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. Evaluating the security of post-quantum cryptography is critically important to ensure resilience against quantum attacks. In this paper, the authors study a complex computational task at the heart of post-quantum cryptography with a hybrid quantum-classical algorithm, heuristically finding some advantages.
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. Evaluating the security of post-quantum cryptography is critically important to ensure resilience against quantum attacks. In this paper, the authors study a complex computational task at the heart of post-quantum cryptography with a hybrid quantum-classical algorithm, heuristically finding some advantages.
The Learning With Errors (LWE) problem constitutes the mathematical foundation of modern Post-Quantum Cryptography (PQC). Cryptanalysis of LWE ranges from classical lattice reduction to machine learning and quantum-classical hybrids. We propose CIM-BDD, a hybrid Bounded-Distance-Decoding solver that reduces LWE to a Quadratic Unconstrained Binary Optimization (QUBO) problem through a strictly \emph{penalty-free} mapping. An algebraic elimination of the secret embeds LWE into a q-ary lattice, absorbing the modular arithmetic and recasting the problem as a Closest Vector Problem (CVP). The squared error norm is then used \emph{directly} as the QUBO energy, so the cryptographic noise is the objective to be minimized rather than a penalized constraint. To realize this general model on current Noisy Intermediate-Scale Quantum (NISQ) devices, we design a special encoding method: a Continuous Relaxed Babai's Nearest Plane (CR-BNP) projection drives an adaptive mixed-radix encoder that greatly reduces both the qubit count and the QUBO coefficient range, so that a single batched hardware submission suffices. We further derive a statistically bounded early-stopping threshold (Tearly) that acts as a one-sided certificate and doubles as a Decision-LWE distinguisher. We validate the framework on the TU Darmstadt LWE Challenge, giving an end-to-end demonstration for both Search- and Decision-LWE of a 40-dimensional instance on the Coherent Ising Machine CPQC-550. This work establishes a new algorithm-hardware co-design paradigm for quantum-classical hybrid cryptanalysis.
The variational quantum algorithm (VQA) is a hybrid classical-quantum algorithm. It can actually run in an intermediate-scale quantum device where the number of available qubits is too limited to perform quantum error correction, so it is one of the most promising quantum algorithms in the noisy intermediate-scale quantum era. In this paper, two ideas for solving the learning with errors problem (LWE) using VQA are proposed. First, after reducing the LWE problem into the bounded distance decoding problem, the quantum approximation optimization algorithm (QAOA) is introduced to improve classical methods. Second, after the LWE problem is reduced into the unique shortest vector problem, the variational quantum eigensolver (VQE) is used to solve it, and the number of qubits required is calculated in detail. Small-scale experiments are carried out for the two LWE variational quantum algorithms, and the experiments show that VQA improves the quality of the classical solutions.
Abstract Variational Quantum Algorithms (VQAs) offer a promising approach to solving optimization problems on quantum computers. In this paper, we investigate the use of VQAs to solve the Learning With Errors (LWE) problem, a crucial challenge in post-quantum cryptography. We propose a VQA-based algorithm and a Quantum Approximate Optimization Algorithm (QAOA) for solving the Learning Parity with Noise (LPN) problem, a special case of LWE. We examine the sample complexity of combinatorial optimization methods for LPN and analyze the performance of these two variational quantum algorithms in terms of quantum resources. For the VQA algorithm, both the number of qubits and the circuit depth scale polynomially with the secret key length n . In the case of QAOA, the number of qubits matches the secret key length n , but the CNOT gate count increases exponentially with n . However, numerical simulations of the VQA-based algorithm show that choosing optimal ansatzes and optimization methods can significantly enhance success rates.
The rapid advancement of quantum computing poses an existential threat to classical public-key cryptographic systems, necessitating the transition to post-quantum cryptography (PQC). However, the deployment of lattice-based PQC schemes such as CRYSTALS-Kyber and CRYSTALS-Dilithium faces significant challenges including computational complexity, noise sensitivity, resource constraints, and security vulnerabilities. This research presents a comprehensive hybrid quantum-classical optimization framework for efficient and noise-tolerant post-quantum cryptographic systems. The framework addresses three primary objectives: (1) developing a unified hybrid quantum-classical optimization architecture that integrates variational quantum algorithms (VQE, QAOA) and QUBO/Ising formulations with classical PQC primitives based on the Learning With Errors (LWE) and Module-LWE problems; (2) designing noise-tolerant PQC protocols through multi-layer noise modeling that characterizes cryptographic, quantum hardware, and classical implementation noise; and (3) enabling resource-aware deployment on constrained platforms including embedded systems, IoT devices, and edge computing nodes. Experimental evaluation demonstrates that the QUBO formulation achieves coefficient ranges suitable for NISQ devices with 99.2% of coefficients within [-1, 1] after normalization. The adaptive partitioning algorithm achieves time-to-solution improvements of 42% for large problem instances compared to classical-only execution. The noise-aware optimization framework achieves decryption failure rate improvements ranging from 2^10.8 to 2^16.8, with combined noise tolerance success rates exceeding 94.56% under severe conditions. Side-channel countermeasures achieve 82.6% reduction in exploitable information leakage, with power leakage correlation reduced from 0.23 to 0.04. The unified Kyber-Dilithium architecture achieves 21% memory savings and 10-11% performance improvements. Reinforcement learning-based scheduling reduces latency by 37% and energy consumption by 39% compared to static scheduling. End-to-end deployment evaluation across smart home, industrial IoT, healthcare, and autonomous vehicle scenarios demonstrates that the hybrid PQC system meets latency and energy requirements while maintaining target security levels. This research provides a practical pathway toward secure communication in the quantum era, bridging the gap between current classical systems and future quantum-enabled solutions.
A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp. estimates) for the number of qubits required per dimension for any lattices (resp. random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately 103 noisy qubits such instances can be tackled.
Very recently, one of the most biggest agenda issues is to provide the proof of quantum computational speedup, particularly with a prospect for near-term uses. However, many quantum algorithms are beyond the reach of noisy intermediate-scale quantum (NISQ) realization. This is mainly because of the requirement of excessively large superposition and massive quantum circuit. Hence, we propose a (say) ``NISQ-compatible'' algorithm for one of the crucial problems in computation and modern cryptography, the learning-with-errors (LWE) problem. We base an approach on the divide-and-conquer, wherein a large core process is subdivided into smaller subprocesses. For a specific error distribution and problem condition, it is shown that the proposed quantum LWE algorithm allows the use of exponentially less-superposed quantum samples and operation overheads are reduced, while achieving polynomial sample complexity.