Multi-angle QAOA for Combinatorial Optimization

• ma-QAOA is a variational generalization of QAOA that replaces shared angles with term-wise angles, enhancing expressivity and yielding higher approximation ratios.
• It achieves reduced circuit depth for problems like MaxCut by optimizing individual parameters according to graph structure and exploiting sparsity.
• Symmetry-based parameter tying and surrogate-assisted training further streamline classical optimization by reducing redundancy and computational overhead.

Multi-angle Quantum Approximate Optimization Algorithm (ma-QAOA, also written MA-QAOA) is a variational generalization of the Quantum Approximate Optimization Algorithm in which the single cost angle and single mixer angle used in each layer of standard QAOA are replaced by term-wise angles attached to the individual operators of the cost and mixer Hamiltonians. Introduced for combinatorial optimization, especially MaxCut, ma-QAOA enlarges the ansatz class without increasing nominal hardware requirements, and thereby trades a higher-dimensional classical optimization problem for greater low-depth expressivity, improved approximation performance, and, in many instances, effectively shallower quantum circuits than standard QAOA (Herrman et al., 2021). Later work established that this additional parameterization is often structured rather than fully essential: graph automorphisms, structured initializations, layerwise retraining, and surrogate-assisted distillation can all reduce optimization overhead while preserving much of the observed benefit (Shi et al., 2022).

1. Variational formulation

In standard QAOA for MaxCut on a graph $G=(V,E)$, the cost and mixer Hamiltonians are

$$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$

and

$B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$

Starting from the uniform superposition

$|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$

a depth-$p$ circuit applies alternating evolutions

$$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$

with

$$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$

The defining restriction is that each layer uses only two scalars, $\gamma_l$ and $\beta_l$, regardless of graph size (Herrman et al., 2021).

ma-QAOA preserves the alternating structure but replaces each shared layer angle by local angles: $$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$

$$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$0

For MaxCut on a graph with $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$1 vertices and $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$2 edges, this yields $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$3 parameters per layer instead of $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$4, or $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$5 parameters rather than $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$6 at depth $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$7 (Herrman et al., 2021). In the equivalent notation used elsewhere,

$$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$8

so each clause $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$9 and each mixer term $B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$0 receives its own tunable angle (Shi et al., 2022).

The quantity optimized is the expected cost, written either as

$B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$1

or as

$B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$2

depending on notation. Performance is typically reported through an approximation ratio, for example

$B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$3

or

$B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$4

where $B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$5 or $B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$6 denotes the optimum MaxCut value (Wilkie et al., 2024).

2. Expressivity, guarantees, and depth reduction

Because standard QAOA is recovered by setting all cost angles equal and all mixer angles equal within each layer, standard QAOA is a special case of ma-QAOA. The optimal performance of ma-QAOA is therefore lower bounded by the conventional ansatz, and the introduction paper proves convergence to the optimal combinatorial solution as $B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$7 by combining this inclusion relation with the corresponding convergence statement for standard QAOA (Herrman et al., 2021).

The increase in expressive power can be strict even at one layer. For MaxCut on star graphs, standard one-layer QAOA tends to approximation ratio $B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$8 as the number of vertices grows, whereas one-layer ma-QAOA achieves approximation ratio $B=\sum_{v\in V(G)} B_v,\qquad B_v=X_v.$9, yielding a $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$0 increase on this infinite family (Herrman et al., 2021). On all connected, non-isomorphic 8-vertex graphs, the reported average approximation ratios are $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$1 for ma-QAOA, $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$2 for 1-QAOA, $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$3 for 2-QAOA, and $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$4 for 3-QAOA, so one layer of ma-QAOA is comparable to three layers of the traditional ansatz and slightly exceeds 3-QAOA on average (Herrman et al., 2021). On larger graph families, the same paper reports, for example, $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$5 versus $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$6 on 50-vertex triangle-free 3-regular graphs and $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$7 versus $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$8 on 100-vertex triangle-free 3-regular graphs for 1-QAOA versus ma-QAOA (Herrman et al., 2021).

