FALQON: Feedback-Based Quantum Optimization

• FALQON is a feedback-based quantum optimization method that replaces classical optimization with online, measurement-based control using Lyapunov feedback.
• The methodology leverages controlled Hamiltonian evolution and commutator measurements to ensure monotonic energy reduction, with variants like TR-FALQON and ITE-FALQON enhancing convergence.
• Practical implementations demonstrate improvements in circuit depth, robustness to barren plateaus, and applicability across many-body physics, combinatorial optimization, and state preparation.

Feedback-based Algorithm for Quantum Optimization (FALQON) is a feedback-driven quantum optimization algorithm in which circuit parameters are not tuned by an external classical optimizer but are generated online from measurements of the evolving quantum state. In its standard form, the state evolves under a controlled Hamiltonian $H(t)=H_p+\beta(t)H_d$, where $H_p$ is the problem Hamiltonian and $H_d$ is a driver Hamiltonian, and the control field is set from the commutator signal $A(t)=\langle i[H_d,H_p]\rangle_t$. With the Lyapunov-style choice $\beta(t)=-kA(t)$ for $k>0$, the cost $C(t)=\langle H_p\rangle_t$ is monotonically non-increasing in the ideal continuous-time limit, so FALQON replaces high-dimensional variational search by deterministic, measurement-based control (Li et al., 2024, Legnini et al., 3 Jul 2025).

1. Formal definition and control-theoretic basis

FALQON is most naturally described as a closed-loop quantum control protocol. The central quantity is the expectation value of the problem Hamiltonian,

$C(t)=\langle \psi(t)|H_p|\psi(t)\rangle,$

and the control objective is to drive this quantity downward without introducing a classical optimizer over a parameter vector. Using the Schrödinger equation for $H(t)=H_p+\beta(t)H_d$, one obtains

$$\frac{dC}{dt}=\beta(t)\,A(t),\qquad A(t)=\langle \psi(t)|i[H_d,H_p]|\psi(t)\rangle.$$

Choosing

$H_p$0

gives

$H_p$1

In this sense, the energy expectation acts as a Lyapunov function, and the feedback signal is the measured commutator expectation itself (Pexe et al., 2 Jun 2026).

On gate-model hardware, the continuous evolution is discretized into layers. A standard first-order implementation applies, at step $H_p$2,

$H_p$3

measures

$H_p$4

and updates the next control through a rule $H_p$5, with the simplest Lyapunov choice $H_p$6. The stopping criterion is typically either $H_p$7 below a threshold, negligible energy variation, or exhaustion of the available circuit depth. A notable structural feature is that the ansatz is not pre-fixed: the circuit grows progressively, and the effective parameter sequence emerges dynamically from the feedback trajectory rather than from an a priori variational template (Pexe et al., 2 Jun 2026).

2. Relation to variational quantum algorithms

FALQON is closely related in circuit form to QAOA-like alternating evolutions, but its optimization logic is different. In QAOA or VQE, one prepares a parameterized state and then searches over a set of angles $H_p$8 or a larger ansatz parameter vector by classical optimization. In FALQON, there is no external loop over a parameter landscape; each new control amplitude is determined by a measurement on the current state. The algorithm therefore replaces static variational optimization by a control trajectory designed to reduce $H_p$9 layer by layer (Pexe et al., 2 Jun 2026).

This difference matters most in regimes where standard variational methods are limited by expressibility–trainability tradeoffs, barren plateaus, rugged energy landscapes, or optimizer instability. A perspective on strongly correlated many-body systems emphasizes that deeper, more expressive ansätze often generate barren plateaus with exponentially small gradients, while noise can further flatten the landscape and shift optima toward mixed states. In that setting, feedback-based quantum algorithms are argued to reduce optimization dimensionality, use local physical information tied directly to the instantaneous energy derivative, and follow “geometrically more robust trajectories” than generic gradient-based variational schemes (Pexe et al., 2 Jun 2026).

A practical consequence is that FALQON typically measures one commutator expectation per step rather than estimating many partial derivatives. This does not eliminate measurement cost, but it changes its structure. The algorithm remains hybrid in the operational sense—measurements are processed classically and fed back into the next layer—but it avoids the classical black-box optimization that defines QAOA and VQE.

3. Variants and algorithmic refinements

Several extensions modify the basic feedback law without abandoning the core idea that measured observables generate the control sequence. Time-Rescaled FALQON (TR-FALQON) introduces a reparameterization of time $H_d$0 and rescales the control according to

$H_d$1

The corresponding discrete protocol rescales both the problem and driver unitaries by $H_d$2. In MaxCut and ANNNI benchmarks, TR-FALQON accelerates convergence in shallow-depth regimes, and in state-preparation settings TR-FQA was reported to reduce the required circuit depth by several hundred layers (Rattighieri et al., 2 Apr 2025).

