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

arXiv:2512.13044 · quant-ph, cond-mat.str-el · Submitted 2025-12-15 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Imaginary-time-enhanced feedback-based quantum algorithms for universal ground-state preparation".

Mira: Here's the gist of the paper. The gist: The ITE-FALQON scheme, which inserts short imaginary-time evolution steps into the feedback loop, suppresses excited-state components and restores monotonic energy descent, achieving reliable ground-state convergence across all fillings in strongly correlated quantum systems.

Kai: Problem with Pure FALQON The feedback-based quantum algorithm (FALQON) fails when applied to systems with degenerate spectra because the commutator signal becomes strongly suppressed, resulting in feedback amplitudes that oscillate weakly around zero and trapping the energy above the true ground state The breakdown appears at half-filling on the 2×2 lattice and extends to both half-filled and doped configurations on the 3×3 lattice This is because particle-hole symmetry generates degenerate manifolds of lowlying states, causing the feedback signal to collapse When the energy trajectory encounters such a manifold, it becomes trapped above the true ground state.

Mira: FALQON Framework and Failure Mechanism The FALQON framework involves a time-dependent Hamiltonian H(t) = Hp + β(t)Hd, where the control field β(t) is determined from the commutator signal A(t) = ih

Hd, Hp: it to enforce a monotonic decrease of the cost function C(t) = hHpit.

Kai: By choosing β(t) = −A(t), the energy expectation decreases monotonically because dC/dt = β(t)A(t) <= 0.

Paper summary: Kai: We just talked about how pure FALQON struggles with spectral degeneracies, and this paper introduces the imaginary-time enhancement scheme as a solution. Essentially, they are showing that when you use this iFALQON method, you can prepare the ground state reliably even in systems where the underlying Hamiltonian has degenerate spectra.

Mira: That's right, it claims that by inserting short imaginary-time evolution steps into the feedback loop, they suppress excited-state components and restore monotonic energy descent. This is important because typically in these scenarios, the feedback signal just collapses and the energy stops decreasing before it finds the true ground state.

Lev: What this means for us is that we're not stuck needing a perfect continuous-time limit; we can use a discrete Trotter expansion with these embedded imaginary time steps to get reliable convergence. It moves away from relying solely on those ideal continuous dynamics where LaSalle's invariance principle would cause the feedback to just halt when the commutator signal hits zero.

Kai: So, the core thesis is that this hybrid real-imaginary dynamics unites Lyapunov stability with dissipative convergence toward the energy minimum, and it achieves high-fidelity ground state preparation across various fillings in strongly correlated systems using models like the Fermi-Hubbard model.

Mira: It really matters because it demonstrates that this type of hybrid approach can work for universal ground-state preparation, not just for specific, simple Hamiltonians. The results show convergence even when starting from half-filled or doped configurations on lattices up to three times three <ref:2512.13044#pg1>.

Lev: From an error correction standpoint, the fact that they're getting errors below ten minus five with this method is promising because it shows a clear path to reaching high fidelity in these complex, strongly correlated regimes without needing extremely precise classical optimization parameters like VQE or QAOA.

Kai: So, if we think about what this means for the broader quantum simulation community, it's that there’s a way to handle the inherent difficulties of degeneracy in finding ground states by modifying the feedback mechanism itself rather than just tweaking the initial Hamiltonian or adding more classical optimization guesswork.

Mira: It's about refining how we use quantum feedback. The authors are Thanh Nguyen Van Long, Lan Nguyen Tran, and Le Bin Ho. They used lattices up to three times three for their testbed in this work on Imaginary-time-enhanced feedback-based quantum algorithms for universal ground-state preparation <ref:2512.13044#pg1,Imaginary-time-enhanced feedback-based quantum algorithms for universal ground-state preparation>.

Lev: And while the paper is solid on these small lattices, we still need to investigate how this iFALQON method performs when we scale the lattice size up significantly or introduce different types of interactions that create even more intricate degeneracies.

Conclusion: Kai: So wrapping this up, the work on "Imaginary-time-enhanced feedback-based quantum algorithms for universal ground-state preparation" shows that pure FALQON has a limitation when spectral degeneracies are present.

Mira: That limitation is that the feedback signal collapses, and energy stops decreasing prematurely because it gets trapped above the true ground state manifold in those degenerate cases.

Lev: The iFALQON scheme solves this by inserting short imaginary-time evolution steps into the feedback loop, which suppresses excited states and restores monotonic energy descent by acting like a dissipative dynamics.

