Adaptive Hamming Weight Operator QAOA

• The paper introduces AHWO-QAOA, a method that replaces standard mixers with adaptive, Hamming weight-preserving operators to maintain strict combinatorial constraints.
• It details encoding techniques for hard constraints via Hamming weight operators and adaptive selection protocols that drive efficient, shallow quantum circuits.
• Empirical benchmarks show that AHWO-QAOA achieves high approximation ratios with 50-80% fewer gate operations, ensuring complete constraint satisfaction in finance and physics applications.

Adaptive Hamming Weight Operator QAOA (AHWO-QAOA) is a variational quantum optimization strategy designed to solve combinatorial problems subject to strict linear or cardinality constraints. Unlike penalty-based QAOA variants, this approach leverages constraint-preserving mixing operators—Hamming Weight Operators—which restrict quantum evolution strictly to the feasible subspace. Adaptive selection of these operators yields shallow circuits, faster convergence, and high approximation quality, as demonstrated on finance and high-energy physics benchmarks, with circuit compactness advantageous for near-term quantum hardware (Hao et al., 4 Jan 2026).

1. Constraint Encoding and Hamming Weight Operators

Central to AHWO-QAOA is the explicit encoding of hard combinatorial constraints via Hamming-weight operators and their associated projectors. For $n$ binary variables, the Hamming-weight operator is defined as $W = \sum_{i=1}^n n_i$, where $n_i = (I - Z_i)/2$ and $Z_i$ is the Pauli Z operator on qubit $i$ (Hao et al., 4 Jan 2026). For fixed-weight constraints ($\sum_i x_i = \kappa$), feasible states lie in the eigenspace $W = \kappa$.

Linear constraints, such as $\sum_i \omega_i x_i = b$, are encoded via the constraint Hamiltonian $$H_s = \sum_i \omega_i n_i = \sum_i \omega_i (I - Z_i)/2$$. The feasible subspace is defined by the eigenspace $H_s = b$, and staying within this subspace throughout circuit evolution ensures constraint satisfaction without penalty terms. The projector onto such feasible subspaces can be constructed via polynomial expressions in $W$.

A similar construction appears in approaches that utilize XY-mixers to enforce one-hot or cardinality constraints by confining mixing operations to those that commute with total $Z$ (Hamming weight), ensuring hard constraint preservation (Wang et al., 2019, LaRose et al., 2021).

2. Mixer Hamiltonians and Constraint Preservation

AHWO-QAOA replaces the standard transverse-field mixer ($H_X = \sum_n X_n$) with sumsof adaptive, constraint-aware Hamming Weight Operators. For fixed-weight or more general linear constraints, these mixers take the form:

$$M_{[i_1...i_n],[j_1...j_m]} = \prod_{r=1}^n (X_{i_r} + i Y_{i_r}) \prod_{k=1}^m (X_{j_k} - i Y_{j_k}) + \mathrm{h.c.}$$

where the indices correspond to qubits whose weight sums or linear combinations are matched (Hao et al., 4 Jan 2026).

The action of $M$ exchanges computational basis states while exactly preserving the constraint. For cardinality constraints, these mixers commute with $W$ and thus restrict evolution to the subspace of fixed Hamming weight, as rigorously shown in (Wang et al., 2019). For generic linear constraints, the swap-like operator construction targets only those transitions between feasible bitstrings, guaranteeing constraint adherence without recourse to penalty energies.

Variants such as XY-mixers ($H_{XY} = \sum_{n<m} [X_n X_m + Y_n Y_m]$) efficiently implement such exchanges for problems like graph coloring and k-hot constraints, providing a general paradigm for constraint-sensitive mixing (LaRose et al., 2021, Wang et al., 2019).

3. Variational Ansatz Structure and Circuit Depth

The full AHWO-QAOA ansatz comprises interleaved applications of cost and (adaptively selected) mixer unitaries:

$$|\psi(\gamma, \beta)\rangle = \left(\prod_{\ell=1}^p e^{-i\beta_\ell H^{(\ell)}_M} e^{-i\gamma_\ell H_C}\right) |\psi_s\rangle$$

where $|\psi_s\rangle$ is any feasible initial bitstring, $H_C$ is the mapped cost Hamiltonian using Pauli Z operators, and $H^{(\ell)}_M$ is the mixer at layer $\ell$ constructed from the subset $S_\ell$ of the operator pool $\mathcal{P}$ (Hao et al., 4 Jan 2026).

For circuit compilation, each Hamming Weight Operator $M_t$ is a Pauli string and may be realized using hardware-native gates (e.g., iSWAP for two-qubit XX+YY exchanges), while higher-order k-body exchanges are compiled using basis rotations, controlled-phase trees, or ancilla-assisted parity detection with gate complexity scaling as $O(k)$. Because only $O(n)$ operators are required per layer, mixer depth grows linearly with $n$, in contrast to $O(n^2)$ scaling of penalty-based QAOA (Hao et al., 4 Jan 2026, LaRose et al., 2021).

The mixer-phaser ansätze described in (LaRose et al., 2021) further unify phase separator and mixer layers into two-parameter gates acting on all edges, allowing compilation at half the depth of standard QAOA on superconducting topologies. Both studies converge on the consistent empirical observation: constraint-preserving mixers and adaptive operator selection yield shallower, hardware-efficient circuits conducive to NISQ implementation (LaRose et al., 2021).