Second-Order FALQON (SO-FALQON) incorporates higher-order commutator information. Besides

$H_d$3

it uses

$H_d$4

and employs the second-order update

$H_d$5

For Max-Cut on 3-regular graphs, this higher-order structure was used to study parameter transfer from small to larger instances; transferred parameters trained on smaller graphs yielded higher approximation ratios than native optimization on larger graphs, because the transferred schedules could safely use more aggressive time steps (Thomaz et al., 5 May 2026).

Imaginary-Time-Enhanced FALQON (ITE-FALQON) augments real-time feedback evolution with short imaginary-time steps,

$H_d$6

or, equivalently, a first-order approximation to $H_d$7. This modification was introduced to overcome a structural failure mode of standard FALQON in spectrally degenerate regimes of the Fermi–Hubbard model. Standard FALQON was shown to stall on the $H_d$8 half-filled lattice and on several $H_d$9 fillings because the commutator signal collapses in degenerate low-energy manifolds; with $A(t)=\langle i[H_d,H_p]\rangle_t$0 and an imaginary-time step inserted every two feedback layers, ITE-FALQON restored monotonic decay of the energy difference and achieved final errors below $A(t)=\langle i[H_d,H_p]\rangle_t$1 across the tested fillings (Long et al., 15 Dec 2025).

Other refinements target robustness or efficiency. A robust variant for coherent control errors introduces a regularized Lyapunov function and yields the feedback law

$A(t)=\langle i[H_d,H_p]\rangle_t$2

trading speed for reduced control amplitude and tighter fidelity bounds under independent coherent errors; the same study proves asymptotic robustness of standard FALQON to systematic multiplicative control errors (Legnini et al., 3 Jul 2025). “Optimal FALQON” departs furthest from the original no-optimizer ethos by treating the per-layer time step $A(t)=\langle i[H_d,H_p]\rangle_t$3 and scaling factor $A(t)=\langle i[H_d,H_p]\rangle_t$4 as classical decision variables, optimizing them layer by layer. On all 94 non-isomorphic 3-regular graphs with 12 vertices, this formulation improved success probability and depth-normalized efficiency over standard FALQON and produced strong warm starts for QAOA (Mancini et al., 8 May 2026).

4. Measurement model and hardware realization

The practical bottleneck in FALQON is not state preparation alone but estimation of the feedback observable $A(t)=\langle i[H_d,H_p]\rangle_t$5, which decomposes into Pauli strings. For MaxCut with

$A(t)=\langle i[H_d,H_p]\rangle_t$6

the commutator becomes a sum of 2-local terms of the form $A(t)=\langle i[H_d,H_p]\rangle_t$7. A measurement study implemented FALQON on the Ket platform and replaced direct observable estimation by classical shadows. Because the relevant observables involve only $A(t)=\langle i[H_d,H_p]\rangle_t$8 and $A(t)=\langle i[H_d,H_p]\rangle_t$9, the authors used a biased classical-shadow ensemble over $\beta(t)=-kA(t)$0 bases and found that, depending on graph geometry, the number of measurements required per layer could be up to 16 times lower than with direct estimation; for complete graphs, the required number of measurements grew logarithmically with the number of observables (Bertuzzi et al., 27 Feb 2025).

Hardware-oriented analyses have focused on the fact that each FALQON layer contains both a problem unitary and a mixer unitary, so two-qubit gate synthesis matters. In a realistic neutral-atom simulation of 2- to 4-qubit MaxCut instances, a small-angle controlled-phase implementation of the $\beta(t)=-kA(t)$1 interaction outperformed a CZ-based implementation under spontaneous emission. For $\beta(t)=-kA(t)$2 rad, the optimized small-angle controlled-phase gate reached fidelity $\beta(t)=-kA(t)$3 in the master-equation simulation with parameters

$\beta(t)=-kA(t)$4

and gate time $\beta(t)=-kA(t)$5. Because each $\beta(t)=-kA(t)$6 term then required one entangling gate instead of two, the total entangling-gate evolution times were reduced from $\beta(t)=-kA(t)$7 to $\beta(t)=-kA(t)$8 for the 2-qubit case and from $\beta(t)=-kA(t)$9 to $k>0$0 for the 3-qubit case, with corresponding gains in approximation ratio and success probability under noise (Li et al., 2024).

5. Application domains and representative results

FALQON has moved well beyond its original combinatorial-optimization setting and is now used as a general state-preparation and control primitive in many-body physics, graph optimization, and domain-specific Ising encodings. The range of applications is broad, but the reported behavior is remarkably consistent: standard one-drive FALQON often lowers energy efficiently, while the most demanding instances require symmetry-aware initialization, multiple drivers, time rescaling, or non-unitary augmentation to obtain high overlap with the true ground state.

