FALQON: Feedback-Based Quantum Optimizer

• FALQON is a feedback-based quantum algorithm that sequentially updates circuit parameters using commutator measurements to ensure monotonic energy descent.
• It employs a Lyapunov control-inspired update rule that replaces classical optimization, thereby constructing the state layer by layer with reduced circuit depth.
• The method has demonstrated robust performance in MaxCut benchmarks and quantum many-body state preparation, with enhanced measurement efficiency via classical shadows.

Searching arXiv for foundational and recent FALQON papers to ground the article with citations. arxiv_search(query="FALQON Feedback-based Algorithm for Quantum Optimization Magann", max_results=10) Searching for the original FALQON preprint and related variants. arxiv_search(query="Magann Feedback-based algorithm for quantum optimization arXiv", max_results=10) Looking up variants and applications discussed in the supplied corpus. arxiv_search(query="FALQON classical shadows MaxCut (Bertuzzi et al., 27 Feb 2025)", max_results=5) Searching for parameter-transfer and robustness work to ensure the article reflects recent developments. arxiv_search(query="FALQON parameter transfer robustness coherent control arXiv", max_results=10) FALQON, the Feedback-based ALgorithm for Quantum OptimizatioN, is a feedback-based quantum algorithm for preparing low-energy states of a problem Hamiltonian without an outer classical variational search. In the later literature it is presented as a purely quantum feedback-based alternative to variational parameter optimization, and several papers explicitly treat it as the optimization-focused member of a broader family of feedback quantum algorithms derived from Lyapunov control ideas (Bertuzzi et al., 27 Feb 2025, Rattighieri et al., 2 Apr 2025). Its defining feature is that circuit parameters are not optimized all at once, as in QAOA, but are generated sequentially from measurements of the current quantum state, typically through the expectation value of a commutator observable involving a driver Hamiltonian and the problem Hamiltonian (Pexe et al., 2 Jun 2026).

1. Conceptual identity and place within feedback-based quantum algorithms

FALQON is formulated for optimization problems whose solution is encoded in the ground state of a Hamiltonian $H_p$. Rather than positing a fixed-depth ansatz with a high-dimensional parameter vector, it constructs a trajectory layer by layer. At each step, the algorithm estimates a feedback observable and uses that estimate to set the next control amplitude, thereby replacing a classical optimizer by a deterministic update rule (Bertuzzi et al., 27 Feb 2025).

The canonical continuous-time control model is

$H_t = H_p + \beta(t) H_d,$

where $H_p$ is the problem Hamiltonian and $H_d$ is a driver or mixer Hamiltonian. The target cost is the problem-energy expectation

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

Using Schrödinger dynamics, the relevant feedback signal is

$$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$

so that

$\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$

With the simplest Lyapunov-compatible choice,

$\beta(t)=-A(t),$

one obtains

$\frac{d}{dt}\langle H_p\rangle = -A^2(t)\le 0,$

which gives monotonic energy descent in the ideal continuous-time picture (Bertuzzi et al., 27 Feb 2025, Rattighieri et al., 2 Apr 2025).

Later work treats this commutator-based update as the defining structural distinction between FALQON and gradient-based VQAs. A perspective article accordingly characterizes FALQON as removing the external classical optimizer in favor of a deterministic update based on measuring $\langle i[H_d,H_p]\rangle$, and places it within a broader class of feedback-based quantum algorithms motivated by Lyapunov control theory (Pexe et al., 2 Jun 2026).

2. Lyapunov-control formulation and digitized circuit structure

To implement FALQON on gate-based hardware, the continuous dynamics are discretized and Trotterized. In the standard first-order form, the layered state is

$H_t = H_p + \beta(t) H_d,$0

with

$H_t = H_p + \beta(t) H_d,$1

The discrete feedback update is

$H_t = H_p + \beta(t) H_d,$2

In this form, each layer consists of a short evolution under the problem Hamiltonian followed by a driver evolution whose amplitude is determined by the previous layer’s commutator measurement (Bertuzzi et al., 27 Feb 2025, Rattighieri et al., 2 Apr 2025).

