Feedback-Based Quantum Optimization

• Feedback-based quantum optimization is defined by sequential, Lyapunov-inspired control updating parameters from commutator measurement outcomes to ensure monotonic energy reduction.
• The method is implemented via layered quantum circuits using digital updates and, with innovations like classical shadows, it significantly lowers measurement overhead.
• Applications span combinatorial problems and ground-state preparation, with recent variants further accelerating convergence and enhancing robustness on noisy hardware.

Feedback-based quantum optimization denotes a family of quantum algorithms in which control parameters are assigned sequentially from measurements performed on the evolving quantum state, rather than by solving a global classical parameter-optimization problem. Its prototypical representative is FALQON, introduced for combinatorial optimization and later generalized to broader ground-state preparation settings. Across this literature, the defining idea is Lyapunov-style closed-loop control: the objective is encoded in a problem Hamiltonian $H_p$, a noncommuting driver $H_d$ generates motion in Hilbert space, and the next control is computed from measured expectation values such as $i[H_d,H_p]$, so that the cost $\langle H_p\rangle$ decreases monotonically in the idealized continuous-time setting (Magann et al., 2021, Larsen et al., 2023).

1. Control-theoretic foundation

The canonical formulation considers the controlled Schrödinger dynamics

$$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$

with objective

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

Differentiation gives

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

The standard Lyapunov feedback law

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

then yields

$\frac{dJ}{dt}=-A(t)^2\le 0,$

which is the basic monotonicity statement behind FALQON and related feedback-based quantum algorithms (Magann et al., 2021).

Digitization produces the layered circuit structure

$$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$

with

$H_d$0

and discrete feedback update

$H_d$1

For sufficiently small $H_d$2, the digitized algorithm inherits the monotonic decrease of the energy expectation, and a sufficient step-size bound quoted in the literature is

$H_d$3

This control-theoretic structure is common to the original MaxCut formulation, later many-body ground-state algorithms, and applications to the ANNNI model (Larsen et al., 2023, Pexe et al., 2024).

The same framework also clarifies what feedback-based optimization is not. It is not global variational training in the QAOA or VQE sense: there is no outer-loop search over a full parameter vector by gradient descent, Nelder–Mead, SPSA, or related methods. The classical role is instead limited to post-processing measurement outcomes and updating the next scalar or layer-local control variable. This distinction is one of the main reasons the literature presents FALQON and its relatives as optimization-free or gradient-free in the variational sense (Magann et al., 2021, Nhi et al., 26 Jan 2026).

2. Measurement architecture and the central cost bottleneck

The practical bottleneck of feedback-based quantum optimization is not the control law itself but the repeated estimation of the observables that feed it. In MaxCut, the unweighted problem Hamiltonian is

$H_d$4

the standard driver is

$H_d$5

and the feedback observable decomposes as

$H_d$6

Thus both the cost and the feedback term are sums of $H_d$7-local Pauli strings. This low locality is structurally favorable, but the number of observables still grows with the number of graph edges, so the per-layer measurement budget can become dominant (Bertuzzi et al., 27 Feb 2025).

The original sampling-complexity comparison with QAOA already makes this point explicit. If $H_d$8 is the number of shots for a two-qubit Pauli string, $H_d$9 the graph degree, $i[H_d,H_p]$0 the circuit depth, and $i[H_d,H_p]$1 the number of classical optimization iterations for QAOA, the paper quotes

$i[H_d,H_p]$2

whereas QAOA requires $i[H_d,H_p]$3 with gradient-free optimization and $i[H_d,H_p]$4 with gradient-based optimization. This shifts the resource question from classical parameter search to measurement overhead inside the feedback loop (Magann et al., 2021).

A major recent development is the use of classical shadows as the measurement primitive inside FALQON. In the MaxCut setting, one prepares $i[H_d,H_p]$5, performs randomized local measurements, and reuses the same data to estimate both

$i[H_d,H_p]$6

instead of measuring each Pauli term separately. Because the relevant observables are $i[H_d,H_p]$7-local and only involve $i[H_d,H_p]$8 and $i[H_d,H_p]$9, a biased shadow protocol can omit the $\langle H_p\rangle$0 basis and reduce the locality factor from $\langle H_p\rangle$1 to $\langle H_p\rangle$2, giving expected scaling

$\langle H_p\rangle$3

for the FALQON-MaxCut observable family (Bertuzzi et al., 27 Feb 2025).

For the threshold $\langle H_p\rangle$4, the reported per-layer budgets in the shadow-based MaxCut study are as follows.

