[Paper Review] A Quantum Approximate Optimization Algorithm

Edward Farhi, Jeffrey Goldstone+1 more|arXiv (Cornell University)|Nov 14, 2014

Quantum Computing Algorithms and ArchitectureComputer Science2 references1,772 citations

TL;DR

This paper introduces the Quantum Approximate Optimization Algorithm (QAOA), a variational quantum algorithm that uses a parameterized quantum circuit to find approximate solutions to combinatorial optimization problems. For MaxCut on 3-regular graphs with p=1, the algorithm guarantees a cut size at least 0.6924 times the optimal value, demonstrating a provable performance advantage over classical methods for fixed p.

ABSTRACT

We introduce a quantum algorithm that produces approximate solutions for combinatorial optimization problems. The algorithm depends on a positive integer p and the quality of the approximation improves as p is increased. The quantum circuit that implements the algorithm consists of unitary gates whose locality is at most the locality of the objective function whose optimum is sought. The depth of the circuit grows linearly with p times (at worst) the number of constraints. If p is fixed, that is, independent of the input size, the algorithm makes use of efficient classical preprocessing. If p grows with the input size a different strategy is proposed. We study the algorithm as applied to MaxCut on regular graphs and analyze its performance on 2-regular and 3-regular graphs for fixed p. For p = 1, on 3-regular graphs the quantum algorithm always finds a cut that is at least 0.6924 times the size of the optimal cut.

Motivation & Objective

• To develop a general-purpose quantum algorithm for finding approximate solutions to NP-hard combinatorial optimization problems.
• To design a parameterized quantum circuit that can be implemented on near-term quantum devices with limited qubit connectivity and coherence.
• To analyze the performance of the algorithm on specific problems like MaxCut, particularly for regular graphs with fixed p.
• To establish a theoretical framework where increasing the parameter p improves the approximation quality, approaching the exact optimum in the limit p→∞.
• To explore classical preprocessing strategies that efficiently determine optimal angles for fixed p, enabling efficient quantum evaluation.

Proposed method

• The algorithm constructs a quantum state |γ,β⟩ using a sequence of unitary operations: alternating applications of U(C,γ) = e^{-iγC} and U(B,β) = e^{-iβB}, where C is the cost Hamiltonian and B is the mixer Hamiltonian.
• The initial state |s⟩ is the uniform superposition over all computational basis states, prepared via Hadamard gates on all qubits.
• The algorithm evaluates the expectation value F_p(γ,β) = ⟨γ,β|C|γ,β⟩, which estimates the average number of satisfied clauses or cut size.
• For fixed p, classical preprocessing computes optimal angles (γ,β) by solving a bounded optimization problem using recursive evaluation of quantum expectation values.
• When p grows with n, the algorithm uses the quantum computer as a subroutine in a classical optimization loop to maximize F_p(γ,β) over the parameter space.
• The algorithm is applied to MaxCut on regular graphs, with analytical results derived for 2-regular and 3-regular graphs using symmetry and perturbation theory.

Experimental results

Research questions

• RQ1 Can a parameterized quantum circuit with fixed p provide a provable approximation ratio for combinatorial optimization problems?
• RQ2 How does the performance of the QAOA scale with increasing p, and does it converge to the exact solution in the limit p→∞?
• RQ3 For fixed p, can classical algorithms efficiently compute the optimal angles (γ,β) that maximize the expectation value F_p(γ,β)?
• RQ4 What is the approximation ratio achievable by QAOA on MaxCut for 3-regular graphs when p=1?
• RQ5 Can the QAOA be interpreted as a sequence of quantum walks on a graph of independent sets, with cost Hamiltonians guiding the evolution?

Key findings

• For p=1 and 3-regular graphs, the QAOA guarantees a cut size of at least 0.6924 times the optimal cut, providing a provable approximation ratio.
• The expectation value F_p(γ,β) is monotonically non-decreasing with increasing p, and lim_{p→∞} M_p = max_z C(z), meaning the algorithm can approach the exact optimum.
• When p is fixed and each qubit participates in a bounded number of clauses, an efficient classical algorithm computes the optimal angles (γ,β) that maximize F_p(γ,β).
• The algorithm’s quantum circuit depth scales linearly with p and the number of constraints, making it suitable for near-term quantum devices.
• For p=1, the algorithm can be interpreted as a continuous-time quantum walk on a graph whose vertices are independent sets, with the mixer Hamiltonian B acting as the adjacency matrix.
• The algorithm achieves a performance gain over classical methods for MaxCut on 3-regular graphs when p=1, demonstrating a quantum advantage in approximation quality.