⚛️ quantum physics

Imaginary-time-enhanced feedback-based quantum algorithms for universal ground-state preparation

This paper introduces an imaginary-time-enhanced feedback-based quantum algorithm (iFALQON) that overcomes the spectral degeneracy limitations of the standard FALQON method, thereby enabling robust and monotonic ground-state preparation for strongly correlated systems like the Fermi-Hubbard model.

Original authors: Thanh Nguyen Van Long, Lan Nguyen Tran, Le Bin Ho

Published 2026-10-23

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CC BY 4.0

Original authors: Thanh Nguyen Van Long, Lan Nguyen Tran, Le Bin Ho

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

In the quest to understand the building blocks of matter, scientists often turn to the strange and powerful world of quantum mechanics. At the heart of this field lies a fundamental challenge: finding the most stable, lowest-energy state of a complex system. This "ground state" is the quiet resting position of atoms and molecules, the configuration they settle into when they are not being disturbed. Knowing this state is crucial because it determines how materials behave, from the way electricity flows through a wire to how chemical reactions unfold. For decades, researchers have tried to use quantum computers to find these states, hoping these machines could solve problems that are too difficult for classical supercomputers. However, the path to the lowest energy is often blocked by a tricky feature of quantum systems: when different energy levels are identical, or "degenerate," the usual methods for guiding a system to its resting place can get stuck, unable to tell which direction to go.

A team of researchers has now proposed a new way to navigate these tricky landscapes, combining two different types of movement to guide a quantum system to its ground state. Their work focuses on a specific algorithm known as FALQON, which acts like a self-correcting compass. Instead of relying on a human or a classical computer to constantly adjust the settings, this algorithm listens to the system itself. It measures a specific signal generated by the quantum state and uses that information to automatically adjust the forces acting on the system, pushing it steadily toward lower and lower energy levels. This approach is attractive because it removes the need for complex, error-prone optimization steps that often plague other quantum methods. However, the researchers discovered that this self-correcting compass has a blind spot. When the system encounters a situation where multiple energy states are exactly the same—a common occurrence in certain materials at specific densities—the signal the algorithm relies on vanishes. Without this signal, the algorithm stops pushing, and the system gets trapped in a state that is not the true lowest energy, leaving the simulation stuck just above the finish line.

To solve this problem, the researchers introduced a hybrid technique they call iFALQON. They realized that while the standard algorithm acts like a smooth, continuous slide down a hill, it sometimes gets stuck on a flat plateau where the slope disappears. To get the system moving again, they inserted short, deliberate bursts of a different kind of evolution known as imaginary-time propagation. Think of this as a gentle, targeted nudge that acts like a filter, quietly draining away the extra energy that keeps the system stuck in higher states. By periodically inserting these nudges into the main feedback loop, the algorithm can break free from the flat plateaus where it would otherwise stall. The researchers tested this new method on a model of interacting particles arranged on grids of various sizes, ranging from small squares to larger three-by-three lattices. In their simulations, the standard algorithm failed to reach the true ground state in half-filled configurations, where the number of particles matched the number of available spots exactly. In these cases, the energy of the system would drop initially but then flatten out, leaving a significant gap between the result and the true answer.

When the team applied the new iFALQON method to the same difficult scenarios, the results changed dramatically. The system was no longer trapped on the plateau. Instead, the energy continued to drop smoothly and steadily until it reached the exact ground state, even in the most complex and highly correlated setups they tested. The method worked consistently across different lattice sizes and particle densities, including the challenging three-by-three grids where the old method had completely stalled. The researchers also compared their approach to other established techniques, such as variational methods that rely on classical computers to tune parameters. They found that their new method produced more accurate results with fewer errors, particularly in the difficult half-filled cases where other methods struggled with instability. The simulations showed that the energy error could be reduced to extremely low levels, far surpassing the performance of the traditional algorithms.

The success of this approach lies in its ability to combine the best of two worlds: the steady, self-correcting guidance of the feedback loop and the powerful, state-filtering capability of the imaginary-time steps. By mixing these two types of dynamics, the algorithm avoids the pitfalls of getting stuck in degenerate states while maintaining the efficiency of a fully quantum-driven process. The researchers emphasize that these findings come from noiseless computer simulations, meaning they represent a proof of concept for how the algorithm behaves in an ideal environment. While real-world quantum computers will face additional challenges like noise and measurement errors, the study demonstrates that the core logic of the method is sound. It offers a promising path forward for simulating strongly correlated quantum systems, which are essential for understanding phenomena like high-temperature superconductivity and exotic magnetic behaviors. By overcoming the specific failure mode that had limited previous feedback-based algorithms, this work suggests a more robust way to prepare quantum states, bringing us closer to using quantum computers to solve some of the most complex problems in physics and chemistry.

