Feedback-Based Quantum Algorithms

• Feedback-based quantum algorithms are adaptive, closed-loop protocols that update circuit parameters using real-time measurement data without an outer classical optimization loop.
• They employ a Lyapunov control mechanism to ensure monotonic energy descent, enabling applications in combinatorial optimization and physical Hamiltonian ground-state preparation.
• Empirical studies show these methods can accelerate convergence and enhance robustness, although they involve tradeoffs in measurement overhead and circuit depth.

Searching arXiv for papers on feedback-based quantum algorithms and key variants mentioned. Searching for the original FALQON paper and extensions on time rescaling, robustness, learned schedules, constrained optimization, and excited states. arXiv search: "feedback-based quantum algorithms FALQON time-rescaled robust excited states constrained optimization" Feedback-based quantum algorithms are gate-model, closed-loop quantum protocols in which a parameterized circuit is grown layer by layer, the evolving state is measured after each layer, and a deterministic feedback law uses those measurement estimates to set the next control parameter. In contrast to variational quantum algorithms such as VQE or QAOA, the circuit parameters are not optimized by an outer classical loop; instead, the optimization logic is embedded into the quantum evolution through measurement-based updates derived from quantum Lyapunov control. In the literature centered on FALQON and its descendants, the objective is typically the expectation value of a problem Hamiltonian, and the control law is designed so that this energy decreases monotonically with depth in the ideal limit (Magann et al., 2021, Pérez et al., 13 Jan 2026).

1. Historical emergence and conceptual scope

A broader genealogy of feedback-based quantum algorithms predates FALQON. In adaptive quantum metrology, a feedback policy maps previous measurement outcomes to future control actions and to a final estimate, and optimized adaptive policies were shown numerically to achieve $V_\text{H} \propto N^{-\alpha_{\text{PSO}}}$ with $\alpha_{\text{PSO}} \approx 1.494$, significantly better than the standard quantum limit benchmark in variance scaling (Hentschel et al., 2011). That line of work already exhibited several features that later became central in quantum optimization: sequential measurements, on-the-fly control updates, and policy learning from either simulated or real-world trials.

The modern optimization-focused formulation emerged with FALQON, introduced as “Feedback-based ALgorithm for Quantum OptimizatioN,” where qubit measurements are used to constructively assign values to circuit parameters so that the quality of the solution improves monotonically with circuit depth, without any classical optimization effort (Magann et al., 2021). In this framework, a FALQON layer looks like a QAOA or digitized quantum annealing Trotter step, but the parameter $\beta_j$ is computed online from measurement data rather than optimized offline (Pérez et al., 13 Jan 2026). This same control-theoretic structure was later extended from combinatorial optimization to ground-state preparation for Fermi–Hubbard Hamiltonians and molecular Hamiltonians represented in second quantization, where the feedback law again replaces classical optimization by a deterministic control update (Larsen et al., 2023).

This trajectory suggests that “feedback-based quantum algorithms” is best understood not as a single algorithm, but as a design paradigm. A problem Hamiltonian encodes an objective, a driver Hamiltonian induces transitions, and measurement-based feedback steers the state toward a target eigenspace or low-energy manifold. In the literature summarized here, the dominant instantiation is Lyapunov-inspired closed-loop control, but the same paradigm now includes constrained optimization, excited-state preparation, time-rescaled protocols, robustness-regularized variants, gradient-accelerated updates, and hybrid unitary–non-unitary schemes (Rahman et al., 2024, Rahman et al., 2024, Rattighieri et al., 2 Apr 2025, Mozakka et al., 12 Feb 2026, Long et al., 15 Dec 2025).

2. Control-theoretic formulation and circuit realization

The canonical continuous-time model is

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

where $H_p$ is the problem Hamiltonian and $H_d$ is a driver Hamiltonian that does not commute with $H_p$ (Magann et al., 2021, Legnini et al., 3 Jul 2025). The Lyapunov function is usually the energy expectation

$$V(\ket{\psi(t)}) = \bra{\psi(t)} H_p \ket{\psi(t)},$$

or, equivalently, a shifted version $V(\ket{\psi}) = \bra{\psi}H_p\ket{\psi} - E_0$ with $E_0 = \lambda_{\min}(H_p)$ (Legnini et al., 3 Jul 2025, Mozakka et al., 12 Feb 2026). Its time derivative can be written as

$\alpha_{\text{PSO}} \approx 1.494$0

The basic Lyapunov feedback law is

$\alpha_{\text{PSO}} \approx 1.494$1

which yields

$\alpha_{\text{PSO}} \approx 1.494$2

This is the core mechanism by which FALQON-style algorithms turn measurement data into monotonic energy descent (Magann et al., 2021, Larsen et al., 2023).

