Hamiltonian-based QAOA: Advances & Applications

• Hamiltonian-based QAOA is a quantum variational method that encodes classical optimization objectives into cost and mixer Hamiltonians for alternating evolution.
• It discretizes adiabatic evolution by alternating controlled Hamiltonian dynamics, bridging the gap between continuous quantum annealing and digital quantum circuits.
• Custom Hamiltonian designs—such as higher-order cost terms and constraint-preserving mixers—enhance its efficiency and adaptability for diverse optimization challenges.

Hamiltonian-based Quantum Approximate Optimization Algorithm (QAOA) denotes the class of alternating-operator variational methods in which both the phase-separation family and the mixer family are generated by Hamiltonians. In the canonical formulation, a classical objective function $f$ is encoded as a diagonal operator $H_f$ through $H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$, and a depth-$p$ state is prepared by alternating the corresponding cost evolution with a mixer evolution, typically from the initial state $\ket{+}^{\otimes n}$. Within the literature, this Hamiltonian picture has been extended well beyond the original MaxCut setting: to higher-order cost Hamiltonians, custom phase operators, feasible-subspace-preserving mixers, counterdiabatic corrections, local-Hamiltonian state preparation, continuous-variable optimization, and even computational universality (Hadfield et al., 2017, Lloyd, 2018).

1. Formal definition and canonical operator structure

In the terminology of "From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz" (Hadfield et al., 2017), Hamiltonian-based QAOA is the subclass in which the phase separator and mixer are both Hamiltonian evolutions,

$\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$

with the output state

$$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$

The same work also distinguishes local Hamiltonian-based QAOA, in which the mixer Hamiltonian is a sum of polynomially many local terms (Hadfield et al., 2017).

For unconstrained binary optimization, the standard mixer is the transverse field,

$H_M=-\sum_{i=1}^n X_i,$

whose ground state is $\ket{+}^{\otimes n}$, prepared by Hadamards on $\ket{0}^{\otimes n}$. For MaxCut, a canonical cost Hamiltonian is

$H_f$0

or equivalently the usual Ising/QUBO form with pairwise $H_f$1 couplings and optional local $H_f$2 fields (Stein et al., 15 Nov 2025, Giovagnoli, 23 Nov 2025).

Hamiltonian construction is not restricted to quadratic objectives. The tutorial "An Introduction to the Quantum Approximate Optimization Algorithm" extends the mapping from QUBO to PUBO, yielding cost Hamiltonians of the form

$H_f$3

with corresponding $H_f$4 gate decompositions for higher-order terms (Giovagnoli, 23 Nov 2025). The graph-coloring study "QAOA of the Highest Order" makes the same point operationally: gate-model QAOA can directly exploit higher-order Pauli gadgets instead of forcing the objective into a quadratic annealing-style encoding (Campbell et al., 2021).

2. Discretized annealing, integrated resources, and schedule physics

A standard interpretation of Hamiltonian-based QAOA is as a discretization of adiabatic evolution. "Quantum Optimization Algorithms" writes the continuous interpolation as

$H_f$5

and approximates the time-ordered evolution by Suzuki–Trotter splitting,

$H_f$6

with $H_f$7 and $H_f$8 giving the familiar QAOA angles (Stein et al., 15 Nov 2025).

"Universal Resources for QAOA and Quantum Annealing" refines this correspondence by introducing integrated coordinates

$H_f$9

so that quantum annealing becomes a path in Hamiltonian space,

$H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$0

In this language, QAOA is a first-order Trotter approximation to the same path, with cumulative angles $H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$1 and $H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$2 identified with the continuous resources (Díez-Valle et al., 3 Jun 2025).

That work also reports that optimized QAOA angles collapse onto universal annealing-like trajectories in the $H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$3 plane and interprets QAOA and QA outputs through a bimodal pseudo-Boltzmann distribution

$H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$4

Within that phenomenology, the coldest temperature scales as $H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$5, while $H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$6 and $H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$7 (Díez-Valle et al., 3 Jun 2025).

A related but distinct schedule-centered analysis appears in quantum chemistry. "Quantum Alternating Operator Ansatz (QAOA) Phase Diagrams and Applications for Quantum Chemistry" uses low-parameter linear ramps,

$H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$8

and studies performance as a function of $H_f\ket{\mathbf{x}}=f(\mathbf{x})\ket{\mathbf{x}}$9 via squared overlap with the target ground state. The resulting phase diagrams show a regime in which non-adiabatic schedules perform better than the adiabatic limit while employing lower quantum circuit depth (Kremenetski et al., 2021).

