Reductions of QAOA Induced by Classical Symmetries: Theoretical Insights and Practical Implications

TL;DR

The paper investigates how classical bit-flip symmetries can be leveraged to reduce QAOA instances for MaxCut and how such reductions reshape the associated dynamical Lie algebras (DLAs). By contrasting standard and free reduced DLAs, it identifies graph-theoretic conditions (parity-separation criteria) that ensure reduced DLAs capture full expressivity, and proves that an explicit graph extension can force equality between standard and free reduced DLAs with quadratic overhead. It provides concrete graph families where reduced DLAs grow quadratically versus exponentially in the full system, and introduces variance-based proxies to diagnose DLA dimensions and potential barren plateaus in practice. The results offer a principled, symmetry-aware preprocessing workflow that can improve trainability while preserving the original optimization landscape, with distinct behavior under Grover-mixer QAOA and practical guidance for choosing reduction vertices.

Abstract

The performance of the Quantum Approximate Optimization Algorithm (QAOA) is closely tied to the structure of the dynamical Lie algebra (DLA) generated by its Hamiltonians, which determines both its expressivity and trainability. In this work, we show that classical symmetries can be systematically exploited as a design principle for QAOA. Focusing on the MaxCut problem with global bit-flip symmetry, we analyze reduced QAOA instances obtained by fixing a single variable and study how this choice affects the associated DLAs. We show that the structure of the DLAs can change dramatically depending on which variable is held fixed. In particular, we construct explicit examples where the dimension collapses from exponential to quadratic, uncovering phenomena that do not appear in the original formulation. Numerical experiments on asymmetric graphs indicate that such reductions often produce DLAs of much smaller dimension, suggesting improved trainability. We also prove that any graph can be embedded into a slightly larger one (requiring only quadratic overhead) such that the standard reduced DLA coincides with the free reduced DLA, in most cases implying exponential dimension and irreducibility on the Hilbert space for reduced QAOA instances. These results establish symmetry-aware reduction as a principled tool for designing expressive and potentially trainable QAOA circuits.

Figures (11)

• Figure 1: Schematic illustration of the QAOA circuit. The initial state preparation unitary $U_\xi$ with $U_{\xi}(\ket{0}^{\otimes n}) = \ket{\xi}$) is followed by $p$ alternating applications of unitaries $U_P(\gamma_j):=e^{-i \gamma_j H_P}$ and $U_M(\beta_j):=e^{-i \beta_j H_M}$, generated by the problem Hamiltonian $H_P$ and the mixer Hamiltonian $H_M$, respectively. At the end of the circuit, the state is measured in the computational basis. Each measurement outcome $x \in \mathbb{B}^n$ is assigned the value $F(x)$ of the objective function, and the empirical mean of these values provides an estimate of $\bra{\psi(\boldsymbol{\beta},\boldsymbol{\gamma})} H_P \ket{\psi(\boldsymbol{\beta},\boldsymbol{\gamma})}$. This estimate is then used in a classical optimization loop to update the parameters $(\boldsymbol{\beta}, \boldsymbol{\gamma})$ with the goal of minimizing the empirical mean.
• Figure 2: Reduction scheme for dynamical Lie algebras. An $n$-bit combinatorial optimization problem gives rise to a QAOA instance with standard dynamical Lie algebra $\mathfrak{g}_{\mathrm{std}}$. Fixing a variable $b_\ell$ produces a reduced optimization problem $\mathrm{COP}_\ell$, a corresponding reduced QAOA instance, and an associated standard reduced DLA $\mathfrak{g}^{\ell}_{\mathrm{std}}$.
• Figure 3: The graph $v-\mathcal{T}_5$, constructed from the path graph $P_5$ (shown in blue). Each vertex is labeled by its index (left) and the parity of its degree (right). Also shown is the reduced graph $\mathcal{T}_5$, obtained by removing the vertex $v$ together with its incident edge.
• Figure 4: A vertex $w \in \mathcal{N}_{v,j}$ that belongs to the subsets $\mathcal{C}^{v}_{j,\mathbf{a}}$, determined by parity-degree data $\mathbf{a}=(a_1,\dots,a_j)$ along a path from $v$ to $w$, and $\mathcal{C}^{v}_{j,\ell,\mathbf{b}}$, determined by parity-degree data $\mathbf{b}=(b_1,\dots,b_\ell)$ along a path extending from $w$ to a vertex $\mathcal{N}_{v,j+\ell}$, which $\ell$ steps further away from $v$.
• Figure 5: The graph $\Gamma'$, where each vertex is labeled by its index (left) and by the parity of its degree (right), and the reduced graph $\Gamma'_v$, obtained by removing the vertex $v$ together with all edges incident to it.

Theorems & Definitions (59)

• Remark 2.1
• Example 2.2
• Remark 2.3
• Definition 3.1
• Example 3.2
• Definition 4.1
• Theorem 4.2
• Definition 4.3
• Theorem 4.4
• Remark 4.5