For MaxCut and many related applications, the initial state is chosen as the $H_t = H_p + \beta(t) H_d,$3-polarized product state

$H_t = H_p + \beta(t) H_d,$4

or, equivalently, the ground state of the transverse-field driver. In the MaxCut instantiation discussed in the measurement paper, this guarantees $H_t = H_p + \beta(t) H_d,$5 (Bertuzzi et al., 27 Feb 2025). The original stopping criterion is based on the cost decrement,

$H_t = H_p + \beta(t) H_d,$6

although some later numerical studies instead fix a sufficiently large number of layers and focus on convergence behavior under finite measurement or noise (Bertuzzi et al., 27 Feb 2025).

This digitized structure is also the starting point for several subsequent generalizations. The optimization-focused literature keeps the same alternating form while modifying either the feedback law, the effective time variable, or the Lyapunov operator being minimized (Rattighieri et al., 2 Apr 2025, Rahman et al., 2024).

3. Canonical MaxCut realization and the measurement bottleneck

A standard FALQON benchmark is MaxCut. One common Hamiltonian convention used in the literature is

$H_t = H_p + \beta(t) H_d,$7

with transverse-field driver

$H_t = H_p + \beta(t) H_d,$8

For the MaxCut choice emphasized in the classical-shadows implementation, the commutator entering the feedback update becomes

$H_t = H_p + \beta(t) H_d,$9

Hence the cost and feedback routines require many low-locality Pauli observables: the cost contains $H_p$0 terms, while the feedback contains $H_p$1 and $H_p$2 terms (Bertuzzi et al., 27 Feb 2025, Li et al., 2024).

This measurement structure is central to FALQON’s practical cost. Earlier work often assumed exact expectation values, but finite-shot execution requires repeated estimation of the feedback observable at every layer. The 2025 classical-shadows study identifies this as a major bottleneck and replaces direct observable-by-observable estimation by a classical-shadows routine that reuses the same randomized measurement data for the cost and feedback observables (Bertuzzi et al., 27 Feb 2025).

For Pauli measurements, the quoted classical-shadows sample-complexity bound is

$H_p$3

where $H_p$4 is the locality of the observables. Because FALQON-for-MaxCut uses only 2-local Pauli strings, the method is favorable in this setting. The same paper also introduces a biased shadow protocol that removes the $H_p$5-basis from the random basis ensemble because the relevant feedback observable contains only $H_p$6 and $H_p$7 Paulis. This changes the locality-dependent prefactor from $H_p$8 to $H_p$9, yielding the expected scaling

$H_d$0

for 2-local observables (Bertuzzi et al., 27 Feb 2025).

Empirically, the reported measurements per layer required to satisfy $H_d$1 are as follows.

$H_d$2	Classical shadows	Direct estimation
4	16,384	16,384
6	32,768	262,144
8	32,768	524,288
10	65,536	1,048,576

For $H_d$3 and $H_d$4, classical shadows use 16× fewer measurements than direct estimation; for $H_d$5, the reduction is 8× (Bertuzzi et al., 27 Feb 2025). The same study numerically confirms logarithmic growth of the required measurements with the number $H_d$6 of feedback observables on complete graphs and fits the empirical law

$H_d$7

which it summarizes conservatively as

$H_d$8

up to an additive constant (Bertuzzi et al., 27 Feb 2025).

A separate neutral-atom simulation study addresses a different MaxCut bottleneck: circuit realization. For 2- to 4-qubit MaxCut instances on a realistic Rydberg platform with spontaneous emission, it reports that implementing the $H_d$9-phase evolutions with an optimized native small-angle controlled-phase gate reduces entangling-gate count and improves FALQON performance relative to a CZ-based decomposition (Li et al., 2024).

4. Variants and generalizations

Several later works retain the commutator-feedback architecture but modify either the control law or the target operator.

Time-rescaled FALQON replaces the physical time $$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$0 by a rescaled variable $$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$1 through $$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$2. In the resulting dynamics,

$$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$3

the feedback law becomes

$$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$4

or, in discrete form,

$$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$5

The reported effect is faster convergence in the early layers on MaxCut and, in the ANNNI state-preparation setting, a reduction of the required circuit depth by several hundred layers (Rattighieri et al., 2 Apr 2025).

