Scientific Reports

A reduction from an LWE problem to maximum independent set problems

Yasuhito Kawano

Published May 2, 2023• 0 citations

Abstract

The learning with errors (LWE) problem is a problem derived from machine learning that is believed to be intractable for quantum computers. This paper proposes a method that can reduce an LWE problem to a set of maximum independent set (MIS) problems, which are graph problems that are suitable for a quantum annealing machine to solve. The reduction algorithm can reduce an n-dimensional LWE problem to several small MIS problems with at most [Formula: see text] nodes when the lattice-reduction algorithm used in the LWE-reduction method successfully finds short vectors. The algorithm is useful for solving LWE problems in a quantum-classical hybrid manner by using an existing quantum algorithm to solve the MIS problems. For example, the smallest LWE challenge problem is reduced to MIS problems with about 40,000 vertices. This result means that the smallest LWE challenge problem will be within the scope of a real quantum computer in the near future.

DOI: 10.1038/s41598-023-34366-7

Related papers

Quantum-classical hybrid algorithm for solving the learning-with-errors problem on NISQ devices

Muxi Zheng, Jinfeng Zeng, Wentao Yang +7 more

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.

arXiv:2408.07936Communications PhysicsMay 20, 20256 citations

Quantum Computing for solving the 3SAT problem by reduction to the MIS combinatorial optimization problem

Samuel Deleplanque

This work tries to anticipate the main issues related to the D-Wave machines topology for solving the 3Sat problem. The polynomial reduction to the Maximum Independent Set Problem (MIS) is employed for solving a 3Sat instance to furnish a less dense graph to the machine. These analog quantum machines, based on the quantum annealing process, takes a Quadratic Unconstrained Binary Optimization model (QUBO) as input, which is a specific mathematical program easy to build considering the MIS.

Apr 17, 20230 citations

Using Variational Quantum Algorithm to Solve the LWE Problem

Lihui Lv, Bao Yan, Hong Wang +4 more

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.

EntropyOct 8, 20229 citations

Quantum Search-to-Decision Reduction for the LWE Problem

Kyohei Sudo, Keisuke Hara, Masayuki Tezuka +1 more

The learning with errors (LWE) problem is one of the fundamental problems in cryptography and it has many applications in post-quantum cryptography. There are two variants of the problem, the decisional-LWE problem, and the search-LWE problem. LWE search-to-decision reduction shows that the hardness of the search-LWE problem can be reduced to the hardness of the decisional-LWE problem. The efficiency of the reduction can be regarded as the gap in difficulty between the problems. We initiate a study of quantum search-to-decision reduction for the LWE problem and propose a reduction that satisfies sample-preserving. In sample-preserving reduction, it preserves all parameters even the number of instances. Especially, our quantum reduction invokes the distinguisher only 2 times to solve the search-LWE problem, while classical reductions require a polynomial number of invocations. Furthermore, we give a way to amplify the success probability of the reduction algorithm. Our amplified reduction is incomparable to the classical reduction in terms of sample complexity and query complexity. Our reduction algorithm supports a wide class of error distributions and also provides a search-to-decision reduction for the learning parity with noise problem. In the process of constructing the search-to-decision reduction, we give a quantum Goldreich-Levin theorem over ℤq where q is a prime. In short, this theorem states that, if a hardcore predicate a・s (mod q) can be predicted with probability distinctly greater than (1/q) with respect to a uniformly random a ∈ ℤqn, then it is possible to determine s ∈ ℤqn.

IEICE Transactions on Fundamentals of Electronics Communications and Computer SciencesAug 15, 20240 citations

First Results Solving Arbitrarily Structured Maximum Independent Set Problems Using Quantum Annealing

Sheir Yarkoni, Aske Plaat, Thomas Bäck

Commercial quantum processing units (QPUs) such as those made by D-Wave Systems are being increasingly used for solving complex combinatorial optimization problems. In this paper, we review a canonical NP-hard problem, the Maximum Independent Set (MIS) problem. We show how to map MIS problems to quadratic unconstrained binary optimization (QUBO) problems, and use a D-Wave 2000Q QPU to solve them. We compare the results from the D-Wave system to classical algorithms such as simulated thermal annealing and the graphical networks package NetworkX. To our knowledge, these are the first results of experiments involving arbitrarily-structured MIS inputs using a D-Wave QPU. We find that the QPU can be used as a heuristic optimizer for randomly generated inputs, but due to physical control errors, can be outperformed by simulated thermal annealing.

IEEE Congress on Evolutionary Computation (CEC)Jul 1, 201829 citations

Quantum Algorithms for Variants of Average-Case Lattice Problems via Filtering

Yuehua Chen, Qipeng Liu, Mark Zhandry

We show polynomial-time quantum algorithms for the following problems: (*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a constant. (*) Learning with errors (LWE) problem given LWE-like quantum states with polynomially large moduli and certain error distributions, including bounded uniform distributions and Laplace distributions. (*) Extrapolated dihedral coset problem (EDCP) with certain parameters. The SIS, LWE, and EDCP problems in their standard forms are as hard as solving lattice problems in the worst case. However, the variants that we can solve are not in the parameter regimes known to be as hard as solving worst-case lattice problems. Still, no classical or quantum polynomial-time algorithms were known for the variants of SIS and LWE we consider. For EDCP, our quantum algorithm slightly extends the result of Ivanyos et al. (2018). Our algorithms for variants of SIS and EDCP use the existing quantum reductions from those problems to LWE, or more precisely, to the problem of solving LWE given LWE-like quantum states. Our main contribution is solving LWE given LWE-like quantum states with interesting parameters using a filtering technique.

arXiv:2108.11015Aug 25, 20215 citations