3. Hamiltonian design: custom phase operators, higher-order costs, and shortcut terms

A major branch of Hamiltonian-based QAOA research treats the Hamiltonians themselves as design variables rather than fixed problem encodings. In "Quantum approximate optimization algorithm with random and subgraph phase operators", the phase Hamiltonian $p$0 is allowed to differ from the original cost Hamiltonian $p$1, with QAOA still optimizing the original objective. For MaxCut, the paper derives a closed-form $p$2 expression for each edge expectation $p$3 in terms of: $p$4, indicating whether $p$5 is present in the custom phase operator; $p$6 and $p$7, counting incident phase terms; and $p$8, counting triangle-condition pairs (Wilkie et al., 2024).

The same study evaluates random, subgraph, triangle-removed, and maximal-degree-edge-removed phase operators on all non-isomorphic 8-vertex graphs. At $p$9, the percentage of tested graphs with at least one custom phase operator outperforming standard QAOA was $\ket{+}^{\otimes n}$0 for random phase operators, $\ket{+}^{\otimes n}$1 for subgraph phase operators, $\ket{+}^{\otimes n}$2 for triangle-removed phase operators, and $\ket{+}^{\otimes n}$3 for maximal-degree-edge-removed phase operators (Wilkie et al., 2024). The paper’s interpretation is structural: triangle terms and high-degree vertices can worsen the one-layer expression, so Hamiltonian pruning can improve approximation ratio while also reducing circuit complexity.

Higher-order Hamiltonian design leads to a different conclusion: some objectives should not be quadratized at all. In the four-corners graph-coloring example of "QAOA of the Highest Order", the native higher-order binary encoding yields a quartic Ising Hamiltonian

$\ket{+}^{\otimes n}$4

implemented with $\ket{+}^{\otimes n}$5 two-qubit gates, compared with $\ket{+}^{\otimes n}$6 for the order-reduced binary formulation and $\ket{+}^{\otimes n}$7 for the unary formulation. The reported QAOA simulations favor the native higher-order encoding in both gate cost and optimization behavior (Campbell et al., 2021).

A third design axis adds explicit auxiliary Hamiltonians. "Shortcuts to Quantum Approximate Optimization Algorithm" introduces S-QAOA, where the standard alternating structure is enriched by an extra two-body interaction

$\ket{+}^{\otimes n}$8

For the MaxCut and SK instances studied there, the $\ket{+}^{\otimes n}$9 interaction performs best numerically. The paper attributes this to counterdiabatic structure visible in the BCH expansion and reports, for weighted 3-regular MaxCut, that at $\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$0 S-QAOA still gives about a $\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$1 fidelity improvement over QAOA, while for the SK model it reaches about $\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$2 fidelity at $\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$3 (Chai et al., 2021).

4. Constraint-preserving mixers and hard-feasibility Hamiltonians

Constrained optimization makes the mixer Hamiltonian a primary design object. The general design criteria in (Hadfield et al., 2017) are that the mixer should preserve the feasible subspace and connect all feasible states. This is the Hamiltonian analogue of replacing soft penalty enforcement by hard subspace-preserving dynamics.

Several concrete constructions instantiate that principle. One family uses mixers tailored to a feasible superposition $\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$4. In the tutorial treatment of constrained QAOA, the Grover mixer satisfies

$\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$5

and at $\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$6 becomes the Grover-like reflection $\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$7 (Stein et al., 15 Nov 2025). Another family uses excitation-preserving Hamiltonians. For fixed-cardinality portfolio optimization, "Constrained Counterdiabatic QAOA for Portfolio Optimization" employs the Hamming-weight-preserving XY mixer

$\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$8

together with the Dicke initial state $\phaseUnitary(\gamma)=e^{-i\gamma H_P},\qquad \mixerUnitary(\beta)=e^{-i\beta H_M},$9, so that the baseline alternating dynamics remain inside the feasible $$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$0 sector (Falla et al., 7 May 2026).

A different approach is to learn the mixer itself. "Quantum constraint learning for quantum approximate optimization algorithm" parameterizes a learned constrained unitary as

$$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$1

with explicit conditions that $$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$2 and that feasible states remain connected under the evolution. The paper also introduces the projection operator $$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$3, the notion of quantum leakage when the learned mixer is imperfect, and the Wasserstein-based metric

$$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$4

to assess constrained performance (Radha, 2021).

