A quantum-classical hybrid algorithm with Ising model for the learning with errors problem

Abstract

The Learning-With-Errors (LWE) problem is a crucial computational challenge with significant implications for post-quantum cryptography and computational learning theory. Here we propose a quantum-classical hybrid algorithm with Ising model (HAWI) to address the LWE problem. Our approach involves transforming the LWE problem into the Shortest Vector Problem (SVP), using variable qubits to encode lattice vectors into an Ising Hamiltonian. We then identify the low-energy levels of the Hamiltonian to extract the solution, making it suitable for implementation on current noisy intermediate-scale quantum (NISQ) devices. We prove that the number of qubits required is less than \(m(3m-1)/2\), where \(m\) is the number of samples in the algorithm. Our algorithm is heuristic, and its time complexity depends on the specific quantum algorithm employed to find the Hamiltonian's low-energy levels. If the Quantum Approximate Optimization Algorithm (QAOA) is used to solve the Ising Hamiltonian problem, and the number of iterations satisfies \(y < O\left(m\log m\cdot 2^{0.2972k}/pk^2\right)\), our algorithm will outperform the classical Block Korkine-Zolotarev (BKZ) algorithm, where \(k\) is the block size related to problem parameters, and \(p\) is the number of layers in QAOA. We demonstrate the algorithm by solving a \(2\)-dimensional LWE problem on a real quantum device with \(5\) qubits, showing its potential for solving meaningful instances of the LWE problem in the NISQ era.