Technical Summary: Imaginary-time-enhanced feedback-based quantum algorithms for ground-state preparation

Problem Statement
Preparing ground states of strongly correlated quantum systems is a central objective in quantum simulation. While hybrid quantum-classical algorithms like the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA) are widely used, they suffer from limitations such as barren plateaus, noise accumulation, and the need for problem-specific ansätze. The Feedback-based Algorithm for Quantum Optimization (FALQON) offers an alternative by using a fully quantum feedback rule to drive Hamiltonian evolution without classical parameter optimization. However, this paper identifies a critical structural failure in FALQON when applied to systems with spectral degeneracies. Specifically, when the energy trajectory encounters a degenerate manifold of low-lying states (common in half-filled Fermi-Hubbard lattices due to particle-hole symmetry), the commutator feedback signal $A(t)$ collapses. This causes the control field $\beta(t)$ to oscillate weakly around zero, preventing the system from escaping the degenerate subspace and reaching the true ground state.

Methodology
To address this stagnation, the authors propose iFALQON (imaginary-time-enhanced FALQON). This hybrid scheme periodically inserts short imaginary-time evolution (ITE) steps into the standard FALQON feedback loop.

FALQON Framework: The system evolves under a time-dependent Hamiltonian $H(t) = H_p + \beta(t)H_d$ , where $H_p$ is the problem Hamiltonian and $H_d$ is a driver. The control field is updated recursively as $\beta(t) = -A(t)$ , where $A(t) = i\langle[H_d, H_p]\rangle_t$ , enforcing a monotonic decrease in the cost function $C(t) = \langle H_p \rangle_t$ .
ITE Integration: The authors introduce a non-unitary update step based on the imaginary-time Schrödinger equation: $|\psi(\tau + \Delta\tau)\rangle \propto (I - H_p \Delta\tau)|\psi(\tau)\rangle$ . This step is inserted periodically (e.g., after every two feedback layers) between standard FALQON Trotter steps.
Mechanism: The ITE step acts as a dissipative mechanism that exponentially suppresses excited-state components. Theoretical analysis shows that this update reduces the cost function whenever the state has non-zero energy variance. Crucially, by redistributing population toward lower-energy sectors, the ITE step reactivates the feedback signal $A(t)$ , which would otherwise vanish in degenerate subspaces, allowing the dynamics to escape local minima and continue toward the ground state.
Implementation Scope: The study utilizes statevector simulations (via Qiskit) to benchmark the algorithmic behavior, avoiding hardware noise to isolate the effects of the feedback and ITE mechanisms. The imaginary-time update is treated as a statevector operation rather than a hardware-executable gate, with a note that practical hardware implementation would require Variational Quantum Imaginary Time Evolution (VarQITE).

Key Results
The authors benchmarked the algorithms on the Fermi-Hubbard (FH) model across various lattice sizes (up to $3\times3$ ) and filling configurations (doped vs. half-filled), as well as on hydrogen chains ( $H_4$ and $H_6$ ).

FALQON Failure in Degenerate Regimes: Simulations confirmed that standard FALQON converges reliably for doped configurations (where particle-hole symmetry is broken) but consistently stalls at half-filling. In half-filled cases (e.g., $2\times2$ and $3\times3$ lattices), the energy error $\Delta E$ plateaus at finite values, and the feedback amplitude $\beta(t)$ collapses to zero oscillations.
iFALQON Success: The iFALQON protocol restored monotonic energy descent across all tested configurations, including the problematic half-filled cases. By inserting ITE steps ( $\Delta\tau = 0.05$ ), the algorithm achieved relative energy errors below $10^{-5}$ (and down to $10^{-7}$ in benchmarks) across all lattice sizes and fillings.
Comparison with Standalone Methods:
- vs. Standalone QITE: iFALQON required fewer ITE updates to reach a target accuracy in doped regimes compared to pure QITE, as the active FALQON feedback assisted the convergence. In half-filled regimes, the reduction was less pronounced, but iFALQON still succeeded where pure FALQON failed.
- vs. VQE and QAOA: In benchmarks against VQE (using EHV ansatz) and QAOA, iFALQON demonstrated superior stability and lower energy errors. While VQE and QAOA exhibited error floors dependent on ansatz expressibility and optimization instability (particularly in the strongly correlated, stretched regimes of hydrogen chains), iFALQON maintained uniform performance with errors in the $10^{-6}$ to $10^{-7}$ range.
Molecular Systems: For hydrogen chains ( $H_4, H_6$ ), iFALQON accurately reproduced Full Configuration Interaction (FCI) potential energy surfaces across both weakly and strongly correlated regimes, whereas VQE-UCCSD showed increasing deviations in the strongly correlated stretched limit.

Significance and Claims
The paper claims that iFALQON provides an effective proof-of-concept framework for improving ground-state convergence in strongly correlated many-body systems. Its primary contribution is demonstrating that a hybrid real-imaginary-time feedback dynamic can overcome the specific limitation of spectral degeneracy that plagues purely unitary feedback algorithms like FALQON.

The authors emphasize that this approach avoids the classical optimization challenges of VQE (barren plateaus, optimizer instability) while extending the applicability of feedback-driven methods to degenerate regimes. The work is modestly framed as a noiseless statevector study focusing on algorithmic behavior; the authors acknowledge that practical hardware implementation will require balancing the improved convergence against the additional measurement and optimization costs associated with implementing the ITE steps (e.g., via VarQITE). The study concludes that iFALQON offers a robust pathway for ground-state preparation in systems where traditional variational or unitary feedback methods struggle.

Technical Summary: Imaginary-time-enhanced feedback-based quantum algorithms for ground-state preparation

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