Number of nodes	Classical Shadows	Direct Measurements
4	16,384	16,384
6	32,768	262,144
8	32,768	524,288
10	65,536	1,048,576

These results show parity at $\langle H_p\rangle$5, but pronounced savings once the graph has enough edges: $\langle H_p\rangle$6 fewer measurements at $\langle H_p\rangle$7, and $\langle H_p\rangle$8 fewer at $\langle H_p\rangle$9 and $$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$0. The same work also reports a numerical confirmation of logarithmic growth of the required number of measurements with the number of observables for complete graphs, consistent with standard classical-shadow theory for many low-locality Pauli observables (Bertuzzi et al., 27 Feb 2025).

This measurement picture corrects a common misconception. Feedback-based methods do not eliminate experimental overhead; they remove global classical parameter optimization and replace it with repeated, structured estimation of commutator expectations. Their practical advantage therefore depends strongly on whether those observables can be estimated efficiently.

3. Canonical problem classes and scientific applications

MaxCut remains the canonical benchmark because its Hamiltonian structure aligns cleanly with the feedback formalism, but the literature quickly expanded beyond combinatorial graph optimization. For ground-state preparation in many-body physics and chemistry, the same feedback law is applied to Hamiltonian decompositions where the driver is chosen from an easily implementable part of the problem. For the Fermi–Hubbard model,

$$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$1

the feedback choice is $$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$2, so the control depends on $$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$3. For second-quantized molecular Hamiltonians,

$$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$4

the driver is taken as $$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$5, giving feedback through $$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$6. In both cases, the reported numerical behavior is monotonic energy decrease with circuit depth and ground-state overlap approaching unity in successful instances (Larsen et al., 2023).

The same control logic has also been used for the transverse-field ANNNI model,

$$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$7

There, feedback-based optimization is used not only for ground-state preparation but also for excited-state targeting by symmetry-sector initialization. Because spin inversion, reflection, and translation commute with both $$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$8 and $$i\frac{d}{dt}\ket{\psi(t)} = \bigl(H_p+\beta(t)H_d\bigr)\ket{\psi(t)},$$9, initializing in a chosen symmetry sector forces the dynamics to remain there, and the algorithm converges to the lowest-energy eigenstate in that sector. This is the paper’s mechanism for extracting excited states without penalty terms or orthogonality constraints (Pexe et al., 2024).

A separate line of work adapts feedback-based optimization to application-driven QUBOs. In drug-combination design, harmful and synergistic drug–drug interactions are encoded into Ising Hamiltonians, and FALQON is used to optimize objectives such as the Maximum Safe Subset and Synergy-Constrained Optimization. The cost Hamiltonians retain the standard diagonal Ising form

$J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$0

while the driver remains

$J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$1

This application emphasizes that feedback-based quantum optimization is not limited to toy graph instances; it can incorporate penalties, rewards, and cardinality constraints inside the same commutator-driven framework (Nhi et al., 26 Jan 2026).

Taken together, these applications show that the feedback-based paradigm is not tied to a single problem family. What persists across them is the same architectural pattern: an objective Hamiltonian, a noncommuting driver, a commutator-derived observable, and a sequential control law that grows the circuit layer by layer.

4. Major algorithmic extensions and accelerations

After the original FALQON formulation, much of the literature focused on correcting its main weaknesses: slow convergence, large depth, and sensitivity to fixed hyperparameters. The resulting variants modify either the control law, the schedule representation, the initialization, or the measurement loop.

Variant	Main modification	Reported effect
CD-FQA	Adds a counterdiabatic-inspired control $J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$2	Accelerates transfer to low-energy states and reduces depth in Ising chains (Malla et al., 2024)
SO-FALQON	Uses a quadratic approximation with $J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$3	Reduces linear depth-scaling coefficient from $J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$4 to $J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$5 on 3-regular MaxCut (Arai et al., 2024)
FOCQS	Perturbatively back-updates previous layers using $J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$6	Improves convergence and required depth over existing feedback control methods (Brady et al., 2024)
GD-QLC	Adds per-layer gradient refinement using $J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$7 and $J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$8	Faster convergence, smoother controls, better robustness to $J(t)=\langle \psi(t)|H_p|\psi(t)\rangle.$9 (Mozakka et al., 12 Feb 2026)
Optimal FALQON	Optimizes per-layer $$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$0 and $$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$1	About $$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$2-fold higher median $$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$3 than standard FALQON on 12-vertex 3-regular graphs (Mancini et al., 8 May 2026)
MGI-FALQON	Reuses measurement outcomes to reinitialize shallow runs	Approaches deep-FALQON energy targets with fixed shallow depths $$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$4 for $$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$5 (Rattighieri et al., 23 Feb 2026)
Learned parameter curves	Replaces online feedback by one-shot classical prediction of $$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$6	Predicted curves remain close to FALQON reference and outperform linear annealing (Pérez et al., 13 Jan 2026)

Counterdiabatic-inspired feedback extends the control Hamiltonian from

$$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$7

to

$$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$8

with both $$\frac{dJ}{dt}=A(t)\beta(t),\qquad A(t)=\langle \psi(t)|\, i[H_d,H_p]\,|\psi(t)\rangle.$$9 and $\beta(t)=-A(t)$0 set by commutator expectations. In one-dimensional Ising chains, the best-performing local choice was $\beta(t)=-A(t)$1, which substantially accelerated low-energy population transfer relative to standard feedback control and was also demonstrated on IBM hardware at shallow depth (Malla et al., 2024).

