SAFE: Surrogate-Assisted QAOA Training

• The paper introduces SAFE as a three-stage framework that integrates LWPP surrogate pre-training, parameter distillation, and exact fine-tuning to reduce optimization cost in ma-QAOA.
• SAFE is a training strategy that enhances the expressivity of ma-QAOA by preconditioning the search and pruning near-zero angles to overcome high-dimensional challenges.
• Empirical findings demonstrate that SAFE with distillation cuts active parameter counts by 64.3% and reduces estimated QPU workload by 94.5% relative to exact-only optimization.

Surrogate-Assisted and Fine-tuning Enhanced (SAFE) is a training framework for multi-angle Quantum Approximate Optimization Algorithm (ma-QAOA) introduced in “SAFE ma-QAOA: Surrogate-Assisted and Fine-tuning Enhanced Multi-Angle QAOA with Parameter Distillation” (Kim et al., 22 May 2026). In this usage, SAFE is a specific three-stage workflow—Low-Weight Pauli Propagation (LWPP) surrogate pre-training, parameter distillation, and exact fine-tuning—designed to make the highly expressive but difficult-to-train ma-QAOA ansatz more practical. Its stated purpose is not to replace exact optimization, but to reduce how much exact optimization is required by preconditioning the search with a scalable classical surrogate and then pruning near-inactive angles before the exact stage (Kim et al., 22 May 2026).

1. Definition and problem setting

SAFE is formulated for ma-QAOA, an extension of standard QAOA in which each cost term and each mixer term receives its own trainable angle at each layer, rather than sharing one global cost angle and one global mixer angle per layer (Kim et al., 22 May 2026). Standard QAOA is written with

$H_C=\hat H,\qquad H_M=\sum_{i=1}^n X_i,$

layer unitaries

$$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$

and ansatz

$$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$

Training minimizes the energy

$$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$

The SAFE framework targets the optimization bottleneck created when this compact parameterization is replaced by ma-QAOA. For a Pauli-decomposed cost Hamiltonian

$H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$

ma-QAOA uses

$$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$

so the number of trainable angles grows from $2p$ to

$p(M_C+n).$

For the Ising-type instances studied in SAFE,

$$H_C=\sum_i h_i Z_i+\sum_{(i,j)\in E} J_{ij} Z_iZ_j.$$

The motivation is therefore operational rather than purely formal. ma-QAOA increases expressivity at fixed depth, but this larger parameterization makes exact optimization expensive because repeated objective and gradient evaluations must be carried out in a much higher-dimensional space. SAFE addresses that difficulty by shifting early-stage exploration to a classical surrogate, then restricting the exact phase to a reduced active subspace (Kim et al., 22 May 2026).

2. Variational structure and the source of training difficulty

The central design tension in SAFE is the same one that defines ma-QAOA more broadly: expressivity improves because different couplings and different qubits can be tuned independently, but the optimization landscape becomes harder to navigate (Kim et al., 22 May 2026). In the reported setting, the exact fine-tuning cost depends on two quantities: the number of active angles and the number of exact optimization steps needed to reach a good solution.

This motivates SAFE’s surrogate stage. The surrogate is based on Pauli propagation in the Heisenberg picture. If

$$\ket{\psi(\boldsymbol{\theta})}=U(\boldsymbol{\theta})\ket{\phi_0},$$

then the expectation of an observable $$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$0 is written as

$$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$1

With Pauli decomposition

$$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$2

each Pauli string is propagated backward through the circuit. For a Pauli rotation

$$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$3

the conjugation rule is

$$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$4

where $$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$5.

Exact propagation can generate exponentially many Pauli strings. SAFE controls this by using Low-Weight Pauli Propagation, which truncates branches whose Pauli weight exceeds a threshold $$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$6. The retained number of strings satisfies

$$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$7

and for fixed $$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$8,

$$U_C(\gamma)=e^{-i\gamma H_C},\qquad U_B(\beta)=e^{-i\beta H_M},$$9

This is the surrogate’s scalability argument: for fixed truncation weight, the retained operator representation grows polynomially rather than exponentially.

The paper is explicit that LWPP is not intended as a calibrated replacement for the exact objective. LWPP alone does not reach the solution quality of exact optimization. Its role is to provide a useful initialization. The authors interpret the truncation as a smoothing of the exact landscape by removing high-weight operator contributions, but that interpretation is presented as heuristic rather than as a theorem (Kim et al., 22 May 2026).