4. Adaptive Operator Selection Protocols

A distinguishing feature of AHWO-QAOA is the layerwise, adaptive selection of mixing operators. At each variational layer, the pool $\mathcal{P}$—comprising all relevant Hamming-weight-preserving operators—is searched for the operator $M_p$ yielding the maximal instantaneous decrease in the objective, according to gradient estimates:

$$g_M = \langle \psi_{p-1} | [H_C, i M] | \psi_{p-1} \rangle$$

$M_p$ is then appended to the current ansatz, and variational parameters are reoptimized (Hao et al., 4 Jan 2026).

Standard adaptive protocols, such as "scanlast," retain a pool of top-performing parameter configurations and seed optimization at each additional layer with those, leveraging empirical angle concentration and circumstantial narrowness in parameter bands across random instances (LaRose et al., 2021). This adaptive strategy consistently accelerates convergence, as measured in quantum resource usage and optimization runtime.

A plausible implication is that adaptive mixer selection, informed by energy gradients, localizes search to promising subregions of Hilbert space, reducing the need for deep alternation required in standard QAOA and yielding rapid approach to high-quality approximate solutions.

5. Performance Benchmarks and Comparative Analysis

AHWO-QAOA, mixer-phaser QAOA, and analogous XY-mixer variants have been benchmarked across domains:

• Portfolio optimization with cardinality and linear constraints;
• Jet clustering in high-energy physics with energy balance constraints (Hao et al., 4 Jan 2026);
• WeightedMaxCutGSP on fully-connected graphs with cardinality constraints (LaRose et al., 2021);
• Graph coloring using one-hot encoding (Wang et al., 2019).

Across these tasks, constraint-preserving approaches enforce feasibility by construction (constraint satisfaction ratio $=1$ for AHWO-QAOA), consistently outpace penalty-based QAOA in both approximation ratio and number of required gate operations. For instance, AHWO-QAOA achieves approximation ratios $\geq 0.95$ at $p=1$ with $50\%$–$80\%$ gate-count reduction compared to penalty QAOA at $p=5$ across $n=8$–$20$ qubits (Hao et al., 4 Jan 2026). In the weighted quadratic binary constraint problem, mixer-phaser ansatz matches standard QAOA performance with half the circuit depth for moderate $p$ ($p \lesssim 10$) (LaRose et al., 2021).

XY mixers dramatically improve low-$p$ performance in one-hot and graph-coloring applications, with exact solution probabilities at $p=1$ much higher than with X-mixer plus penalties, and generalized $W$-state initializations yielding further boosts (Wang et al., 2019).

Problem Class	Method	Gate Count*	Approximation Ratio*	Constraint Satisfaction
Portfolio Optimization	Penalty-QAOA p=5	~960–1270	0.89–0.98	≤ 0.4
Portfolio Optimization	AHWO-QAOA p=1	~470	≥ 0.99	1.0
WeightedMaxCutGSP	Standard QAOA	2p(N−1)	≥ 0.90 (p≥6)	1.0 (via mixers)
WeightedMaxCutGSP	Mixer-phaser	p(N−1)	≥ 0.90 (p≥6)	1.0 (via mixers)

*Values extracted from (Hao et al., 4 Jan 2026) and (LaRose et al., 2021); representative for n = 16–20.

6. Compilation and Implementation on Quantum Hardware

AHWO-QAOA and related mixer-phaser circuits are designed for direct hardware compatibility. On superconducting platforms, native iSWAP or $XY$ couplers enable efficient realization of XX+YY (Hamming weight exchange) gates. Capitalizing on linearly-scaling gate counts per layer, AHWO-QAOA and unified mixer-phaser ansätze are compiled using a single SWAP network per layer, halving the number of two-qubit operations relative to conventional alternating-layer QAOA (LaRose et al., 2021).

Single-layer AHWO-QAOA matches or exceeds multi-layer penalty QAOA, making it particularly well-suited to implementation on NISQ devices where reduced circuit depth mitigates decoherence and gate infidelity challenges—especially relevant for large, fully-connected instances.

Preparation of initial feasible states is straightforward: select any computational basis vector in the feasible subspace (fixed Hamming weight or desired linear sum). For one-hot encoded coloring problems, generalized $W$-state initialization is optimal, and can be realized analytically or via local gates (Wang et al., 2019).

7. Related Methods, Generalizations, and Outlook

The constraint-preserving QAOA paradigm encompasses AHWO-QAOA, mixer-phaser ansätze, and adaptive bias-field QAOA (ab-QAOA), all characterized by explicit tailoring of the mixing operator to the problem constraints. The adaptive bias-field protocol modifies single-qubit mixers using feedback from measured local magnetizations, yielding mean-field Hamming-weight bias and marked reduction in layer count to reach target accuracy (Yu et al., 2021). This suggests further generalizability to problems with complex constraints via local field adaptation.

XY-mixer techniques, including "complete" and "ring" topology implementations, generalize to arbitrary linear constraints and suggest that efficient circuit constructions can be obtained for most constrained combinatorial problems (Wang et al., 2019). The convergence of methods in (LaRose et al., 2021) and (Hao et al., 4 Jan 2026) points to a broad, scalable framework for constraint-aware quantum optimization.

A plausible implication is that future quantum algorithms for combinatorial optimization will universally incorporate constraint-preserving mixers, adaptive operator selection, and hardware-efficient compilation strategies as standard for approaching practical hard-constrained problems.