Second-order and gradient-refined variants address slow convergence more directly. SO-FALQON expands the per-step energy change to second order in $\beta(t)=-A(t)$2 and replaces the first-order law $\beta(t)=-A(t)$3 by

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

with $\beta(t)=-A(t)$5, $\beta(t)=-A(t)$6, and $\beta(t)=-A(t)$7 defined by nested commutators. On 3-regular MaxCut, the number of layers required to reach the Goemans–Williamson ratio $\beta(t)=-A(t)$8 still scales linearly with problem size, but the fitted slope drops from $\beta(t)=-A(t)$9 for first-order FALQON to $\frac{dJ}{dt}=-A(t)^2\le 0,$0 for SO-FALQON (Arai et al., 2024). GD-QLC keeps the feedback structure but performs a small number of local gradient steps for each new layer using the observables $\frac{dJ}{dt}=-A(t)^2\le 0,$1 and $\frac{dJ}{dt}=-A(t)^2\le 0,$2, yielding faster convergence and smoother control amplitudes than vanilla FALQON in classical simulations (Mozakka et al., 12 Feb 2026).

FOCQS targets a different limitation: FALQON’s purely forward-greedy character. It approximates the final-time control gradient

$\frac{dJ}{dt}=-A(t)^2\le 0,$3

from locally measured quantities $\frac{dJ}{dt}=-A(t)^2\le 0,$4 and $\frac{dJ}{dt}=-A(t)^2\le 0,$5, then retroactively updates earlier layers. In the reported tests on random Ising and MIS instances, this perturbative back-update improves the final approximation ratio over FALQON, and repeated application improves it further, albeit with diminishing returns (Brady et al., 2024).

Other variants change the outer structure rather than the local law. Measurement-Guided Initialization repeatedly runs shallow FALQON, extracts single-qubit marginals from dominant output bitstrings, and uses those marginals to prepare a biased product-state restart. For weighted MaxCut on complete graphs, the paper reports that similar energy targets that standard FALQON reaches only at depths of order $\frac{dJ}{dt}=-A(t)^2\le 0,$6 to $\frac{dJ}{dt}=-A(t)^2\le 0,$7 can be approached by MGI-FALQON using fixed depths

$\frac{dJ}{dt}=-A(t)^2\le 0,$8

trading coherent depth for repeated measurement-and-restart cycles (Rattighieri et al., 23 Feb 2026).

Two recent directions partly relax the original anti-variational stance. Optimal FALQON retains the FALQON structure but treats the per-layer timestep $\frac{dJ}{dt}=-A(t)^2\le 0,$9 and scaling factor $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$0 as decision variables in a two-dimensional classical subproblem. On all $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$1 non-isomorphic $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$2-regular graphs with $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$3 vertices, the median success probabilities reported for standard versus optimized FALQON are $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$4 versus $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$5 for the first-order form, and $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$6 versus $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$7 for the second-order form (Mancini et al., 8 May 2026). A different compromise uses supervised learning to predict the entire FALQON parameter curve $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$8 from the problem instance in one shot, shifting the feedback burden to offline data generation; in the tested weighted 3-regular MaxCut setting, the predicted curves remained close to FALQON reference curves and outperformed digitized linear annealing schedules (Pérez et al., 13 Jan 2026).

5. Hardware implementations, noise, and robustness

The original experimental demonstration of FALQON was a MaxCut instance on IBM’s superconducting processor ibmq_manila. For a three-qubit chain graph and $$|\psi_\ell\rangle = U_d(\beta_\ell)U_p\cdots U_d(\beta_1)U_p|\psi_0\rangle,$$9 layers with $H_d$00, the experiment observed monotonic decrease of $H_d$01 and monotonic increase of the ground-state success probability up to layer $H_d$02, after which hardware noise prevented further improvement even though noiseless simulations continued to improve (Magann et al., 2021). A later counterdiabatic-inspired variant was also run on IBM cloud hardware for a four-qubit mixed-field Ising model, where shallow-depth performance favored the local $H_d$03-control version among the tested CD-FQA protocols (Malla et al., 2024).