3. SAFE workflow

SAFE is organized as a sequential three-stage procedure (Kim et al., 22 May 2026).

Stage	Mechanism	Role
1	LWPP surrogate pre-training	Optimize full ma-QAOA parameter vector cheaply
2	Parameter distillation	Remove angles that remain near zero
3	Exact fine-tuning	Refine only the remaining active parameters

In the first stage, SAFE optimizes the full ma-QAOA parameter vector with LWPP for 500 optimization steps. The reported truncation settings are

$$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$0

and all optimization stages use Adam with learning rate 0.02 and default betas $$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$1 (Kim et al., 22 May 2026). Implementation is in Python/PyTorch/CUDA with a custom GPU-accelerated Pauli propagation library.

The second stage is parameter distillation. After the 500 LWPP steps, SAFE inspects the learned angle vector and prunes parameters whose absolute values are below a chosen threshold. The paper studies three thresholds: 0, 0.01, and 0.3. Threshold $$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$2 means no pruning; thresholds $$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$3 and $$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$4 remove angles satisfying the implied rule

$$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$5

The active parameter count after this step is denoted

$$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$6

Because ma-QAOA assigns separate angles to individual $$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$7, $$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$8, and $$\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})} = U_B(\beta_p)U_C(\gamma_p)\cdots U_B(\beta_1)U_C(\gamma_1)\ket{+}^{\otimes n}.$$9 terms in each layer, this pruning is term-wise or gate-wise rather than merely layer-wise.

The third stage is exact fine-tuning on the original ma-QAOA energy objective

$$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$0

This exact phase runs for 100 additional steps. The baseline “exact-only” method uses the same 100-step exact budget but skips both LWPP and distillation. SAFE without distillation performs “500 LWPP steps + 100 exact steps” while retaining all parameters; SAFE with distillation performs the same sequence but prunes near-zero angles before the exact stage (Kim et al., 22 May 2026).

A central implication is that SAFE changes both the initialization quality and the dimensionality of the exact search problem. The paper does not prove that near-zero surrogate-learned angles are globally irrelevant; it presents this as an empirically effective heuristic.

4. Evaluation protocol and metrics

The SAFE evaluation spans three Ising-type optimization families (Kim et al., 22 May 2026):

1. Sherrington–Kirkpatrick spin glass: fully connected, $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$1, $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$2.
2. Two-dimensional square-lattice spin glass: nearest-neighbor couplings from $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$3, next-nearest-neighbor from $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$4, and fields $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$5.
3. Max-Cut: Erdős–Rényi random graphs with 30% edge probability, mapped by $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$6, $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$7 on edges.

System sizes are $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$8 for Sherrington–Kirkpatrick and Max-Cut, and $$E(\boldsymbol{\gamma},\boldsymbol{\beta}) = \bra{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}H_C\ket{\psi(\boldsymbol{\gamma},\boldsymbol{\beta})}.$$9, $H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$0, $H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$1 for the two-dimensional lattice, corresponding to 12, 16, 20 qubits. Each size-family pair uses five random instances. Depths are

$H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$2

Initialization covers 11 settings: five random seeds from $H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$3, five constant-magnitude starts with magnitudes 0.01, 0.05, 0.1, 0.2, 0.4, and one QAOA-Relax warm start using

$H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$4

with discretized schedule

$H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$5

Three metrics organize the reported analysis. The first is the normalized approximation ratio

$H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$6

where $H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$7 and $H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$8 are computed exactly by brute force. The second is the first-hit step

$H_C=\sum_{\alpha=1}^{M_C} c_\alpha P_\alpha,$9

which measures when a method reaches its own near-optimal regime during exact fine-tuning. The third is the hardware-oriented exact fine-tuning cost proxy

$$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$0

The paper emphasizes that $$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$1 is only a ballpark estimate rather than a literal QPU runtime, because actual hardware cost depends on shot count, measurement grouping, gradient-estimation rules, and device overhead (Kim et al., 22 May 2026).

5. Empirical findings

The headline empirical result is that SAFE with distillation provides the strongest overall resource reduction relative to exact-only while maintaining strong final approximation ratios (Kim et al., 22 May 2026). In the aggregate results, SAFE with distillation yields a 64.3 percent reduction in active parameter count and a 94.5 percent reduction in estimated QPU workload relative to exact-only. The active parameter count is reported to drop on average from

$$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$2

The workload reduction combines fewer active parameters with fewer exact steps needed to reach the near-optimal regime. SAFE without distillation uses about 26.6% of the exact-only $$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$3, while SAFE with distillation uses about 5.5% of exact-only. Within the SAFE workflow, adding distillation further reduces the optimizer steps to the near-optimal regime by 44.4 percent relative to SAFE without distillation (Kim et al., 22 May 2026).

