[Paper Review] Quantum Speedup for the Maximum Cut Problem

Weng-Long Chang, Renata Wong+4 more|arXiv (Cornell University)|May 26, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes a novel quantum algorithm for the maximum cut problem that achieves a quadratic speedup over classical exact algorithms by combining Grover's search with a custom quantum circuit to count edges in cuts. The algorithm runs in $ O( ilde{2}^{n/2}/r) $ time and $ O(m^2) $ space, and is experimentally validated on IBM's quantum simulator, demonstrating correct identification of the maximum cut with probability >1/2 for a 3-vertex graph.

ABSTRACT

Given an undirected, unweighted graph with $n$ vertices and $m$ edges, the maximum cut problem is to find a partition of the $n$ vertices into disjoint subsets $V_1$ and $V_2$ such that the number of edges between them is as large as possible. Classically, it is an NP-complete problem, which has potential applications ranging from circuit layout design, statistical physics, computer vision, machine learning and network science to clustering. In this paper, we propose a quantum algorithm to solve the maximum cut problem for any graph $G$ with a quadratic speedup over its classical counterparts, where the temporal and spatial complexities are reduced to, respectively, $O(\sqrt{2^n/r})$ and $O(m^2)$. With respect to oracle-related quantum algorithms for NP-complete problems, we identify our algorithm as optimal. Furthermore, to justify the feasibility of the proposed algorithm, we successfully solve a typical maximum cut problem for a graph with three vertices and two edges by carrying out experiments on IBM's quantum computer.

Motivation & Objective

• To develop a quantum algorithm that provides a provable speedup for the NP-complete maximum cut problem.
• To design a quantum circuit that efficiently computes the number of edges in a cut using auxiliary qubits and controlled operations.
• To integrate the cut evaluation circuit into Grover’s search framework for solution amplification.
• To validate the feasibility of the algorithm on a real quantum platform using IBM’s quantum simulator.
• To demonstrate that the algorithm is optimal among oracle-based quantum approaches for NP-complete problems.

Proposed method

• The algorithm uses a quantum circuit (CFE and CSE) to compute the number of edges crossing a cut by evaluating each edge’s contribution via controlled logic operations on qubit pairs.
• Auxiliary qubits $ z_{i,j} $ store the cumulative count of edges in a cut after processing each edge, enabling scalable counting.
• The algorithm integrates these circuits into Grover’s amplitude amplification procedure to amplify the probability of measuring maximum cut states.
• The initial state is prepared with Hadamard gates on qubit registers for $ x_d $, $ z_{i,j} $, and auxiliary qubits, followed by iterative Grover steps.
• A CNOT gate conditioned on the target cut size $ z_{m,t} $ flips the auxiliary qubit to mark solutions, which are then amplified via diffusion.
• The algorithm uses quantum counting to estimate the number of maximum cuts $ R $, which determines the number of Grover iterations.

Experimental results

Research questions

• RQ1 Can a quantum algorithm achieve a quadratic speedup for the exact solution of the maximum cut problem?
• RQ2 How can the number of edges in a cut be efficiently computed using quantum circuits with controlled operations?
• RQ3 Is the proposed quantum algorithm optimal among oracle-based approaches for NP-complete problems?
• RQ4 Can the algorithm be implemented and validated on a real quantum computing platform?
• RQ5 What is the required number of Grover iterations for convergence, and how does it scale with the number of vertices?

Key findings

• The proposed quantum algorithm achieves a quadratic speedup, reducing time complexity from $ O(2^n) $ classically to $ O(\sqrt{2^n / r}) $, where $ r $ is the number of maximum cut solutions.
• The algorithm requires $ O(m^2) $ space, making it efficient in terms of qubit usage for sparse graphs.
• The algorithm is optimal among oracle-based quantum algorithms for NP-complete problems, as it matches the theoretical lower bound of Grover’s search.
• The algorithm was successfully implemented and tested on IBM’s quantum simulator using Qiskit, correctly identifying the maximum cut with probability >1/2 for a 3-vertex graph.
• The experimental results confirmed that both bitstrings 010 and 101 (representing equivalent maximum cuts) were measured with high probability, indicating correct solution identification.
• The full circuit, including state preparation, oracle marking, uncomputation, and diffusion, was executed in a single run with $ \frac{\pi}{4} \sqrt{2^n / R} $ iterations, where $ R = 2 $ for the example graph.