Hardware-aware compilation has become increasingly important because feedback-based algorithms typically require many layers. On a realistic neutral-atom Rydberg platform, FALQON phase-separation blocks for MaxCut can be implemented more efficiently with an optimized small-angle controlled-phase gate than with standard CZ-based decompositions. For the benchmark small-angle gate with $H_d$04 rad, corresponding to $H_d$05, the reported maximum fidelity is approximately $H_d$06 including spontaneous emission and above $H_d$07 without spontaneous emission. In 2- and 3-qubit FALQON simulations, this small-angle implementation outperformed CZ-based compilation under the master-equation noise model because it halves the entangling-gate count per $H_d$08 block and reduces total two-qubit evolution time from $H_d$09 to $H_d$10 for 2 qubits and from $H_d$11 to $H_d$12 for 3 qubits (Li et al., 2024).

Robustness analysis has also been developed at the control-theoretic level. Under coherent multiplicative control errors,

$H_d$13

FALQON is shown to be asymptotically robust to bounded systematic errors satisfying $H_d$14: under the stated nondegeneracy, gap, and connectivity assumptions, the state converges to the ground state, or at least the expected cost converges to the optimal ground energy in the degenerate case (Legnini et al., 3 Jul 2025). For independent coherent errors, the same paper derives the fidelity bound

$H_d$15

which motivates a regularized Lyapunov objective

$H_d$16

and the robust control law

$H_d$17

In the reported MaxCut simulations, the regularized version improves performance under independent coherent errors, while standard FALQON remains asymptotically robust to systematic coherent errors (Legnini et al., 3 Jul 2025).

These results sharpen the usual NISQ narrative. Feedback-based quantum optimization does not merely face generic circuit noise; it faces a distinctive combination of repeated measurement, deepening circuits, and control-law sensitivity. The literature’s response has therefore been correspondingly hybrid: improved measurement primitives, hardware-native gate design, and explicit robustness regularization.

6. Classical counterparts, quantum advantage, and unresolved questions

A major recent development is the explicit construction of classical counterparts to feedback-based quantum optimization. Using quantum–classical correspondence for spin systems, the exact classical analogue of FALQON is written as

$H_d$18

and the exact classical analogue of inhomogeneous FALQON as

$H_d$19

This framework makes it possible to compare feedback-based quantum dynamics against directly corresponding classical spin dynamics rather than against unrelated heuristics (Hattori et al., 13 May 2026).

That comparison yields a mixed verdict. In the small-scale 2-SAT benchmarks reported in the classical-counterpart study, quantum algorithms can be advantageous to classical algorithms in terms of the quality of solutions, while classical algorithms tend to show faster convergence than quantum ones. FALQON outperformed its exact classical counterpart CC-FALQON, but iFALQON performed worse than CC-iFALQON. In the broader comparison, FALQON, CC-iFALQON, CACAO, HOT-CACAO, and HOT-CACAO+ ended with similar low final energy densities on average, while FALQON obtained lower-energy solutions in $H_d$20 out of $H_d$21 instances, which the paper presents as evidence for the possibility of quantum advantage rather than a blanket superiority claim (Hattori et al., 13 May 2026).

The other side of that comparison is the power of classical quantum-inspired descendants. CACAO replaces the full quantum dynamics by classical spin dynamics derived from local counterdiabatic feedback ideas,

$H_d$22

and scales to $H_d$23 spins on the reported benchmarks (Hatomura, 10 Jun 2025). The later classical-counterpart paper extends this direction to HOT-CACAO and HOT-CACAO+, where higher-order counterdiabatic-inspired terms are added; HOT-CACAO+ is reported to show significant scalability for higher-order unconstrained binary optimization problems, including large 3-SAT/HUBO instances (Hattori et al., 13 May 2026).

These classical results expose the main unresolved question surrounding the field. Feedback-based quantum optimization clearly defines a distinct algorithmic paradigm: measurement-conditioned control in place of global variational training. But its practical value depends on a three-way comparison that is still incomplete. First, against QAOA-like global optimization, feedback methods trade outer-loop training for repeated online estimation and often greater depth. Second, against improved feedback variants, the balance shifts among depth, measurement cost, and classical side computation. Third, against classical counterparts, any quantum advantage appears problem-dependent rather than universal (Magann et al., 2021, Hattori et al., 13 May 2026).

Several limitations remain explicit across the literature. Strong general convergence guarantees are still lacking outside restrictive settings; finite-shot estimation, decoherence, and control noise remain central practical constraints; and broad large-scale empirical validation on hardware is limited. A plausible implication is that feedback-based quantum optimization is best regarded not as a single algorithm but as a growing control-theoretic design space, spanning pure Lyapunov descent, counterdiabatic assistance, perturbative back-updates, learned schedule surrogates, shallow-run state refinement, and classical counterparts. The common core is unchanged: optimization is moved inside the dynamical feedback loop, and the decisive research question becomes how much structure that loop can exploit before its measurement and depth costs dominate.