Kai: In simple terms, this means we can use hybrid real-imaginary time feedback dynamics to reliably find the ground state in strongly correlated quantum systems where degeneracies are an issue. It's a robust way to prepare those states without relying on classical parameter optimization guesswork.

Mira: It's about moving beyond just continuous time dynamics by using that imaginary-time component to ensure convergence toward the true energy minimum, even when the system is stuck in degenerate subspaces.

Lev: The iFALQON framework offers a promising path forward because it demonstrates linear geometric convergence for systems with spectral gaps and high fidelity results on the Fermi-Hubbard model simulations.

Kai: Ultimately, this paper suggests that hybrid real-imaginary time feedback dynamics are a scalable approach for ground state preparation in strongly correlated quantum systems. This opens up new avenues for designing algorithms that can handle the complexities of realistic physical Hamiltonians.

University of Science, Vietnam National University · Frontier Research Institute for Interdisciplinary Sciences, Tohoku University

quant-ph, cond-mat.str-el

Submitted: 2025-12-15

Updated: 2026-10-22

Comments: 11 pages, 7 figures

Journal ref: Physical Review Research (2026)

DOI: 10.1103/cbcv-fnm2

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 86/100

The gist: The gist: The ITE-FALQON scheme, which inserts short imaginary-time evolution steps into the feedback loop, suppresses excited-state components and restores monotonic energy descent, achieving

Key concepts

Pure FALQON Failure: The standard feedback algorithm (FALQON) stops working when the system encounters spectral degeneracies—multiple states with the same energy. In this scenario, the feedback signal becomes too weak to drive the system toward lower energy, causing it to get trapped above the true ground state manifold.
Imaginary-Time Enhancement (ITE): This technique introduces short steps of imaginary-time evolution into the feedback process. This acts like a natural dissipation mechanism that actively suppresses unwanted excited states and increases the probability of finding the system in lower energy sectors, helping it escape degenerate traps.
Monotonic Energy Descent: The goal is to ensure that every step taken by the algorithm leads to a lower total energy. ITE-FALQON achieves this by combining real-time feedback with imaginary-time steps, which guarantees that the cost function (energy) strictly decreases until it reaches the absolute minimum, which is the ground state.

Terminology

Degeneracies — 1× in this paper · explained in 2 episodes
Feedback Loop — 1× in this paper · explained in 2 episodes
Hubbard Model — 1× in this paper · explained in 5 episodes

Summary

The gist: The ITE-FALQON scheme, which inserts short imaginary-time evolution steps into the feedback loop, suppresses excited-state components and restores monotonic energy descent, achieving reliable ground-state convergence across all fillings in strongly correlated quantum systems.

Problem with Pure FALQON

The feedback-based quantum algorithm (FALQON) fails when applied to systems with degenerate spectra because the commutator signal becomes strongly suppressed, resulting in feedback amplitudes that oscillate weakly around zero and trapping the energy above the true ground state The breakdown appears at half-filling on the 2×2 lattice and extends to both half-filled and doped configurations on the 3×3 lattice This is because particle-hole symmetry generates degenerate manifolds of lowlying states, causing the feedback signal to collapse When the energy trajectory encounters such a manifold, it becomes trapped above the true ground state.

FALQON Framework and Failure Mechanism

The FALQON framework involves a time-dependent Hamiltonian H(t) = Hp + β(t)Hd, where the control field β(t) is determined from the commutator signal A(t) = ih[Hd, Hp]it to enforce a monotonic decrease of the cost function C(t) = hHpit. By choosing β(t) = −A(t), the energy expectation decreases monotonically because dC/dt = β(t)A(t) <= 0. In the ideal continuous-time limit, LaSalle’s invariance principle implies that dynamics may enter an invariant set where h[Hd, Hp]it = 0, leading to β(t) = 0 and the energy stopping its descent. This typically occurs once the evolution has reached the ground-state manifold.

Imaginary-Time Enhancement (ITE)

To resolve this obstruction, ITE-FALQON introduces an effective dissipative dynamics that naturally suppresses excited-state amplitudes and enhances overlap with the ground state. This is achieved by periodically inserting short imaginary-time evolution steps generated by Hp into the feedback loop. Specifically, Eq. (6) shows that ψki = N (I − Hp∆τ)e −iHp∆t e −iβk−1Hd∆t ψk−1i, where the term (I − ∆τHp) redistributes the population toward lower-energy sectors. This hybrid real-imaginary dynamics unites Lyapunov stability with dissipative convergence toward the energy minimum.

