Quantum Alternating Operator Ansatz (QAOA+)

• Quantum Alternating Operator Ansatz (QAOA+) is a variational framework that constructs mixers preserving hard constraints by confining state evolution to feasible subspaces.
• It alternates phase-separator and specialized mixer unitaries to explore targeted regions of the Hilbert space, thereby optimizing combinatorial objectives.
• Advanced strategies like Adaptive Mixer Allocation and Progressive Quantum Algorithm reduce circuit depth and improve approximation ratios in applications such as the Maximum Independent Set.

The Quantum Alternating Operator Ansatz (QAOA+ or QAO-Ansatz) is a variational quantum circuit framework that generalizes the original Quantum Approximate Optimization Algorithm (QAOA) by enabling systematic solutions to constrained combinatorial optimization problems (CCOPs) through the construction of feasibility-preserving alternating unitaries. Unlike the original QAOA, which only alternates phase and transverse-field mixing operators and explores the entire Hilbert space—including infeasible states—QAOA+ constructs mixers that preserve problem constraints exactly, confining the state evolution to the feasible subspace. This Ansatz finds broad application in combinatorial optimization, resource allocation, sampling, quantum state preparation, and statistical physics.

1. Mathematical Structure and General Framework

QAOA+ prepares a layered variational state of depth $p$ by alternately applying two families of unitaries—phase-separators and mixers—to a fixed feasible initial state $|\phi_0\rangle$. The state is

$$|\psi(\boldsymbol{\gamma},\boldsymbol{\beta})\rangle = U_M(\beta_p)U_C(\gamma_p)\cdots U_M(\beta_1)U_C(\gamma_1)|\phi_0\rangle,$$

where:

• $U_C(\gamma_\ell) = \exp(-i \gamma_\ell H_C)$ is the phase-separation unitary, with $H_C$ diagonal and encoding the objective function (cost Hamiltonian) on the computational basis.
• $U_M(\beta_\ell) = \exp(-i \beta_\ell H_M)$ is the mixer unitary, where $H_M$ is designed so for all $\beta_\ell$ it preserves feasibility and connects all feasible basis states.
• The sequence of variational parameters ${\gamma_1,...,\gamma_p}$ and ${\beta_1,...,\beta_p}$ is optimized classically to (approximately) extremize the expectation value of the cost Hamiltonian:

$$F_p(\boldsymbol{\gamma},\boldsymbol{\beta}) = \langle \psi(\boldsymbol{\gamma},\boldsymbol{\beta}) |\, H_C \,| \psi(\boldsymbol{\gamma},\boldsymbol{\beta}) \rangle.$$

In the $p\to\infty$ limit, QAOA+ becomes equivalent to continuous adiabatic evolution, but in practice small $p$ is used for NISQ-era feasibility (Hadfield et al., 2017).

2. Constraint-Preserving Mixers and Feasibility by Construction

The central innovation of QAOA+ is the design of mixing unitaries $U_M(\beta)$ that exactly preserve the feasible subspace associated with hard constraints. This is achieved by decomposing $H_M$ into partial “controlled-flip” or “swap” operators, which act only when a bit-flip or swap preserves feasibility:

• For graph-based problems such as Maximum Independent Set (MIS), each partial mixer $B_u$ acts on qubit $u$ and is enabled only when none of $u$’s neighbors are occupied (i.e., $\forall v\in \mathcal N(u): x_v=0$),

$$B_u = \frac{1}{2} X_u \prod_{j=1}^\ell (I+Z_{v_j}).$$

Here, $e^{-i\beta B_u}$ flips $x_u$ only if all neighbors are unoccupied, maintaining independence (Saleem, 2019, Ni et al., 2024, Ni et al., 2024).

This general principle applies to a wide array of constraint types, including fixed Hamming weight (using “XY” or Johnson graph mixers (1910.13483)), one-hot or permutation constraints (using swap or null-swap operators in assignment and scheduling problems (Hadfield et al., 2017, Fingerhuth et al., 2018)), and even asymmetric linear constraints (with specially constructed “merge” operators (Goldstein-Gelb et al., 2024)).

