Provable avoidance of barren plateaus for the Quantum Approximate Optimization Algorithm with Grover mixers

TL;DR

This work analyzes GM-QAOA, a variant of QAOA that uses the Grover mixer, by explicitly characterizing its dynamical Lie algebra (DLA). The authors prove that the DLA is isomorphic to $\mathfrak{su}(d) \oplus \mathfrak{u}(1) \oplus \mathfrak{u}(1)$ when $d<2^n$ (and to $\mathfrak{su}(d) \oplus \mathfrak{u}(1)$ if $d=2^n$), where $d$ counts nonzero projections of the initial state onto energy eigenspaces of the problem Hamiltonian $H_P$; GM-QAOA thus has the largest commutant among QAOA variants with the same initial state, while its associative closure is minimal. The paper derives an explicit formula for the loss-function variance under GM-QAOA and shows an inverse-polynomial lower bound for a broad class of $s$-local problems, implying that barren plateaus are avoided at sufficient depth. Applications to Grover search, MaxCut and Weighted MaxCut, SAT variants, and TSP illustrate the practical reach of the theory, alongside numerical simulations confirming the anticipated scaling. Overall, the results guide mixer design to mitigate trainability issues and offer a rigorous framework for analyzing the expressive power of GM-QAOA relative to standard QAOA.

Abstract

We analyze the dynamical Lie algebras (DLAs) associated with the Grover-mixer variant of the Quantum Approximate Optimization Algorithm (GM-QAOA). When the initial state is the uniform superposition of computational basis states, we show that the corresponding DLA is isomorphic to $\mathfrak{su}(d) \oplus \mathfrak{u}(1)\oplus \mathfrak{u}(1)$, where $d$ denotes the number of distinct values of the objective function. We also establish an analogous result for other choices of initial states and Grover-type mixers. Furthermore, we prove that the DLA of GM-QAOA has the largest possible commutant among all QAOA variants initialized with the same state, corresponding physically to the maximal set of conserved quantities. We derive an explicit formula for the variance of the GM-QAOA loss function in terms of the objective function values, and we show that for a broad class of optimization problems, GM-QAOA with sufficiently many layers avoids barren plateaus.

Figures (5)

• Figure 1: Summary of the main results of the paper. We determine the dynamical Lie algebras associated with Grover-mixer QAOA circuits. The DLA is isomorphic to $\mathfrak{su}(d) \oplus \mathfrak{u}(1) \oplus \mathfrak{u}(1)$, where $d<2^n$ is the number of distinct objective function values on the initial state $\ket\xi$. For most combinatorial optimization problems, $d$ grows polynomially in the number of qubits $n$, implying that the DLA dimension is polynomial in $n$. As a consequence, the variance of the loss function admits a lower bound of order $1/\mathrm{poly}(n)$, showing that GM-QAOA avoids barren plateaus for high enough circuit depth. Finally, we prove that among all QAOA variants initialized with the same initial state, the GM-QAOA DLA admits the largest commutant, and establish an explicit formula for its dimension in terms of the multiplicities $n_1,\dots,n_r$ of the objective function values.
• Figure 2: Schematic illustration of the QAOA circuit. The initial state preparation unitary $U_\xi$ 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 3: Variance and expectation of the loss function across depths. We sampled five random graphs $G_i$ ($1\leq i \leq 5$) with $n=8$ vertices, and computed the expectation and variance of the loss function for $100$ random sampled parameters, with circuit depths $p$ from $1$ up to $30$. Left panel. As a function of $p$, the graph of the expectation seems convex up, which suggests it is a decreasing function with a slowing rate of decrease. Right panel. The variance shows a sharp growth for small $p$, after which it flattens and stays elevated. The maximum and minimum values in this plot represent the bounds corresponding to $n=8$ in Fig. \ref{['fig:var_range']}.
• Figure 4: Upper and lower bounds for the variance of the loss function. As a function of $n$, the variance is bounded from below by the analytical lower bound $\frac{1}{3n^4}$. For each $n$ up to $10$, we sampled $5$ random graphs, ran the QAOA circuit with randomly sampled $100$ parameters per graph, computed the variance of the loss function, and recorded the minimum and maximum variance for all depths up to $p=30$.
• Figure 5: House graph.

Theorems & Definitions (25)

• Theorem 3.1
• Remark 3.2
• Corollary 3.3
• Theorem 3.4
• Corollary 3.5
• Theorem 3.6
• Corollary 3.7
• Corollary 3.8
• Theorem 3.9
• Theorem 4.1