A central practical observation is that many optimized ma-QAOA angles are zero. On 8-vertex graphs, about $|s\rangle=\frac{1}{\sqrt{2^n}}\sum_z |z\rangle,$9 of $p$0 and $p$1 of $p$2 are zero, so the corresponding gates can be removed from the circuit entirely (Herrman et al., 2021). This zero-angle sparsity links expressivity to circuit compression rather than to a monotone increase in physical depth.

Scalability for $p$3 was analyzed explicitly in later work. On seven MaxCut data sets of random connected graphs, the average-case target performance required 5 QAOA layers but only 2 MA-QAOA layers, corresponding to a depth reduction factor of $p$4. In worst-case comparisons across the data sets, the reduction varied from $p$5 to $p$6, with examples including 8 layers versus 2 and 7 versus 2 (Gaidai et al., 2023). The same analysis, however, distinguishes depth from total QPU time. Using the model

$p$7

where $p$8 is the number of QPU calls required by the classical optimizer, the paper concludes that MA-QAOA is not optimal for minimization of the total QPU time, even though it is effective as a depth-reduction strategy on NISQ devices (Gaidai et al., 2023).

3. Symmetry, redundancy, and structured parameter sharing

A major development in the study of ma-QAOA concerns the observation that not all of its angles are always necessary. For MaxCut, graph automorphisms $p$9 partition vertices and edges into orbits, and ma-QAOA parameters can be tied across the corresponding orbit classes. Rather than optimizing every vertex and edge angle independently, one optimizes one parameter per vertex orbit and one per edge orbit under a chosen symmetry (Shi et al., 2022).

This reduction is numerically substantial. On all 7,565 connected, non-isomorphic 8-node graphs with nontrivial symmetry, the paper reports that symmetry can be used to reduce the number of parameters with no decrease in the objective in $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$0 of the graphs, with an average parameter-reduction ratio of $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$1. It further reports that $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$2 of the graphs admit this reduction using only the largest symmetry. When the largest-symmetry reduction does decrease the objective, it still reduces the parameter count by $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$3 at the cost of only a $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$4 decrease in the objective (Shi et al., 2022). A random baseline with the same number of parameters but arbitrary grouping performs much worse, indicating that the improvement is not explained by compression alone but by compression aligned with graph automorphisms (Shi et al., 2022).

These results motivated symmetry-aware middle-ground parameterizations. In hypergraph optimization, AA-QAOA is described as sharing common angles among gates in the same automorphism or symmetry class; if the graph has no symmetries, AA-QAOA collapses back to MA-QAOA (Camilleri et al., 2 May 2026). Other structure-aware variants use partial multi-angle freedom rather than full term-wise independence. For additive product graphs, one low-girth MaxCut study assigns a vector of cost angles $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$5, one per edge type induced by the atom graphs, while keeping a single scalar mixer angle $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$6 per layer (Li et al., 2024). On several additive product graphs, standard QAOA outperforms the best-known classical local algorithms by $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$7 to $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$8, and the structure-aware ma-QAOA improves on QAOA by an additional $$|\gamma,\beta\rangle = U(B,\beta_p)U(C,\gamma_p)\cdots U(B,\beta_1)U(C,\gamma_1)|s\rangle,$$9 to $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$0 (Li et al., 2024).

4. Optimization, initialization, and trainability

The enlarged parameter space of ma-QAOA intensifies a difficulty already present in standard QAOA: the objective landscape is nonconvex and multimodal. In a QAOA parameter-optimization study on graph clustering, the landscape is described as containing many low-quality, nondegenerate local optima, and optimization becomes harder as $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$1 grows (Shaydulin et al., 2019). This suggests that ma-QAOA, with many more parameters per layer, should be especially sensitive to initialization and training protocol.

