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

• URL: http://arxiv.org/abs/2108.11015v2
• Date: Wed, 6 Oct 2021 06:28:53 GMT
• Title: Quantum Algorithms for Variants of Average-Case Lattice Problems via Filtering
• Authors: Yilei Chen and Qipeng Liu and Mark Zhandry
• Abstract summary: We show-time quantum algorithms for the following problems. Our main contribution is solving LWE given LWE-like quantum states with interesting parameters using a filtering technique.
• Score: 13.70673475846529
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• A quantum-classical hybrid algorithm with Ising model for the learning with errors problem [13.06030390635216]
  We propose a quantum-classical hybrid algorithm with Ising model (HAWI) to address the Learning-With-Errors (LWE) problem. We 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. Our algorithm is iterations, and its time complexity depends on the specific quantum algorithm employed to find the Hamiltonian's low-energy levels.
  arXiv  Detail & Related papers  (2024-08-15T05:11:35Z)
• Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints [76.53316706600717]
  Recently proposed quantum algorithm arXiv:2206.14999 is based on semidefinite programming (SDP) We generalize the SDP-inspired quantum algorithm to sum-of-squares. Our results show that our algorithm is suitable for large problems and approximate the best known classicals.
  arXiv  Detail & Related papers  (2024-08-14T19:04:13Z)
• Hybrid classical-quantum branch-and-bound algorithm for solving integer linear problems [0.0]
  Quantum annealers are suited to solve several logistic optimization problems expressed in the QUBO formulation. The solutions proposed by the quantum annealers are generally not optimal, as thermal noise and other disturbing effects arise when the number of qubits involved in the calculation is too large. We propose the use of the classical branch-and-bound algorithm, that divides the problem into sub-problems which are described by a lower number of qubits.
  arXiv  Detail & Related papers  (2023-11-16T09:19:01Z)
• A hybrid quantum-classical algorithm for multichannel quantum scattering of atoms and molecules [62.997667081978825]
  We propose a hybrid quantum-classical algorithm for solving the Schr"odinger equation for atomic and molecular collisions. The algorithm is based on the $S$-matrix version of the Kohn variational principle, which computes the fundamental scattering $S$-matrix. We show how the algorithm could be scaled up to simulate collisions of large polyatomic molecules.
  arXiv  Detail & Related papers  (2023-04-12T18:10:47Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• End-to-end resource analysis for quantum interior point methods and portfolio optimization [63.4863637315163]
  We provide a complete quantum circuit-level description of the algorithm from problem input to problem output. We report the number of logical qubits and the quantity/depth of non-Clifford T-gates needed to run the algorithm.
  arXiv  Detail & Related papers  (2022-11-22T18:54:48Z)
• Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis [59.545114016224254]
  Quantum algorithms for topological data analysis seem to provide an exponential advantage over the best classical approach. We show that the central task of TDA -- estimating Betti numbers -- is intractable even for quantum computers. We argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices.
  arXiv  Detail & Related papers  (2022-09-28T17:53:25Z)
• An entanglement perspective on the quantum approximate optimization algorithm [0.0]
  We study the entanglement growth and spread resulting from randomized and optimized QAOA circuits. We also investigate the entanglement spectrum in connection with random matrix theory.
  arXiv  Detail & Related papers  (2022-06-14T17:37:44Z)
• Adiabatic Quantum Graph Matching with Permutation Matrix Constraints [75.88678895180189]
  Matching problems on 3D shapes and images are frequently formulated as quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard. We propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware. The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures.
  arXiv  Detail & Related papers  (2021-07-08T17:59:55Z)
• Fast algorithm for quantum polar decomposition, pretty-good measurements, and the Procrustes problem [0.0]
  We show that the problem of quantum polar decomposition has a simple and concise implementation via the quantum singular value QSVT. We focus on the applications to pretty-good measurements, a close-to-optimal measurement to distinguish quantum states, and the quantum Procrustes problem.
  arXiv  Detail & Related papers  (2021-06-14T17:50:41Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.