could be a practical tool for quantum hardware in the near term.

Ultimately, this work demonstrates that we do not need to wait for perfect, error-free quantum computers to begin solving some of the hardest problems in condensed matter physics. By embracing the probabilistic nature of quantum measurement and using clever sampling techniques, researchers can extract the essential information about how electrons move in complex materials. The method transforms a problem that was previously thought to be intractable into one that is manageable, opening the door to a new generation of simulations that could help us understand and design the advanced materials of the future. The findings suggest that with the right algorithm, even the noisy quantum computers of today can begin to shed light on the deepest mysteries of strongly correlated matter.

Technical Summary: Green's Functions from Sample-based Krylov Quantum Diagonalization

Problem Statement
Strongly correlated materials, such as transition metal oxides and cuprates, require methods that treat Coulomb repulsion and kinetic energy on equal footing, rendering single-particle approximations inadequate. While Dynamical Mean-Field Theory (DMFT) maps lattice problems onto a single interacting impurity in a self-consistent bath, the "impurity solver" step remains a bottleneck. Classical solvers face severe limitations: Exact Diagonalization (ED) is restricted to small clusters due to exponential Hilbert space growth; Quantum Monte Carlo (QMC) suffers from the fermionic sign problem and requires ill-conditioned analytic continuation; and tensor networks struggle with entanglement growth in higher dimensions or at long times. Consequently, current DMFT implementations often rely on bath discretizations with very few sites, limiting resolution and accuracy, particularly near metal-insulator transitions.

Quantum algorithms offer a potential bypass, but existing approaches for calculating Green's functions often require ancilla qubits (e.g., for Hadamard tests), deep circuits, or complex optimization prone to barren plateaus. Furthermore, it was unclear whether sample-based methods, which work well for ground states, could be extended to the full Lehmann representation of spectral functions, which involves poles spanning two different particle-number sectors ( $N \pm 1$ ).

Methodology
The authors generalize the Sample-based Krylov Quantum Diagonalization (SKQD) method to compute single-particle Green's functions. The approach retains the shallow-circuit, ancilla-free character of SKQD by shifting the computational burden of matrix element evaluation to classical post-processing.

Krylov Subspace Construction: The method constructs unitary Krylov subspaces in the $N$ , $N+1$ , and $N-1$ particle-number sectors. Starting from a reference state $|\phi_{\text{ref}}\rangle$ (either a non-interacting bare state or a Hartree-Fock state), the system undergoes short-time real-time evolution $e^{-iH\Delta t}$ for $K$ steps.
Sampling: Instead of measuring overlaps via ancilla-assisted Hadamard tests, the quantum device prepares these evolved states and measures them in the computational basis. The resulting bitstrings are sampled $M$ times to estimate the probability weights $|\langle b | \psi_k \rangle|^2$ .
Classical Reconstruction:
- Ground State: The sampled bitstrings define a truncated subspace. The Hamiltonian is projected onto this subspace and diagonalized classically to obtain the ground state energy $E_0$ and eigenvector $|\Psi_0\rangle$ .
- Green's Function Assembly: To satisfy sum rules, the response subspaces for the addition ( $N+1$ ) and removal ( $N-1$ ) branches are augmented. The authors construct states $|\chi_+\rangle = c^\dagger_{0\uparrow}|\Psi_0\rangle$ and $|\chi_-\rangle = c_{0\uparrow}|\Psi_0\rangle$ within these augmented spaces.
- Spectral Function: The Green's functions $G_+(\omega)$ and $G_-(\omega)$ are reconstructed using a Lanczos continued fraction expansion driven by the sampled basis, avoiding the need for the full Lehmann representation summation. The spectral function is then $A(\omega) = -\frac{1}{\pi} \text{Im}[G_+(\omega) + G_-(\omega)]$ .

Key Contributions

Extension to Green's Functions: The paper demonstrates that SKQD can be generalized from ground-state energy calculations to the evaluation of frequency-dependent single-particle Green's functions.
Ancilla-Free Implementation: The method eliminates the need for ancilla qubits and complex interference measurements, relying solely on state preparation and computational basis sampling, which is more compatible with near-term quantum hardware (NISQ).
Basis Compression: The study identifies that the Hartree-Fock (HF) basis and the non-interacting bare basis are superior starting points for SKQD compared to real-space site bases. In these rotated bases, the spectral weight is concentrated in a small fraction of the Hilbert space, allowing for effective truncation.
Augmentation Strategy: A specific protocol is introduced to augment the Krylov subspaces for the $N \pm 1$ sectors to ensure the conservation of spectral weight (sum rules) without requiring the full Hilbert space.

Results
The method was applied to the particle-hole-symmetric single-impurity Anderson model (SIAM) in a chain geometry, representing a DMFT bath.

Accuracy: For system sizes up to $L=12$ (12 sites, 24 spin-orbitals, Hilbert space dimension $\approx 8.5 \times 10^5$ ), the SKQD method successfully reproduced the impurity spectral function $A(\omega)$ across interaction strengths $U=2, 4, 6$ , covering the metal-insulator transition.
Subspace Efficiency: The method achieved high accuracy (error $< 0.05$ ) using a subspace comprising only a small fraction of the full Hilbert space. For $L=12$ , the effective subspace size was reduced by three to four orders of magnitude compared to the full dimension.
Scaling Trends:
- The required pool size (number of sampled configurations) is non-monotonic with respect to interaction strength $U$ , peaking at intermediate coupling ( $U=4$ ) where both coherent quasiparticle and incoherent Hubbard bands coexist.
- Crucially, the ratio of the required pool size to the total Hilbert space dimension decreases as the system size increases ( $L=8, 10, 12$ ). This suggests that the method becomes relatively more efficient for larger bath discretizations, a necessary condition for practical DMFT applications.
Basis Dependence: The HF and bare orbital bases significantly outperformed real-space site bases. The latter failed to concentrate spectral weight, requiring subspace sizes comparable to the full Hilbert space to achieve convergence.

Significance and Claims
The authors claim that this work establishes a viable pathway for using near-term quantum hardware as impurity solvers within DMFT loops. By demonstrating that accurate spectral functions can be recovered from a small fraction of the Hilbert space using shallow, ancilla-free circuits, the method potentially allows for DMFT calculations with a larger number of bath sites than currently practical with classical solvers.

The paper remains modest regarding scalability, noting that while the trend of decreasing relative subspace size is encouraging, a controlled finite-size scaling analysis was not performed due to classical simulation limits. The authors identify the principal challenge for future implementation as the circuit depth required to diagonalize the quadratic part of the Hamiltonian (the "bare" or HF basis) for more complex lattice models with long-range interactions, though they argue this cost remains modest for the single-impurity case. The work positions SKQD as a complementary, sampling-based alternative to existing quantum algorithms for real-frequency spectral functions.

Technical Summary: Green's Functions from Sample-based Krylov Quantum Diagonalization

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