Several initialization strategies have therefore been proposed. For MA-QAOA at $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$2, the strongest initialization result in one comparative study is QAOA Relax: optimized QAOA angles, obtained using the best QAOA initializer in that paper, are lifted into the MA-QAOA parameter space and then refined. The same work reports that Constant is the best QAOA initializer among those tested, with $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$3 and $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$4, and that QAOA Relax consistently and significantly outperforms random initialization used in previous MA-QAOA studies. By contrast, Interp, which is effective for standard QAOA, does not transfer well to MA-QAOA and by $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$5 can perform worse than random initialization (Gaidai et al., 2023).

A different line of work exploits angle regularities. By examining optimized ma-QAOA parameters, one paper finds that many angles cluster around multiples of $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$6, especially on 8-vertex graphs. It therefore initializes all angles as random multiples of $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$7 in $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$8 and uses the resulting vector as the starting seed for one run of BFGS. The reported average approximation ratios are $$U(C,\gamma)=e^{-i\gamma C}, \qquad U(B,\beta)=e^{-i\beta B}.$$9, $\gamma_l$0, and $\gamma_l$1 for $\gamma_l$2, compared with $\gamma_l$3, $\gamma_l$4, and $\gamma_l$5 for one-start random-seed BFGS. A variant that sets angles associated with maximum-degree vertices to zero and initializes the rest to random multiples of $\gamma_l$6 yields $\gamma_l$7, $\gamma_l$8, and $\gamma_l$9 (Wilkie et al., 2024). The same paper reports that for $\beta_l$0, about $\beta_l$1 of 4-vertex optimized angles and $\beta_l$2 of 8-vertex optimized angles are multiples of $\beta_l$3 within numerical tolerance (Wilkie et al., 2024).

Training protocols have also been reworked. Orbit-QAOA, introduced as a cyclic layerwise MA-QAOA training method, concludes that optimizing one complete layer per epoch is an efficient granularity, and that selectively retraining layers while freezing stabilized parameters can achieve final cost equivalent to standard MA-QAOA with lower update overhead. Across diverse graph benchmarks, Orbit-QAOA reduces training steps by up to $\beta_l$4, reduces approximation-ratio error by up to $\beta_l$5 compared to unified stop condition-applied enhanced LMA-QAOA, and achieves equivalent approximation performance compared to standard MA-QAOA (Jang et al., 27 Jan 2026).

Surrogate-assisted training pushes the same idea further. SAFE ma-QAOA first uses Low-Weight Pauli Propagation as a classical surrogate for pre-training, then applies parameter distillation by removing angles near zero, and finally fine-tunes the remaining parameters with the exact objective. On Sherrington-Kirkpatrick, two-dimensional square-lattice spin glass, and Max-Cut instances, SAFE with distillation reports a $\beta_l$6 reduction in active parameter count and a $\beta_l$7 reduction in estimated QPU workload relative to exact-only optimization; within the SAFE workflow, distillation further reduces the optimizer steps to the near-optimal regime by $\beta_l$8 relative to the version without distillation (Kim et al., 22 May 2026).

5. Continuous-time quantum walks and universality

ma-QAOA also has a non-variational interpretation. It has been shown to be equivalent to a restriction of continuous-time quantum walks on dynamic graphs, where a dynamic graph is a sequence

$\beta_l$9

and the evolution is generated by the corresponding adjacency matrices. The key observation is that if

$$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$0

then choosing angles so that

$$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$1

makes the ma-QAOA unitary coincide with the associated continuous-time quantum walk step (Herrman, 2022).

In that framework, ma-QAOA is shown to be universal for quantum computation. The universality construction uses a modified diagonal operator

$$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$2

and a mixer

$$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$3

and then realizes the universal gate set $$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$4 through suitable angle choices (Herrman, 2022). The result is conceptually significant because it places ma-QAOA between the gate model and continuous-time quantum walks: it is a layered variational circuit, a restricted dynamic-graph evolution, and a universal computational model.