Mixers are required to satisfy:

• Feasibility preservation: for any $\beta$, $U_M(\beta)$ maps the feasible subspace onto itself.
• Connectivity: any pair of feasible basis states is reachable by a sequence of applications of $U_M(\beta)$.
• Efficient implementability: partitionability or decomposition into lower-weight operations to reduce circuit depth and accommodate NISQ constraints (Hadfield et al., 2017).

3. Dependence on Circuit Depth, Initial State, and Symmetry

Simulation results for canonical problems such as MIS demonstrate that as $p$ increases, probability concentrates on optimal feasible solutions and vanishes elsewhere (Saleem, 2019). Specifically:

• For symmetric graphs (vertex-transitive), all maximum independent sets are sampled with equal probability in the $p\to\infty$ limit.
• For asymmetric graphs, finite $p$ leads to strong preference towards higher-cardinality maxima because the action of $H_M$ depends on local degree, inducing an asymmetry in the feasible state probabilities (Saleem, 2019):

$$P_p(x^{(3)}) \gg P_p(x^{(2)}) \quad \text{for moderate }p$$

(e.g., for $K_{2,3}$, the size-3 MIS dominates at practical depths).

The choice of initial state $|\phi_0\rangle$ substantially affects low-$p$ performance:

• Initializing with a high-cost feasible state (e.g., a maximal independent set for MIS) yields higher approximation ratio at small $p$ than starting from the empty set or a singleton (Saleem, 2019). For instance, at $p=1$, the normalized ratios can be $1.0$ (from a maximal IS), $0.89$ (empty set), $0.68$ (size-1 IS).

However, as depth increases, the dependence on initial state diminishes, as mixing distributes amplitude across the entire feasible subspace.

4. Advanced Mixer Strategies and Circuit Resource Reduction

The overhead of constraint-preserving mixers (multi-controlled rotations) can be considerable in terms of gate count and depth. Adaptive methods have been developed to mitigate these costs:

• Adaptive Mixer Allocation (AMA): Instead of activating all partial mixers at each layer, AMA adaptively selects a subset of the most promising mixers based on their estimated contribution to cost-function improvement, reducing circuit depth and CNOT count. For instance, for the MIS on random graphs, AMA achieves higher optimal approximation ratios (+1.82% on ER, +3.02% on 3-regular graphs) while using only 29–34% circuit depth and 15–18% of the CNOT gates relative to standard QAOA+ (Ni et al., 2024).
• Progressive Quantum Algorithm (PQA): PQA incrementally grows a subgraph likely to contain the optimal solution, running QAOA+ on the current subgraph at each step, and halting when optimization plateaus. This approach yields orders-of-magnitude savings in required qubits and runtime (e.g., $5.565\%$/ $2.170\%$ of qubits and $17.59\%$/$6.430\%$ runtime for random ER/3-regular graphs at AR of 0.95) (Ni et al., 2024).

These strategies facilitate QAOA+ deployment on larger instances within the limitations of NISQ hardware, exploiting problem structure and dynamic resource allocation.

5. Applications Across Problem Classes

QAOA+ is applicable to:

• Hard-constrained combinatorial optimization: e.g., Maximum Independent Set (Saleem, 2019, Ni et al., 2024), Maximum $k$-Vertex Cover (1910.13483), Minimum Exact Cover (Wang et al., 2022), portfolio optimization (Hodson et al., 2019), lattice protein folding (Fingerhuth et al., 2018).
• Quantum local search: QAOA+ is integrated into hybrid quantum-classical algorithms, applying depth-1 QAOA+ to small subgraphs and combining their results to approach globally optimal solutions on hardware-limited qubit counts (Tomesh et al., 2021).
• Sampling and quantum state engineering: Grover-mixer QAOA variants ensure uniformity across objective level sets, useful for sampling and approximate counting, though upper bounds show this only yields quadratic improvement over uniform sampling, requiring exponential circuit depth for scalably sizable instances (Xie et al., 2024, Drapeau et al., 10 Mar 2025).
• Quantum control and state preparation: QAOA sequences are used in NMR to prepare long-lived singlet states by alternating free evolution and tailored RF-pulse Hamiltonians, yielding robust protocols for quantum sensing and spectroscopy (Hullamballi et al., 2024).