The quality of the final solutions remains close to the no-distillation setting and often close to, or even sometimes exceeding, the exact-only best reference. The paper emphasizes that LWPP-only is clearly weaker than SAFE followed by exact refinement; the gain comes from using surrogate optimization as initialization rather than as a substitute for the exact stage.

Several individual settings illustrate the mechanism. For Sherrington–Kirkpatrick at $$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$4,

• at $$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$5, SAFE without distillation starts exact fine-tuning at approximation ratio 0.645 and reaches 1.000, while SAFE with distillation starts at 0.851 and also reaches 1.000;
• at $$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$6, SAFE without distillation starts at 0.565, while SAFE with distillation starts at 0.836, and both end at 1.000.

These examples support the paper’s claim that distillation can improve the starting point for exact optimization, not merely shrink the search dimension. There are also settings where the first-hit step is zero, meaning the surrogate-guided initialization is already inside the method’s own near-optimal regime during exact fine-tuning; the paper gives Max-Cut at $$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$7 without distillation and Max-Cut at $$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$8 with distillation as examples (Kim et al., 22 May 2026).

The parameter-pattern analysis also matters to the authors’ interpretation. SAFE measures circular cosine similarity between cost-angle vectors,

$$U_C^{(\ell)}=\prod_{\alpha=1}^{M_C} e^{-i\gamma_{\alpha,\ell}P_\alpha},\qquad U_B^{(\ell)}=\prod_{i=1}^{n} e^{-i\beta_{i,\ell}X_i},$$9

which equals $2p$0 when angles agree modulo $2p$1. High similarity between LWPP-learned and exact-refined angles is taken as evidence that the surrogate often captures a meaningful coarse parameter structure that exact optimization refines rather than discards.

6. Interpretation, limitations, and nomenclature

The authors’ main interpretation is that LWPP behaves as a smoothed surrogate of the exact ma-QAOA landscape by suppressing high-weight Pauli contributions generated during backward propagation (Kim et al., 22 May 2026). This suggests that surrogate optimization can move the parameters into a good basin before expensive exact evaluations begin. The second mechanism is structural sparsification: many surrogate-trained angles remain near zero and can be removed with limited loss, reducing the dimension of the exact problem.

Several limitations are explicit. The choice of truncation weight $2p$2 is problem dependent: $2p$3 often works well for Sherrington–Kirkpatrick and the two-dimensional grid, whereas Max-Cut is reported to be more stable at $2p$4, and denser Max-Cut cases outside the main suite sometimes required $2p$5. The pruning threshold is also a hyperparameter; the reported comparisons use 0, 0.01, and 0.3. The empirical study is limited to 12–20 qubits, the exact workload estimate is based on the proxy $2p$6, and the exact fine-tuning experiments are emulator-based rather than executed on real QPUs. The paper also states that it does not claim a formal barren-plateau mitigation theorem (Kim et al., 22 May 2026).

Within the QAOA literature, SAFE should therefore be understood as a training strategy rather than a new ansatz. Its contribution is the combination of a scalable classical surrogate, pruning of near-inactive term-wise angles, and a short exact refinement phase. A plausible implication is that SAFE is most relevant where low-depth expressive ansätze are desirable but exact optimization is the dominant resource bottleneck.

The acronym is specific to this ma-QAOA framework and should not be conflated with other unrelated SAFE usages. In arXiv literature, SAFE has also denoted “Solution And Fitness Evolution” in coevolutionary model discovery (Sipper et al., 2022), “Selective Adapter FrEezing” in parameter-efficient transformer fine-tuning (Son et al., 2024), “Semantic-Aware FinE-tuning” for few-shot CLIP adaptation (Zhu et al., 2023), and “Sparse Autoencoder-Based Framework for Robust Query Enrichment and Hallucination Mitigation in LLMs” (Abdaljalil et al., 4 Mar 2025). In contrast, Surrogate-Assisted and Fine-tuning Enhanced SAFE refers specifically to the ma-QAOA workflow centered on LWPP pre-training, parameter distillation, and exact fine-tuning (Kim et al., 22 May 2026).