Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices

• URL: http://arxiv.org/abs/2409.04010v2
• Date: Mon, 9 Sep 2024 02:57:20 GMT
• Title: Quantum multi-row iteration algorithm for linear systems with non-square coefficient matrices
• Authors: Weitao Lin, Guojing Tian, Xiaoming Sun,
• Abstract summary: We propose a quantum algorithm inspired by the classical multi-row iteration method. Our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems.
• Score: 7.174256268278207
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the field of quantum linear system algorithms, quantum computing has realized exponential computational advantages over classical computing. However, the focus has been on square coefficient matrices, with few quantum algorithms addressing non-square matrices. Towards this kind of problems defined by $ Ax = b $ where $ A $$ \in\mathbb{R}^{m \times n} $, we propose a quantum algorithm inspired by the classical multi-row iteration method and provide an explicit quantum circuit based on the quantum comparator and Quantum Random Access Memory (QRAM). The time complexity of our quantum multi-row iteration algorithm is $ O(K \log m) $, with $ K $ representing the number of iteration steps, which demonstrates an exponential speedup compared to the classical version. Based on the convergence of the classical multi-row iteration algorithm, we prove that our quantum algorithm converges faster than the quantum one-row iteration algorithm presented in [Phys. Rev. A, 101, 022322 (2020)]. Moreover, our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems and quadratic optimization problems.

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)
• The Algorithm for Solving Quantum Linear Systems of Equations With Coherent Superposition and Its Extended Applications [8.8400072344375]
  We propose two quantum algorithms for solving quantum linear systems of equations with coherent superposition. The two quantum algorithms can both compute the rank and general solution by one measurement. Our analysis indicates that the proposed algorithms are mainly suitable for conducting attacks against lightweight symmetric ciphers.
  arXiv  Detail & Related papers  (2024-05-11T03:03:14Z)
• Generalized quantum Arimoto-Blahut algorithm and its application to quantum information bottleneck [55.22418739014892]
  We generalize the quantum Arimoto-Blahut algorithm by Ramakrishnan et al. We apply our algorithm to the quantum information bottleneck with three quantum systems. Our numerical analysis shows that our algorithm is better than their algorithm.
  arXiv  Detail & Related papers  (2023-11-19T00:06:11Z)
• A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games [102.46640028830441]
  We introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as $mathcalO(d/epsilon)$ to $epsilon$-Nash equilibria. This quadratic speed-up sets a new benchmark for computing $epsilon$-Nash equilibria in quantum zero-sum games.
  arXiv  Detail & Related papers  (2023-11-17T20:38:38Z)
• Qubit-Efficient Randomized Quantum Algorithms for Linear Algebra [3.4137115855910767]
  We propose a class of randomized quantum algorithms for the task of sampling from matrix functions. Our use of qubits is purely algorithmic, and no additional qubits are required for quantum data structures.
  arXiv  Detail & Related papers  (2023-02-03T17:22:49Z)
• Quantum speedup of leverage score sampling and its application [0.0]
  In this paper, we propose a quantum algorithm to accelerate the computation of leverage scores. As an application, we propose a new quantum algorithm for rigid regression problems with vector solution outputs.
  arXiv  Detail & Related papers  (2023-01-15T14:40:18Z)
• Solving the semidefinite relaxation of QUBOs in matrix multiplication time, and faster with a quantum computer [0.20999222360659603]
  We show that some quantum SDO solvers provide speedups in the low-precision regime. We exploit this fact to exponentially improve the dependence of their algorithm on precision. A quantum implementation of our algorithm exhibits a worst case running time of $mathcalO left(ns + n1.5 cdot textpolylog left(n, | C |_F, frac1epsilon right)$.
  arXiv  Detail & Related papers  (2023-01-10T23:12:05Z)
• Quantum Clustering with k-Means: a Hybrid Approach [117.4705494502186]
  We design, implement, and evaluate three hybrid quantum k-Means algorithms. We exploit quantum phenomena to speed up the computation of distances. We show that our hybrid quantum k-Means algorithms can be more efficient than the classical version.
  arXiv  Detail & Related papers  (2022-12-13T16:04:16Z)
• Quantum Algorithm For Estimating Eigenvalue [0.0]
  We provide a quantum algorithm for estimating the largest eigenvalue in magnitude of a given Hermitian matrix. Our quantum procedure can also yield exponential speedup compared to classical algorithms that solve the same problem.
  arXiv  Detail & Related papers  (2022-11-11T13:02:07Z)
• 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)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

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.