Hard-feasibility can also be encoded algebraically in the driver Hamiltonian. Choco-Q defines a commute Hamiltonian $$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$5 satisfying $$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$6, where $$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$7 is the operator form of the linear constraint. Its universal formulation is

$$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$8

and the paper claims $$\ket{\boldsymbol \beta,\boldsymbol \gamma} = e^{-i \beta_p H_M} e^{-i \gamma_p H_P}\cdots e^{-i \beta_1 H_M} e^{-i \gamma_1 H_P}\ket{s}.$$9 in-constraints rate, more than $H_M=-\sum_{i=1}^n X_i,$0 algorithmic improvement in successfully finding the optimal solution, and $H_M=-\sum_{i=1}^n X_i,$1 end-to-end acceleration compared to prior QAOA designs (Xiang et al., 31 Mar 2025).

Constraint preservation can be combined with counterdiabatic augmentation. In CCD-QAOA, approximate adiabatic gauge potentials derived from nested commutators of the Ising portfolio Hamiltonian and the XY mixer are inserted into each layer,

$H_M=-\sum_{i=1}^n X_i,$2

and the paper reports consistently higher approximation ratios than standard XY-mixer QAOA, Grover-mixer QAOA, and penalty-based QAOA at fixed depth, while also noting leakage and circuit-overhead tradeoffs introduced by the CD terms (Falla et al., 7 May 2026).

5. Extensions to chemistry, local Hamiltonians, continuous variables, and other nonclassical objectives

Hamiltonian-based QAOA is not limited to finding good classical bit strings. In quantum chemistry, the target is a many-body quantum ground state, not a computational-basis optimum. "Quantum Alternating Operator Ansatz (QAOA) Phase Diagrams and Applications for Quantum Chemistry" therefore chooses

$H_M=-\sum_{i=1}^n X_i,$3

with the Hartree–Fock ground state as the initial state, so that the ansatz becomes

$H_M=-\sum_{i=1}^n X_i,$4

The paper studies $H_M=-\sum_{i=1}^n X_i,$5, $H_M=-\sum_{i=1}^n X_i,$6, $H_M=-\sum_{i=1}^n X_i,$7, and $H_M=-\sum_{i=1}^n X_i,$8, with $H_M=-\sum_{i=1}^n X_i,$9 values around $\ket{+}^{\otimes n}$0–$\ket{+}^{\otimes n}$1 qubits, and uses ASCI to simulate the relevant subspace (Kremenetski et al., 2021).

A more direct many-body generalization appears in "A Quantum Approximate Optimization Algorithm for Local Hamiltonian Problems", which introduces a specialized algorithm called HamQAOA for Local Hamiltonian Problems and Quantum MaxCut. For a general 2-local Hamiltonian, it uses four drivers,

$\ket{+}^{\otimes n}$2

and the depth-$\ket{+}^{\otimes n}$3 state

$\ket{+}^{\otimes n}$4

For Quantum MaxCut on Heisenberg systems, the paper reports rigorous high-girth guarantees, improved ground-energy-density bounds with depth, and numerical evidence that linear-depth HamQAOA can deterministically prepare exact ground states of 1-dimensional antiferromagnetic Heisenberg spin chains; it explicitly finds exact ground states for $\ket{+}^{\otimes n}$5 at $\ket{+}^{\otimes n}$6 and $\ket{+}^{\otimes n}$7 at $\ket{+}^{\otimes n}$8 (Kannan et al., 2024).

Continuous-variable Hamiltonian-based QAOA goes further by replacing bit strings with positions $\ket{+}^{\otimes n}$9. In CV-QAOA, the cost Hamiltonian is

$\ket{0}^{\otimes n}$0

and the default mixer is kinetic,

$\ket{0}^{\otimes n}$1

The key Heisenberg-picture update derived in (Verdon et al., 2019) is

$\ket{0}^{\otimes n}$2

so each layer acts like gradient descent with momentum, and for broad initial superpositions the algorithm becomes "gradient descent in superposition". The same framework incorporates equality and inequality constraints through penalty potentials and was numerically tested on the Styblinski–Tang function (Verdon et al., 2019).

Problem-specific Hamiltonian synthesis has also been developed in domains such as digital communications. For ML detection with Gray-labelled constellations, the objective is transformed into a pseudo-Boolean polynomial and then into a diagonal Hamiltonian

$\ket{0}^{\otimes n}$3

with the paper showing that for Gray-labelled rectangular MQAM the in-phase and quadrature qubits are independent in the Hamiltonian (Cui et al., 2022).