Convergence Analysis and Results

The theorem for ITE-FALQON proves that the normalized cost function C(Φ∆τ (ψ)) obeys a bound C(Φ∆τ (ψ)) <= E − 2∆τ(1 − 1/2h) / (1 - ∆τh) squared V. This upper bound implies that the cost function is strictly decreasing until ψk reaches the ground state of Hp, i.e., V = 0. The convergence of energies shows that for any non-zero variance, Ek cannot converge to any excited eigenvalue. Furthermore, if a spectral gap exists, the ITE-FALQON iteration contracts the distance to ψgsi by at least the fixed factor e − ∆τ∆ per step, establishing a linear (geometric) convergence rate.

Benchmarking on Fermi-Hubbard Model

The Fermi-Hubbard (FH) model is used as a controlled testbed, capturing the interplay between kinetic hopping and on-site repulsion. The simulations were performed using the Qiskit statevector simulator, providing a noiseless benchmark of algorithmic performance without decoherence. For the [3×3] lattice, pure FALQON fails to concentrate probability near the ground state because it cannot drive transitions across the densely packed low-lying spectrum. In contrast, ITE-FALQON shows that both half-filled and doped configurations converge monotonically toward the exact ground state energy with final errors below 10−5. This uniform behavior demonstrates that hybrid real-imaginarytime feedback reliably restores ground-state convergence.

Conclusion

The ITE-FALQON scheme overcomes the obstruction caused by spectral degeneracies in FALQON by gradually reducing excited-state contributions and achieving high-fidelity ground-state preparation across all tested lattices. This hybrid real-imaginarytime framework therefore offers a robust and scalable approach to ground-state preparation in strongly correlated quantum systems.

Acknowledgments

This paper is supported by JSPS KAKENHI Grant Number 23K13025, and the Tohoku Initiative for Fostering Global Researchers for Interdisciplinary Sciences (TI-FRIS) of MEXT’s Strategic Professional Development Program for Young Researchers. L.N.T is funded by National Foundation for Science and Technology Development (NAFOSTED) Grant Number 103.01-2024.06. The data availability states that no data were created or analyzed in this study.

References

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Improvements for AI systems

Ground state preparation for strongly correlated quantum systems can be reliably achieved on any lattice size up to 3x3 by using an imaginary-time-enhanced feedback algorithm called ITE-FALQON. This allows the system to reach the true ground state even in cases where the pure feedback method fails due to spectral degeneracies.
The improved AI system can perform robust quantum simulation of fermionic lattice models, such as the Fermi-Hubbard model, across various fillings and lattice sizes without getting stuck in excited state manifolds. This capability is critical for studying exotic phases like Mott insulators and unconventional superconductivity.
The algorithm ensures monotonic energy descent by incorporating short imaginary-time evolution steps into the feedback loop to suppress excited-state components and escape degenerate subspaces. This guarantees convergence to the exact ground state energy, as shown by the theorem proving C(PhiDelta tau) is strictly decreasing until V = 0.
The system can efficiently handle complex fermionic operators through Jordan-Wigner mapping, transforming the Hamiltonian into a qubit representation that allows direct simulation on quantum registers. This enables high-fidelity ground state preparation using current NISQ hardware simulators like Qiskit without requiring classical parameter optimization.

Abstract

Preparing ground states of strongly correlated quantum systems is a central goal in quantum simulation and optimization. The feedback-based quantum algorithm (FALQON) provides an attractive alternative to variational methods with a fully quantum feedback rule, but it fails in the presence of spectral degeneracies, where the feedback signal collapses and the evolution cannot reach the ground state. Using the Fermi-Hubbard model on lattices up to 3x3, we show that this breakdown appears at half-filling on the 2x2 lattice and extends to both half-filled and doped configurations on the 3x3 lattice. We then introduce an imaginary-time-enhanced FALQON (i FALQON) scheme, which inserts short imaginary-time evolution steps into the feedback loop. This proposed method suppresses excited-state components, escapes degenerate subspaces, restores monotonic energy descent and improves ground-state convergence across the tested fillings. We further benchmark its performance against existing methods, showing improved results for iFALQON, demonstrating the potential of hybrid real-imaginary-time feedback dynamics for ground-state preparation in strongly correlated quantum systems.

Sources

A Quantum Approximate Optimization Algorithm