Quantum-Classical Hybrid Algorithm for Solving the Learning-With-Errors Problem on NISQ Devices

TL;DR

This paper introduces a hybrid quantum-classical algorithm leveraging Ising models to solve the Learning-With-Errors problem on NISQ devices, transforming it into a lattice problem and demonstrating feasibility on small quantum hardware.

Contribution

It presents a novel hybrid approach that encodes LWE into an Ising Hamiltonian and demonstrates its application on near-term quantum devices, addressing scalability and noise issues.

Findings

Successfully solved a 2D LWE problem on a 5-qubit device.

Demonstrated the method's suitability for noisy intermediate-scale quantum hardware.

Analyzed the performance of the algorithm with different quantum eigensolvers.

Abstract

The Learning-With-Errors (LWE) problem is a fundamental computational challenge with implications for post-quantum cryptography and computational learning theory. Here we propose a quantum-classical hybrid algorithm with Ising model to address LWE, transforming it into the Shortest Vector Problem and using variable qubits to encode lattice vectors into an Ising Hamiltonian. By identifying low-energy Hamiltonian levels, the solution is extracted, making the method suitable for noisy intermediate-scale quantum devices. The required number of qubits is less than $m(m+1)$, where $m$ is the number of samples. Our heuristic algorithm's time complexity depends on the specific quantum eigensolver used to find low-energy levels, and the performance when using the Quantum Approximate Optimization Algorithm is investigated. We validate the algorithm by solving a $2$-dimensional LWE problem on a $5$-qubit quantum device, demonstrating its potential for solving meaningful LWE instances on near-term quantum devices.