6. Applications and extensions beyond the original MaxCut setting

Although ma-QAOA was introduced in the MaxCut setting, later work placed it in broader application pipelines. A hierarchical vehicle-routing method for 13-location problems uses K-means clustering to decompose the instance into three balanced clusters of 4 nodes each, solves intra-cluster OTSP subproblems with standard QAOA, and solves inter-cluster routing with MA-QAOA on a 12-qubit formulation. For that inter-cluster stage, the paper reports 66 two-qubit cost parameters, 12 one-qubit cost parameters, and 12 mixer parameters per layer, for a total of 90 parameters per layer. Over 10 synthetic VRP data sets, the reported MA-QAOA approximation ratios range from $$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$5 to $$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$6 relative to Gurobi (Dash et al., 1 Nov 2025).

In hypergraph optimization, MA-QAOA is treated as the maximally expressive endpoint of a parameterization spectrum that also includes SA-QAOA, AA-QAOA, and $$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$7A-QAOA. In that setting, each individual parameterized gate in each layer has its own separate angle, which makes MA-QAOA the reference ansatz for expressivity but also associates it with more function evaluations, more iterations, and a higher risk of barren plateaus (Camilleri et al., 2 May 2026). The same work uses stabilizer Rényi entropy to analyze non-stabilizerness and reports that MA-QAOA tends to generate more magic than $$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$8A-QAOA, reinforcing the view that higher expressivity may entail additional classical and quantum resource costs (Camilleri et al., 2 May 2026).

Low-girth graph families supply another extension. On planar tiling graphs, the same low-girth study reports that QAOA outperforms classical local algorithms by about $$U(C,\vec{\gamma_l}) = e^{-i\sum_a C_a \gamma_{l,a}} = \prod_a e^{-i\gamma_{l,a}C_a},$$9 to $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$00, and ma-QAOA improves that advantage by another $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$01 to $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$02 (Li et al., 2024). The broader implication is that multi-angle parameterization is not restricted to complete term-wise freedom; it can be tailored to edge types, automorphism classes, or hyperedge orders while still being understood as part of the ma-QAOA design space.

7. Limitations, misconceptions, and current assessment

Several recurring misconceptions have been corrected by the literature. One is that full term-wise parameterization implies that all angles are necessary. Symmetry-based studies show the opposite: many angles can be tied without changing the objective on a large fraction of instances, and random tying does not reproduce the same effect (Shi et al., 2022). Another is that greater expressive power automatically lowers total computational cost. Depth may decrease substantially, but MA-QAOA is not optimal for minimizing total QPU time once the cost of classical optimization and repeated circuit evaluations is included (Gaidai et al., 2023).

A further misconception is that a single successful QAOA heuristic should transfer unchanged to ma-QAOA. Interp is an example of a standard-QAOA heuristic that does not work well in the multi-angle regime, whereas QAOA Relax, angle-rounding seeds, layerwise cyclic retraining, and surrogate-assisted distillation are specifically adapted to the geometry of the enlarged parameter space (Gaidai et al., 2023). This indicates that ma-QAOA should be treated not merely as “QAOA with more parameters,” but as a distinct optimization problem.

The current assessment is therefore bifurcated. When maximum expressivity at low depth is the priority, MA-QAOA remains the benchmark ansatz: standard QAOA is a special case, substantial improvements over fixed-angle QAOA have been demonstrated, and universality results place the framework on firm theoretical ground (Herrman et al., 2021). When classical optimization cost, function evaluations, or symmetry structure dominate, reduced or structured variants such as automorphism-based tying, angle distillation, AA-QAOA, or $$C=\sum_{jk\in E(G)} C_{jk}, \qquad C_{jk}= \frac{1}{2}(I-Z_jZ_k),$$03A-QAOA become natural alternatives (Shi et al., 2022). The defining research problem is no longer whether ma-QAOA is more expressive than standard QAOA, but how to deploy that expressivity in a way that preserves quantum advantage under realistic optimization and hardware constraints.