Second-order FALQON augments the first commutator signal with nested commutators. In the MaxCut-on-3-regular-graphs transfer study, the second-order operators are

$$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$6

with update

$$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$7

That work then studies parameter transfer and finds an empirical optimal-step scaling

$$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$8

very close to $$C(t)=\langle H_p\rangle = \langle \psi(t)|H_p|\psi(t)\rangle.$$9, and reports that schedules learned on the smallest source graphs frequently outperform native optimization on larger 3-regular targets (Thomaz et al., 5 May 2026).

Imaginary-time-enhanced FALQON addresses a different failure mode: spectral degeneracy and dense low-energy structure. In the Fermi–Hubbard study, standard FALQON can stall when the commutator signal collapses inside or near degenerate subspaces. ITE-FALQON inserts periodic normalized short imaginary-time steps,

$$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$0

into the feedback loop. The paper proves that each such step strictly lowers the energy whenever the state has nonzero energy variance and reports restored monotonic convergence across all tested fillings up to $$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$1 Fermi–Hubbard lattices (Long et al., 15 Dec 2025).

Constrained variants replace the Lyapunov target by an operator whose ground state is the optimal feasible solution. FALQON-C introduces

$$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$2

and updates controls from $$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$3 while still evolving under the simpler cost unitary $$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$4. The stated advantage is that constraints are encoded for control rather than by exponentiating a penalty-augmented Hamiltonian, reducing circuit depth relative to applying ordinary FALQON to the penalty-converted QUBO (Rahman et al., 2024). FALQON-IC extends this idea to invalid-configuration constraints through two constructions: a deflation operator of the form

$$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$5

and a folded-spectrum operator

$$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$6

both designed so that the optimal feasible configuration becomes the ground state without introducing slack-variable overhead (Rahman et al., 20 Feb 2025).

FOCQS, finally, keeps the same feedback philosophy but retroactively updates earlier layers using a perturbative approximation to the global control gradient $$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$7. In the paper’s formulation, the basic FALQON signal

$$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$8

is supplemented by a one-step-ahead quantity $$A(t)\equiv \langle \psi(t)|\,i[H_d,H_p]\,|\psi(t)\rangle,$$9, and the estimated global gradient is

$\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$0

The reported effect is improved convergence and reduced required depth over plain FALQON without introducing a full classical variational search (Brady et al., 2024).

5. Application domains

Although first developed for combinatorial optimization, FALQON has been extended to several distinct domains.

In quantum many-body state preparation, a 2024 study applies a FALQON-aligned feedback quantum algorithm to the transverse-field ANNNI model. The protocol retains the same commutator-based update law but uses symmetry-sector initialization to target not only ground states but also excited states. The prepared states are then used for finite-size scaling, excitation-gap analysis, correlation functions, and structure factors, showing that the FALQON principle can serve as a state-preparation tool rather than only a classical-cost optimizer (Pexe et al., 2024). A later perspective article treats this extension as evidence that feedback-based quantum algorithms may be relevant for strongly correlated many-body problems such as deconfined quantum criticality and topological phase transitions, while explicitly presenting that assessment as interpretive rather than as a formal theorem (Pexe et al., 2 Jun 2026).

In bioinformatics, DNA assembly has been formulated as a QUBO over read orderings and then solved with standard FALQON, SO-FALQON, and TR-FALQON. For long-read SARS-CoV-2 and human mitochondrial DNA instances, the paper reports that both SO-FALQON and TR-FALQON improve convergence to the ground state and increase success probabilities at reduced circuit depths relative to standard FALQON (Prado et al., 24 Feb 2026).

In high-energy physics, FALQON has been used for jet-to-parent assignment in fully hadronic $\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$1 events. On a 6-qubit parton-level benchmark, the reported matching accuracy is about $\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$2 at depth $\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$3, outperforming standard QAOA at $\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$4 and the hemisphere baseline, while remaining slightly below the best ma-QAOA result on the same task (Scott et al., 2024).

In graph-structured learning and computer vision, FALQON has been used as the optimization engine for minimum spanning tree formulations. For prototype selection in Optimum-Path Forest classifiers, a PUBO-MST encoding combined with FALQON produced prototype sets whose average OPF classification accuracy matched classical Prim exactly on the four reported datasets, despite imperfect ground-state success rates in the underlying quantum optimization stage (Pexe et al., 20 May 2026). A separate MST study aimed at vision systems reports a sharper algorithmic lesson: standard one-driver FALQON can reduce expected energy while failing to concentrate amplitude on the MST state, whereas multi-driver FALQON and especially TR-FALQON over multi-drive give better ground-state fidelity and higher solution-state probability on the tested small synthetic graphs (Pexe et al., 21 Mar 2026).