On a gate-model device, the evolution is discretized with time step $\alpha_{\text{PSO}} \approx 1.494$3 and first-order Trotterization. One defines

$\alpha_{\text{PSO}} \approx 1.494$4

and the depth-$\alpha_{\text{PSO}} \approx 1.494$5 state

$\alpha_{\text{PSO}} \approx 1.494$6

The discrete feedback law is

$\alpha_{\text{PSO}} \approx 1.494$7

Operationally, the commutator is expanded into Pauli strings, those strings are measured, and the measured expectations determine the next control parameter (Magann et al., 2021, Larsen et al., 2023). The resulting circuit family is layered and adaptive: one prepares the initial state, applies the first $\alpha_{\text{PSO}} \approx 1.494$8 layers, measures the feedback observable, computes $\alpha_{\text{PSO}} \approx 1.494$9, and then extends the circuit by one more layer. This produces a parameter sequence specific to the problem instance being solved (Pérez et al., 13 Jan 2026).

The framework is explicitly contrasted with open-loop variational methods. QAOA and VQE treat the full parameter vector as a set of free variables optimized by a classical routine, while FALQON and related feedback algorithms use measurement results on the current state to set the next parameter by direct evaluation of a control law (Magann et al., 2021, Legnini et al., 3 Jul 2025). The practical implication is that feedback-based algorithms do not search a global parameter landscape; they instead follow a state-dependent trajectory defined by measured commutators and by a Lyapunov decrease condition.

3. Principal algorithmic families and extensions

The most direct generalization of ground-state FALQON to excited states is the weighted feedback-based quantum algorithm for excited-state calculation. Its construction is inspired by weighted SSVQE: one evolves $\beta_j$0 mutually orthogonal initial states under a common control, defines a weighted Lyapunov function

$\beta_j$1

and chooses weights with $\beta_j$2 for $\beta_j$3, so that the minimum is attained at $\beta_j$4. The associated feedback law uses a weighted sum of commutator expectations, and by modifying the weights one can either prepare the lowest $\beta_j$5 eigenstates or isolate the $\beta_j$6-th excited state (Rahman et al., 2024).

Constrained optimization led to FALQON-C, which addresses quadratic constrained binary optimization by introducing a Hermitian operator $\beta_j$7 whose ground state is the optimal feasible solution. The notable design choice is that $\beta_j$8 enters the Lyapunov function and feedback law, while the circuit itself continues to implement the simpler unconstrained cost Hamiltonian $\beta_j$9. This separates circuit complexity from constraint handling and, in the formulation given, reduces circuit depth relative to applying original FALQON to a QUBO-converted constrained problem (Rahman et al., 2024).

Time-rescaled feedback algorithms modify the evolution by a time reparametrization $$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$0, so that the rescaled Hamiltonian is $$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$1. For TR-FQA and TR-FALQON, the feedback law becomes

$$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$2

and the discrete circuit uses rescaled unitaries

$$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$3

This construction preserves the Lyapunov decrease condition while accelerating convergence in shallow-depth regimes (Rattighieri et al., 2 Apr 2025). A related development, formulated specifically in DNA assembly, is TR-FALQON together with SO-FALQON, where a second-order expansion of the energy change per layer is used to derive a hybrid control rule that can allow larger $$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$4 and fewer layers (Prado et al., 24 Feb 2026).

Several works explicitly target the limitations of purely local Lyapunov feedback. FOCQS develops an analytic perturbative framework that takes the local greedy information collected by Lyapunov feedback control and uses it to perturbatively update previous control layers, similar to the global optimal control achievable using Pontryagin optimal control (Brady et al., 2024). A different route is GD-QLC, which augments Lyapunov feedback with per-layer gradient estimation based on

$$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$5

and then performs local gradient descent on $$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$6 while preserving the low-dimensional feedback structure (Mozakka et al., 12 Feb 2026).

The most consequential extension for strongly correlated systems is the insertion of imaginary-time evolution into the feedback loop. ITE-FALQON modifies the layer update by interleaving the unitary FALQON step with a normalized first-order imaginary-time map

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

which suppresses excited-state components and restores monotonic descent in degenerate or symmetry-trapped settings (Long et al., 15 Dec 2025). In the thermofield-double setting of the Maldacena–Qi model, this idea is combined with time rescaling to form ITE-TR-FALQON, which integrates the imaginary-time evolution present in imaginary-time-enhanced FALQON with the time-rescaling mechanism (Pexe et al., 2 Jul 2026).