6. Universality and the limits of computational expressivity

Hamiltonian-based QAOA is not merely an optimization heuristic. "Quantum approximate optimization is computationally universal" proves that the alternating-Hamiltonian template can implement universal quantum computation when the layer times are treated as control parameters rather than variational angles (Lloyd, 2018).

The construction uses

$\ket{0}^{\otimes n}$4

and a specially engineered one-dimensional $\ket{0}^{\otimes n}$5-type Hamiltonian

$\ket{0}^{\otimes n}$6

with coefficients $\ket{0}^{\otimes n}$7 chosen to be not rationally related. By choosing a time $\ket{0}^{\otimes n}$8 so that three unwanted phases nearly wrap around the circle while one desired term accumulates the target angle, the paper obtains effective evolutions such as

$\ket{0}^{\otimes n}$9

and similarly for the other components (Lloyd, 2018).

With these effective interactions, the alternating evolution synthesizes pairwise operations

$H_f$00

acting in parallel on alternating nearest-neighbor pairs. The same paper further notes that appropriate global $H_f$01-rotations give access to a $H_f$02-type Hamiltonian through

$H_f$03

This combination is then identified with a broadcast quantum cellular automaton architecture, implying that the QAOA-generated dynamics can simulate arbitrary quantum circuits to arbitrary accuracy, given enough layers and sufficiently precise timing (Lloyd, 2018).

The significance of this result is conceptual as much as constructive. The original variational interpretation remains intact, but the same alternating template

$H_f$04

can serve either as a heuristic optimizer or as a universal quantum processor, depending on how the times are chosen (Lloyd, 2018).

7. Parameter landscapes, optimizer behavior, and implementation strategies

The optimization problem induced by Hamiltonian-based QAOA can itself be studied analytically. For MaxCut at $H_f$05, "Quantum Approximate Optimization Algorithm for MaxCut: A Fermionic View" derives an explicit edgewise formula in terms of the degrees of the endpoints and the number of triangles containing the edge, showing that the one-layer contribution depends only on local graph structure (Wang et al., 2017). For the 1D antiferromagnetic ring, the paper maps QAOA to control of an ensemble of independent pseudospins via a Jordan–Wigner transform, identifies symmetry-reduced critical manifolds such as

$H_f$06

and reports that no local optima were observed numerically on the relevant reduced landscape for the ring case (Wang et al., 2017).

Practical optimization under hardware constraints has motivated a separate literature. "The QAOA with Few Measurements" studies the standard Hamiltonian formulation

$H_f$07

for MaxCut and shows that classical optimization is possible even with $H_f$08 shot per objective evaluation on a $H_f$09 problem with $H_f$10 active qubits. The reported optimizers are dual annealing, which used $H_f$11 parameter evaluations, and natural evolution strategies, which used $H_f$12, $H_f$13, $H_f$14 generations, and $H_f$15 total quantum evaluations (Polloreno et al., 2022).

Other implementation strategies modify the Hamiltonian between shallow runs. Loop-QAOA keeps the circuit at very small depth, typically $H_f$16, but updates the MaxCut Hamiltonian weights using the measured shallow-circuit output distribution. The paper reports that under bit-flip, phase-flip, and depolarizing noise, loop-QAOA continues improving as the number of loops increases, whereas conventional QAOA improves only up to a small depth and may then degrade (Duan et al., 2021).

Symmetry reduction offers another route. Automorphism-assisted QAOA identifies edge-equivalence classes under graph automorphisms and replaces many repeated Pauli terms by weighted representatives in a reduced Hamiltonian. On tree-structured graphs, the paper reports nearly unchanged approximation ratios together with large runtime and memory reductions; for example, on a $H_f$17 binary tree at $H_f$18, optimization time drops from $H_f$19 s for the full Hamiltonian to $H_f$20 s for the reduced Hamiltonian, with peak memory dropping from $H_f$21 GB to $H_f$22 GB (Prakash, 2024). The same work notes a caveat: if the reverse causal cone at the chosen depth already covers all relevant gates, Hamiltonian term reduction alone may not produce a comparable speedup (Prakash, 2024).

Taken together, these results present Hamiltonian-based QAOA as a family of alternating Hamiltonian control schemes whose performance depends on far more than the nominal depth $H_f$23. Hamiltonian choice, locality, symmetry, commutator structure, schedule design, measurement budget, and feasible-subspace engineering all directly shape the attainable state manifold and the trainability of the resulting variational problem.