In drug-combination design, FALQON is used to minimize Ising Hamiltonians encoding harmful and synergistic drug–drug interactions. The paper applies it to the Maximum Safe Subset and Synergy-Constrained Optimization tasks and reports that plain FALQON identifies the correct low-energy solutions while ITE-FALQON concentrates probability more sharply on the optimal drug combinations (Nhi et al., 26 Jan 2026).

6. Limitations, robustness, and current assessment

FALQON’s main practical limitation is that eliminating the classical optimizer does not eliminate hybrid overhead. The algorithm still requires repeated estimation of commutator expectations, layer by layer, and therefore inherits measurement burden, readout sensitivity, and classical–quantum latency. A perspective survey stresses that progressive circuit-depth growth, measurement errors, and feedback-loop latency are central engineering obstacles, and notes that unmitigated readout bias can steer the trajectory in the wrong direction (Pexe et al., 2 Jun 2026).

Several failure modes are now explicit in the literature. One is degeneracy-induced stagnation: if the state enters an invariant set where

$\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$5

the control collapses and standard FALQON can plateau above the ground state. This mechanism is demonstrated concretely in the Fermi–Hubbard study and is one reason for the development of ITE-FALQON (Long et al., 15 Dec 2025). Another is symmetry-protected stagnation, emphasized in the many-body perspective: if the mixer fails to connect the relevant symmetry sectors, the feedback signal can vanish prematurely even though the ground state has not been reached (Pexe et al., 2 Jun 2026).

A second limitation is problem dependence of measurement efficiency. The classical-shadows acceleration works because FALQON-for-MaxCut uses many 2-local Pauli observables. The same study repeatedly stresses that classical shadows are advantageous when one wants many observables of small locality, but not for full tomography or high-weight operators (Bertuzzi et al., 27 Feb 2025).

Robustness to coherent control error has also been analyzed. Under multiplicative systematic coherent errors,

$\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$6

with $\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$7, the Lyapunov decrease keeps the same sign, and the paper proves asymptotic convergence of standard FALQON under its stated nondegeneracy assumptions. For independent coherent control errors, it derives the fidelity bound

$\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$8

where

$\frac{d}{dt}\langle H_p\rangle = A(t)\beta(t).$9

The same paper proposes a robustified update based on the regularized Lyapunov function $\beta(t)=-A(t),$0, yielding

$\beta(t)=-A(t),$1

which reduces control amplitudes and improves empirical robustness under independent coherent errors (Legnini et al., 3 Jul 2025).

A current mitigation strategy for the measurement burden is parameter transfer. On 14-node MaxCut recipients, direct reuse of donor schedules learned on 8-, 10-, or 12-node graphs performs particularly well for dense Erdős–Rényi recipients, with reported final approximation ratios around $\beta(t)=-A(t),$2, $\beta(t)=-A(t),$3, and $\beta(t)=-A(t),$4 for recipient densities $\beta(t)=-A(t),$5, $\beta(t)=-A(t),$6, and $\beta(t)=-A(t),$7, respectively. The same study concludes that transfer success is governed much more by the recipient graph than by donor size, and argues that this can significantly reduce recipient-side measurement overhead (Fumaco et al., 28 May 2026).

Taken together, these results place FALQON in a distinct position within quantum optimization. It is not a standard variational algorithm, because its parameters are generated online by feedback rather than trained globally. It is not measurement-free, because the feedback signal itself must be estimated at every stage. Its strongest demonstrated use cases are those in which the commutator observable decomposes into many low-locality Pauli strings, the control objective benefits from monotone energy descent, and shallow-depth improvements matter more than asymptotic optimality. The subsequent literature can therefore be read as a progressive attempt to retain FALQON’s optimizer-free structure while addressing its concrete bottlenecks: measurement cost, finite-depth stagnation, degeneracy, constraint handling, hardware noise, and transferability.