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

• URL: http://arxiv.org/abs/2408.07936v1
• Date: Thu, 15 Aug 2024 05:11:35 GMT
• Title: A quantum-classical hybrid algorithm with Ising model for the learning with errors problem
• Authors: Muxi Zheng, Jinfeng Zeng, Wentao Yang, Pei-Jie Chang, Bao Yan, Haoran Zhang, Min Wang, Shijie Wei, Gui-Lu Long,
• Abstract summary: 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.
• Score: 13.06030390635216
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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.

Related papers

• 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)
• Depth scaling of unstructured search via quantum approximate optimization [0.0]
  Variational quantum algorithms have become the de facto model for current quantum computations. One such problem is unstructured search which consists on finding a particular bit of string. We trotterize a CTQW to recover a QAOA sequence, and employ recent advances on the theory of Trotter formulas to bound the query complexity.
  arXiv  Detail & Related papers  (2024-03-22T18:00:03Z)
• Two quantum algorithms for solving the one-dimensional advection-diffusion equation [0.0]
  Two quantum algorithms are presented for the numerical solution of a linear one-dimensional advection-diffusion equation with periodic boundary conditions. Their accuracy and performance with increasing qubit number are compared point-by-point with each other.
  arXiv  Detail & Related papers  (2023-12-30T21:23:15Z)
• Quantum Algorithms for the Pathwise Lasso [1.8058773918890538]
  We present a novel quantum high-dimensional linear regression algorithm based on the classical LARS (Least Angle Regression) pathwise algorithm. Our quantum algorithm provides the full regularisation path as the penalty term varies, but quadratically faster per iteration under specific conditions. We prove, via an approximate version of the KKT conditions and a duality gap, that the LARS algorithm is robust to errors.
  arXiv  Detail & Related papers  (2023-12-21T18:57:54Z)
• 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)
• Accelerating the training of single-layer binary neural networks using the HHL quantum algorithm [58.720142291102135]
  We show that useful information can be extracted from the quantum-mechanical implementation of Harrow-Hassidim-Lloyd (HHL) This paper shows, however, that useful information can be extracted from the quantum-mechanical implementation of HHL, and used to reduce the complexity of finding the solution on the classical side.
  arXiv  Detail & Related papers  (2022-10-23T11:58:05Z)
• A single $T$-gate makes distribution learning hard [56.045224655472865]
  This work provides an extensive characterization of the learnability of the output distributions of local quantum circuits. We show that for a wide variety of the most practically relevant learning algorithms -- including hybrid-quantum classical algorithms -- even the generative modelling problem associated with depth $d=omega(log(n))$ Clifford circuits is hard.
  arXiv  Detail & Related papers  (2022-07-07T08:04:15Z)
• 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)
• Entanglement and coherence in Bernstein-Vazirani algorithm [58.720142291102135]
  Bernstein-Vazirani algorithm allows one to determine a bit string encoded into an oracle. We analyze in detail the quantum resources in the Bernstein-Vazirani algorithm. We show that in the absence of entanglement, the performance of the algorithm is directly related to the amount of quantum coherence in the initial state.
  arXiv  Detail & Related papers  (2022-05-26T20:32:36Z)
• Discriminating Quantum States with Quantum Machine Learning [0.0]
  We propose, implement and analyze a quantum k-means (qk-means) algorithm with a low time complexity. Discriminating quantum states allows the identification of quantum states from low-level in-phase and quadrature signal (IQ) data. In order to reduce dependency on a classical computer, we use the qk-means to perform state discrimination on the IBMQ Bogota device.
  arXiv  Detail & Related papers  (2021-12-01T07:09:14Z)
• Space-efficient binary optimization for variational computing [68.8204255655161]
  We show that it is possible to greatly reduce the number of qubits needed for the Traveling Salesman Problem. We also propose encoding schemes which smoothly interpolate between the qubit-efficient and the circuit depth-efficient models.
  arXiv  Detail & Related papers  (2020-09-15T18:17:27Z)

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.