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

TL;DR

The paper identifies a breakdown of fully quantum feedback optimization (FALQON) for degenerate spectra, showing reliable ground-state convergence in doped Fermi-Hubbard configurations but stagnation at half-filling due to spectral degeneracies. It introduces imaginary-time-enhanced FALQON (ITE-FALQON), which periodically interleaves short imaginary-time steps with the quantum feedback loop to damp excited-state components and recover monotonic energy descent. A formal convergence theorem is derived, and the method is benchmarked on Fermi-Hubbard lattices up to $3\times3$, demonstrating monotonic convergence to the ground state across all fillings and lattice sizes, with a faster relaxation due to the imaginary-time steps. The approach provides a robust, scalable, fully quantum route to ground-state preparation in strongly correlated systems, leveraging a hybrid real-imaginary-time dynamics to overcome degeneracy-induced stagnation.

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 (ITE-FALQON) scheme, which inserts short imaginary-time evolution steps into the feedback loop. The hybrid method suppresses excited-state components, escapes degenerate subspaces, and restores monotonic energy descent. The ITE-FALQON achieves a reliable ground-state convergence across all fillings, providing a practical route to scalable ground-state preparation in strongly correlated quantum systems.

Figures (4)

• Figure 1: Dynamics of the FALQON without ITE. (a) Time evolution of the energy difference $\Delta E(t)=E(t)-E_0$ for various FH lattices and fillings. Doped systems $([1\!\times\!3]$, $[1\!\times\!4]$, $[2\!\times\!3])$ exhibit smooth exponential convergence toward the exact ground energy, while half-filled configurations $([2\times2]$, $[3\times3])$ stagnate at finite energy errors. (b) Corresponding feedback amplitude $\beta(t)$ as defined by the Lyapunov control law $\beta(t)=-A(t)$. The early peak in $\beta(t)$ accelerates energy descent, but in half-filled lattices the feedback collapses to zero prematurely, halting further optimization. The inset shows long-time oscillations of $\beta(t)$ for the $2\!\times\!2$ and $3\!\times\!3$ cases, demonstrating symmetric fluctuations around zero and confirming feedback quenching due to degeneracy. Together, these results reveal that pure FALQON effectively prepares ground states only for non-degenerate systems, motivating the introduction of an imaginary-time enhancement. In the legend, $[i\!\times\!j]$ stands for the lattice site and $(k,l)$ stands for $(N_\uparrow, N_\downarrow )$.
• Figure 2: Energy spectra and population dynamics for the doped and half-filled $[2\!\times\!2]$ FH systems under FALQON without ITE. (a,d) Mean energy $E(t)$ (dashed red) overlaid on the energy spectra $E_n$ of $H_{\rm FH}$. In the doped case (a), $E(t)$ descends toward the ground state without crossing any degenerate levels. In the half-filled case (d), $E(t)$ crosses a manifold of degenerate excited states, suppressing the feedback response and halting further relaxation. (b, c) Probability distributions $P(E)$ at $t=0$ and $t=10^3$ for the doped system, showing collapse into the ground state. (e, f) Corresponding distributions for the half-filled system, where significant weight remains in low-lying excited states due to trapping within a subspace.
• Figure 3: Energy convergence of the ITE-FALQON. Time evolution of the energy difference $\Delta E(t)=E(t)-E_0$ for various FH lattices. An imaginary-time step of $\Delta\tau=0.05$ is inserted after every two feedback layers. All configurations now converge monotonically toward the exact ground energy, reaching relative errors below $10^{-5}$. The exponential decay across more than six decades confirms that ITE-FALQON restores stable, universal ground-state preparation.
• Figure 4: Energy distributions for the $[3\!\times\!3]$ FH lattice. (a, b) Initial energy populations for doped and half-filled configurations, showing dominant support at high excited levels above the ground state (red dashed line). (c, d) Final distributions under pure FALQON remain spread over excited states, indicating failure to reach the ground state. (e, f) ITE-FALQON collapses the distributions onto the ground state, demonstrating successful convergence.