An Introduction to the Quantum Approximate Optimization Algorithm

TL;DR

The paper presents a principled, first-principles tutorial on the Quantum Approximate Optimization Algorithm (QAOA) for solving combinatorial problems formulated as QUBO and PUBO. It derives the Ising-like problem Hamiltonian, outlines the layer-wise, parameterized circuit structure, and provides explicit gate-decomposition strategies (including $R_{ZZ}$ and higher-order $R_{Z^k}$ gates) while highlighting symmetries that reduce the variational search space. Through concrete Max-Cut and Knapsack examples, it analyzes the energy landscape and demonstrates how parameter scaling enhances optimization, offering practical guidance for near-term quantum devices. The work further generalizes QAOA to PUBO problems, detailing how higher-order interactions map to quantum circuitry and discussing practical considerations and symmetries. Overall, the article consolidates the theoretical foundations, circuit constructions, and practical implications of QAOA in the NISQ era.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a promising variational quantum algorithm introduced to tackle classically intractable combinatorial optimization problems. This tutorial offers a comprehensive, first-principles introduction to QAOA and its properties, focusing on its application to Quadratic and Polynomial Unconstrained Binary Optimization (QUBO and PUBO) problems. The tutorial begins by outlining variational quantum circuits and QUBO problems, focusing on their key properties and the encoding of problem constraints through quadratic penalty terms. Next, it explores the QAOA in detail, covering its Hamiltonian formulation, gate decomposition, and example applications, along with their implementation and performance results. This is followed by an analysis of the algorithm's energy landscape, where proofs are provided for its symmetry and periodicity, and where a resulting parameter space reduction is proposed. Finally, the tutorial extends these concepts to PUBO problems by generalizing the results to higher-order Hamiltonians and discussing the associated symmetries and circuit construction.

Figures (6)

• Figure 1: Pictorial representation of the Adiabatic Theorem. Starting from a system in the lowest energy state of a simple Hamiltonian, a unitary evolution is performed until the Hamiltoinian of interest is reached. According to the adiabathic theorem, now the system will be in the lowest energy state of the target Hamiltonian, provided that there is a gap between the lowest energy state and the next one.
• Figure 2: Example of one specific implementation of the QAOA for the Max Cut problem with a simple square graph with four vertices $V = \{ 0,1,2,3 \}$ connected by the edges $E = \{(0,1), (1,2), (2,3), (3,0)\}$. The two layers $U_{\mathrm{i}}$ and $U_{\mathrm{f}}$ together represent the main QAOA layer $L_p$, which here has been repeated only once, thus $p=1$. The layer $L_p$ can be repeated multiple times to achieve a better approximation of the lowest energy eigenstate.
• Figure 3: Example of one generic implementation of the QAOA circuit with one layer for the Knapsack problem with $n=4$ variables. The coefficients are $a_p^{ij} = \gamma_p a^{ij}$ and $b^i_p = \gamma_p b^i$, with $a^{ij}, b^i$ as defined in Equation (\ref{['eq:KNP-a-b']}).
• Figure 4: Plots of the energy landscape of one QAOA layer. Row 1 and row 2: plots of the energy landscape after one QAOA layer for, respectively, the Max Cut and Knapsack problem. Column A: three dimensional plot of the energy landscape of the scaled Hamiltonian $\mathcal{H}_\text{f}/k$. Column B: two dimensional plot of the energy landscape of the scaled Hamiltonian $\mathcal{H}_\text{f}/k$. Column C: two dimensional plot of the energy landscape of the Hamiltonian $\mathcal{H}_\text{f}$.
• Figure 5: Plots of the results of the optimization. Row 1 and row 2: plots of the results of the QAOA circuit for, respectively, the Max Cut and the Knapsack problem run ten times each with different sets of initial angles. Column A: plots of the cost function $\braket{\mathcal{H}_\text{f}}$ decreasing during the optimization of the QAOA circuit. The standard deviation across the ten initializations is represented by the shaded area. Column B: histogram of the sampled output state, where each binary string is associated with the scaled Hamiltonian evaluated on that string. The binary strings are organized from the lowest energy (left) to the highest energy (right). The histogram takes into account the same ten runs.

Theorems & Definitions (17)

• Theorem 2.1: Adiabatic Theorem
• Proposition 2.1
• proof
• Definition 3.1: Energy landscape
• Proposition 3.1: QAOA Symmetry for QUBO problems
• proof
• Lemma 3.1
• proof
• Proposition 3.2: QAOA Periodicity for QUBO problems
• proof