4. Application domains

Combinatorial optimization remains the most visible application domain. The original FALQON paper treated MaxCut, both numerically and experimentally, on a superconducting processor (Magann et al., 2021). Later work broadened this to weighted MaxCut, MAX-CLIQUE, MIN-COVER, and quadratic constrained binary optimization (Mozakka et al., 12 Feb 2026, Rahman et al., 2024). The QCBO formulation shows that the feedback paradigm can accommodate explicit equality and inequality constraints through a Lyapunov operator rather than by embedding all penalties directly into the cost unitary (Rahman et al., 2024).

Ground-state preparation for physical Hamiltonians forms a second major branch. Feedback-based quantum algorithms were formulated for Fermi–Hubbard Hamiltonians and for molecular Hamiltonians represented in second quantization, with $$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$8 chosen as the hopping or one-body term and the feedback law derived from the commutator with the interaction sector (Larsen et al., 2023). The same formalism was later applied to the ANNNI model, where TR-FQA was shown to reduce the required circuit depth by several hundred layers relative to standard FQA (Rattighieri et al., 2 Apr 2025). Strongly correlated and highly entangled targets motivate the imaginary-time-enhanced variants: ITE-FALQON restores reliable convergence for Fermi–Hubbard lattices up to $$i \frac{d}{dt} \ket{\psi(t)} = \bigl(H_p + \beta(t) H_d\bigr)\ket{\psi(t)},$$9 in situations where pure FALQON breaks down, and ITE-TR-FALQON prepares the TFD-like ground state of the Maldacena–Qi model with fidelities close to unity (Long et al., 15 Dec 2025, Pexe et al., 2 Jul 2026).

A particularly distinctive case study is de novo DNA assembly. Long-read DNA fragments are mapped to a QUBO and then to an Ising Hamiltonian whose ground state encodes the optimal read ordering. In this formulation, fixing read 0 in position 0 reduces the qubit count so that 4 reads require 9 qubits, 5 reads require 16 qubits, and 6 reads require 25 qubits (Prado et al., 24 Feb 2026). The feedback algorithms studied there are standard FALQON, SO-FALQON, and TR-FALQON, tested on long-read DNA fragments from SARS-CoV-2 and human mitochondrial DNA (Prado et al., 24 Feb 2026).

The field has also generated quantum-inspired and hybrid offloading approaches. CACAO is a counterdiabaticity-assisted classical algorithm for optimization derived from the feedback and counterdiabatic structure of FALQON and CD-FQA, and it was studied on systems up to $H_p$0 spins (Hatomura, 10 Jun 2025). A different direction replaces online feedback by a learned open-loop schedule: a teacher–student model maps a MaxCut instance to an associated FQA parameter curve in a single classical inference step, thereby attempting to remove the layer-wise sampling overhead associated with measuring $H_p$1 (Pérez et al., 13 Jan 2026).

5. Empirical performance, robustness, and resource tradeoffs

The original hardware demonstration established the operational viability of the paradigm on a superconducting processor. For a 3-qubit MaxCut instance on ibmq_manila, with $H_p$2, depth $H_p$3, and $H_p$4 shots per circuit, the experiment observed a monotonic decrease of $H_p$5 and a monotonic increase of the ground-state probability up to layer $H_p$6, after which hardware noise prevented further improvement even though the noiseless simulation continued to descend (Magann et al., 2021). In noise-free MaxCut simulations on unweighted, connected 3-regular graphs with $H_p$7, the mean number of layers required to reach $H_p$8 or $H_p$9 grew roughly linearly with $H_d$0 (Magann et al., 2021).

Time rescaling consistently improves the shallow-depth regime. In MaxCut, TR-FALQON accelerates convergence to the optimal solution in the early layers of the circuit and significantly outperforms its standard counterpart in shallow-depth regimes (Rattighieri et al., 2 Apr 2025). In ANNNI ground-state preparation, TR-FQA demonstrates superior convergence, reducing the required circuit depth by several hundred layers (Rattighieri et al., 2 Apr 2025). The DNA-assembly study gives a more problem-specific picture: all algorithms were run up to 300 layers; for the SARS-CoV-2 instance with 5 reads mapped to 16 qubits, TR-FALQON 1 achieved the lowest final energy and the highest success probability, exceeding 50%, while TR-FALQON 2 converged fastest in terms of layers and saturated around 40%; for human mitochondrial DNA with 5 reads, TR-FALQON 1 again achieved the lowest final energy and highest solution probability, about 60%; and for 6 reads, TR-FALQON 1 surpassed 40% while standard FALQON remained below 10% after 300 layers (Prado et al., 24 Feb 2026).