Domain	Representative result	Paper
ANNNI model and criticality	Ground and excited states, finite-size scaling, correlation functions, and structure factors were computed without classical optimization by choosing symmetry-adapted initial states	(Pexe et al., 2024)
Strongly correlated many-body systems	A perspective argues that feedback-based quantum algorithms are promising for deconfined quantum criticality, topological phase transitions, strange metals, many-body localization, and quantum spin liquids because they avoid high-dimensional optimization and preserve physical symmetries	(Pexe et al., 2 Jun 2026)
Fermi–Hubbard ground states	Standard FALQON fails in degenerate low-energy sectors, while ITE-FALQON restores monotonic convergence across $k>0$1 to $k>0$2 lattices at $k>0$3	(Long et al., 15 Dec 2025)
Thermofield-double preparation	In the Maldacena–Qi model, standard FALQON and TR-FALQON fail from trivial product states; ITE-TR-FALQON reaches fidelities close to unity and reproduces von Neumann and Rényi entropy spectra of the exact TFD state	(Pexe et al., 2 Jul 2026)
DNA assembly	Standard FALQON, SO-FALQON, and TR-FALQON were applied to de novo assembly instances; TR-FALQON and SO-FALQON improved convergence to the ground state and increased success probabilities at reduced circuit depths	(Prado et al., 24 Feb 2026)
Minimum spanning tree and vision graphs	For an MST Hamiltonian, one-drive FALQON lowered expected energy but failed to concentrate amplitude on the MST; multi-drive FALQON improved solution-state probability, and TR-FALQON over multi-drive gave the best fidelity in the tested instances	(Pexe et al., 21 Mar 2026)
Drug-combination design	FALQON optimized Ising encodings of the Maximum Safe Subset and Synergy-Constrained Optimization problems using DDI data, and ITE-FALQON reached exact ground-state energies in the reported small clinically motivated instances	(Nhi et al., 26 Jan 2026)

These studies also clarify a recurring theme: FALQON is especially effective when the problem structure can be embedded into the choice of $k>0$4, $k>0$5, and the initial state. Gauge constraints in lattice gauge models, particle-number sectors in Fermi–Hubbard systems, and graph-specific encodings in MST or MaxCut all benefit from this co-design perspective.

6. Limitations, misconceptions, and open problems

FALQON’s most common misconception is that “no classical optimizer” implies “no classical overhead.” The literature does not support that reading. A plausible implication is that FALQON should instead be understood as eliminating black-box classical optimization while retaining classical post-processing of measurements, real-time feedforward, and, in some variants, nontrivial parameter management. Measurement of commutator observables, grouping of Pauli strings, and calibration of $k>0$6 or related hyperparameters remain central practical issues.

Several structural limitations are now well documented. The choice of driver $k>0$7 is crucial: if $k>0$8 is ineffective or vanishes in relevant subspaces, the control law stalls. Degeneracies and dense low-energy spectra can suppress the commutator signal even when the system is not in the true ground state, which is why ITE-FALQON was introduced for the Fermi–Hubbard problem (Long et al., 15 Dec 2025). Standard one-drive FALQON can also lower expected energy without concentrating amplitude on the desired solution state, as observed in the MST study, where amplitude redistribution required multi-drive controls and time rescaling (Pexe et al., 21 Mar 2026).

Robustness is similarly nuanced. For coherent control errors, asymptotic robustness to systematic multiplicative errors can be proven, but independent errors degrade fidelity according to bounds involving the time-integrated norm of $k>0$9, and regularization of the control law can improve this trade-off (Legnini et al., 3 Jul 2025). Noise-induced barren-plateau arguments are milder for FALQON than for highly expressive variational ansätze, yet long feedback trajectories still accumulate gate noise, readout noise, and latency. This is especially relevant because circuit depth grows with the number of feedback steps, and the strongest many-body applications in the literature often require deep circuits or non-unitary primitives that remain difficult to realize faithfully on current hardware (Pexe et al., 2 Jun 2026).

Open questions are correspondingly broad. The many-body perspective identifies rigorous convergence conditions for generic $C(t)=\langle H_p\rangle_t$0, scaling laws for required depth and accuracy, systematic noise-robust control modifications, and extensions to excited states and low-energy bands as central unresolved problems (Pexe et al., 2 Jun 2026). The recent proliferation of variants—TR-FALQON, SO-FALQON, ITE-FALQON, robust FALQON, Optimal FALQON, and multi-drive formulations—suggests that FALQON is less a single fixed algorithm than a family of feedback-based control schemes whose practical performance depends strongly on circuit co-design, symmetry structure, and the geometry of the target energy landscape.