6. Theoretical Analyses, Guarantees, and Parameterization

Analyzing QAOA+ performance has leveraged both continuous adiabatic and discrete adiabatic theorems (Kremenetski et al., 2021, Kremenetski et al., 2023). For unconstrained or symmetric constrained problems with regular feasible-subspace graphs, convergence to the optimal solution is guaranteed as the number of layers $p \to \infty$. For general linearly constrained subspaces with asymmetric mixing graphs, similar convergence results are established by ensuring the mixers form a “mixing family” that is feasibility-preserving, stoquastic, and fully connects the feasible subspace (Goldstein-Gelb et al., 2024).

Parameter-setting heuristics and classical proxies further reduce the cost of variational optimization by exploiting the observed near-homogeneity of QAOA amplitudes w.r.t. cost value for many classes of instances (Sud et al., 2022).

7. Design and Implementation Guidelines

Successful implementation of QAOA+ relies on the following principles (Saleem, 2019, Hadfield et al., 2017, Goldstein-Gelb et al., 2024):

• Initial state selection: Choose a feasible state with large objective value to enhance shallow circuit performance.
• Mixer construction: Ensure the mixer preserves feasibility and the feasible state interaction graph is connected; partition mixers into layers to optimize for hardware connectivity.
• Phase-separator choice: Cost Hamiltonian should be diagonal and encode all soft constraints/objectives, imprinted via $Z$-rotations.
• Adaptivity: Where possible, customize the activation of mixers per layer (e.g., AMA) or apply subgraph selection (PQA) to manage circuit width and depth.
• Parameter optimization: Employ schedule-informed initialization (e.g., linear ramp, adiabatic-inspired interpolation) to navigate barren plateaus in the parameter landscape.
• Application-aware mapping: Carefully encode hard constraints into mixer design, leveraging assignment-specific operators for problems such as max-clique, coloring, or scheduling.

Table: Core QAOA+ Components for MIS

Component	Construction	Role
Initial state $|\phi_0\rangle$	Feasible bitstring (e.g., empty or large IS)	Seeds feasible subspace
Phase Hamiltonian $H_C$	$\frac{1}{2}\sum_u (I - Z_u)$ (counts set size)	Rewards objective
Mixer Hamiltonian $H_M$	$\sum_u \frac{1}{2} X_u \prod_{j}(I+Z_{v_j})$	Preserves feasibility
Layered circuit	Alternate $U_M(\beta_\ell), U_C(\gamma_\ell)$	Variational ansatz
Optimization	Classical maximization of $\langle H_C \rangle$	Parameter learning

• Max Independent Set and Quantum Alternating Operator Ansatz (Saleem, 2019)
• An Adaptive Mixer Allocation Algorithm for the Quantum Alternating Operator Ansatz (Ni et al., 2024)
• Progressive Quantum Algorithm for Maximum Independent Set with Quantum Alternating Operator Ansatz (Ni et al., 2024)
• The Quantum Alternating Operator Ansatz on Maximum k-Vertex Cover (1910.13483)
• From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz (Hadfield et al., 2017)
• Convergence guarantee for linearly-constrained combinatorial optimization with a quantum alternating operator ansatz (Goldstein-Gelb et al., 2024)
• Quantum Local Search with the Quantum Alternating Operator Ansatz (Tomesh et al., 2021)
• Quantum Alternating Operator Ansatz for the Preparation and Detection of Long-Lived Singlet States in NMR (Hullamballi et al., 2024)
• Performance Upper Bound of Grover-Mixer Quantum Alternating Operator Ansatz (Xie et al., 2024)

QAOA+ represents a flexible, constraint-respecting generalization of quantum variational algorithms, blending efficient circuit constructions with rigorous theoretical guarantees for a broad class of constrained optimization and sampling problems.