Robustness has been analyzed explicitly for coherent control errors. FALQON is asymptotically robust with respect to systematic coherent control errors in the multiplicative model $H_d$1, and the robustness analysis yields a fidelity bound for independent errors in terms of a problem-dependent Lipschitz constant

$H_d$2

A regularized Lyapunov function

$H_d$3

leads to the robust feedback law

$H_d$4

which reduces the magnitude of $H_d$5 and improves robustness to independent coherent control errors. Simulations on an 8-node MaxCut problem with $H_d$6 showed that the robust version with $H_d$7 had significantly smaller final cost error than standard FALQON with $H_d$8 under independent errors (Legnini et al., 3 Jul 2025).

Measurement overhead, however, remains a central tradeoff. In hardware, each FALQON layer requires estimating $H_d$9, so the total number of shots scales like $H_p$0 (Pérez et al., 13 Jan 2026). This motivates schedule distillation by machine learning. In the teacher–student study, weighted 3-regular graphs with $H_p$1 and weights sampled from $H_p$2 were used to train a graph neural network to output an FQA parameter curve of length $H_p$3. The total dataset contained 2,240 instances, and the predicted curves produced similar results to FALQON reference curves while outperforming linear quantum annealing schedules (Pérez et al., 13 Jan 2026). Gradient-accelerated feedback offers a different compromise: GD-QLC increases per-layer measurement work by requiring both $H_p$4 and $H_p$5, but in the reported weighted MAX-CUT, MAX-CUT, MAX-CLIQUE, and MIN-COVER experiments it achieved significantly faster convergence and better-behaved control parameters than FALQON, while avoiding the extremely large $H_p$6 spikes observed in SO-FALQON (Mozakka et al., 12 Feb 2026).

6. Limitations, misconceptions, and emerging directions

A recurring misconception is that “optimization-free” means “resource-light.” The literature does not support that interpretation. Feedback-based quantum algorithms eliminate an outer classical optimization loop, but they replace it with layer-wise measurement and classical processing of observables such as $H_p$7 and, in more advanced variants, higher commutators or learned auxiliary quantities (Magann et al., 2021, Pérez et al., 13 Jan 2026, Mozakka et al., 12 Feb 2026). This is why sampling cost, shot allocation, and measurement grouping remain central implementation questions.

Another misconception is that monotonic energy descent is equivalent to universal convergence. Several papers explicitly show otherwise. In strongly correlated systems, FALQON can fail in the presence of spectral degeneracies, where the feedback signal collapses and the evolution cannot reach the ground state (Long et al., 15 Dec 2025). In the Maldacena–Qi model, standard FALQON and TR-FALQON face severe kinetic limitations, failing to converge to the highly entangled ground state when initialized in trivial product states, because of symmetry traps and insufficient entanglement growth (Pexe et al., 2 Jul 2026). These results make a more general point: a Lyapunov law can ensure local descent of an objective while still leaving the dynamics confined to an invariant set that does not contain the desired target state.

This suggests a sharper characterization of the present frontier. The field is moving from first-order, purely unitary, single-parameter feedback toward richer hybrids: regularized control laws for robustness (Legnini et al., 3 Jul 2025); perturbative correction of previous controls in the spirit of Pontryagin optimal control (Brady et al., 2024); per-layer gradient acceleration (Mozakka et al., 12 Feb 2026); and hybrid unitary–non-unitary schemes that insert imaginary-time evolution to break degeneracies and symmetry traps (Long et al., 15 Dec 2025, Pexe et al., 2 Jul 2026). The TFD study states the implication most strongly: the introduction of non-unitary dynamics is strictly necessary to break symmetry traps and filter out excited states, while time-rescaling drastically accelerates algorithm convergence (Pexe et al., 2 Jul 2026).

Scalability remains unresolved. Demonstrations include MaxCut up to 20 qubits, DNA assembly up to 25 qubits, Fermi–Hubbard lattices up to $H_p$8, and classical analogues up to $H_p$9 spins, but the cost of repeated measurement of commutators and double commutators grows with Hamiltonian complexity (Magann et al., 2021, Prado et al., 24 Feb 2026, Long et al., 15 Dec 2025, Hatomura, 10 Jun 2025). A plausible implication is that the long-term development of feedback-based quantum algorithms will depend on three coupled advances: more informative yet cheaper feedback observables, schedule-compression schemes that reduce or amortize sampling, and hardware-aware implementations of non-unitary or counterdiabatic enhancements. Within that trajectory, feedback-based quantum algorithms are no longer merely an alternative to VQE or QAOA; they have become a broader control-theoretic framework for adaptive quantum state preparation, quantum optimization, and instance-specific schedule generation across both classical and quantum objectives (Pérez et al., 13 Jan 2026, Brady et al., 2024).