An Efficient Quantum Algorithm for Linear System Problem in Tensor Format

• URL: http://arxiv.org/abs/2403.19829v1
• Date: Thu, 28 Mar 2024 20:37:32 GMT
• Title: An Efficient Quantum Algorithm for Linear System Problem in Tensor Format
• Authors: Zeguan Wu, Sidhant Misra, Tamás Terlaky, Xiu Yang, Marc Vuffray,
• Abstract summary: We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension.
• Score: 4.264200809234798
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Solving linear systems is at the foundation of many algorithms. Recently, quantum linear system algorithms (QLSAs) have attracted great attention since they converge to a solution exponentially faster than classical algorithms in terms of the problem dimension. However, low-complexity circuit implementations of the oracles assumed in these QLSAs constitute the major bottleneck for practical quantum speed-up in solving linear systems. In this work, we focus on the application of QLSAs for linear systems that are expressed as a low rank tensor sums, which arise in solving discretized PDEs. Previous works uses modified Krylov subspace methods to solve such linear systems with a per-iteration complexity being polylogarithmic of the dimension but with no guarantees on the total convergence cost. We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA and perform a detailed analysis of the circuit depth of its implementation. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension, which is comparable to the per-iteration complexity of the classical heuristic methods.

Related papers

• Shadow Quantum Linear Solver: A Resource Efficient Quantum Algorithm for Linear Systems of Equations [0.8437187555622164]
  We present an original algorithmic procedure to solve the Quantum Linear System Problem (QLSP) on a digital quantum device. The result is a quantum algorithm avoiding the need for large controlled unitaries, requiring a number of qubits that is logarithmic in the system size. We apply this to a physical problem of practical relevance, by leveraging decomposition theorems from linear algebra to solve the discretized Laplace Equation in a 2D grid.
  arXiv  Detail & Related papers  (2024-09-13T15:46:32Z)
• A Catalyst Framework for the Quantum Linear System Problem via the Proximal Point Algorithm [9.804179673817574]
  We propose a new quantum algorithm for the quantum linear system problem (QLSP) inspired by the classical proximal point algorithm (PPA) Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing texttimattQLSP_solver. By carefully choosing the step size $eta$, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches.
  arXiv  Detail & Related papers  (2024-06-19T23:15:35Z)
• Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems [0.8192907805418583]
  We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems. By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to $K$. We validate our methods with classical simulations and actual IBM quantum computer implementation, showcasing their applicability for solving physical problems such as heat transfer.
  arXiv  Detail & Related papers  (2023-09-20T16:27:16Z)
• Quantum Relaxation for Linear Systems in Finite Element Analysis [0.0]
  We present Quantum Relaxation for Linear System (qRLS) as an iterative approach for gate-based quantum computers. The well-conditioned system enables a practical iterative solution of finite element problems using the state-of-the-art Quantum Signal Processing (QSP) variant of QLSAs. This represents an exponential efficiency gain, offering a new approach for iterative finite element problem-solving on quantum hardware.
  arXiv  Detail & Related papers  (2023-08-02T18:31:33Z)
• An Inexact Feasible Quantum Interior Point Method for Linearly Constrained Quadratic Optimization [0.0]
  Quantum linear system algorithms (QLSAs) have the potential to speed up algorithms that rely on solving linear systems. In our work, we propose an Inexact-Feasible QIPM (IF-QIPM) and show its advantage in solving linearly constrained quadratic optimization problems.
  arXiv  Detail & Related papers  (2023-01-13T01:36:13Z)
• 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)
• First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems [88.58409977434269]
  We consider the problem of computing an equilibrium in a class of nonlinear generalized Nash equilibrium problems (NGNEPs) Our contribution is to provide two simple first-order algorithmic frameworks based on the quadratic penalty method and the augmented Lagrangian method. We provide nonasymptotic theoretical guarantees for these algorithms.
  arXiv  Detail & Related papers  (2022-04-07T00:11:05Z)
• Detailed Account of Complexity for Implementation of Some Gate-Based Quantum Algorithms [55.41644538483948]
  In particular, some steps of the implementation, as state preparation and readout processes, can surpass the complexity aspects of the algorithm itself. We present the complexity involved in the full implementation of quantum algorithms for solving linear systems of equations and linear system of differential equations.
  arXiv  Detail & Related papers  (2021-06-23T16:33:33Z)
• Fixed Depth Hamiltonian Simulation via Cartan Decomposition [59.20417091220753]
  We present a constructive algorithm for generating quantum circuits with time-independent depth. We highlight our algorithm for special classes of models, including Anderson localization in one dimensional transverse field XY model. In addition to providing exact circuits for a broad set of spin and fermionic models, our algorithm provides broad analytic and numerical insight into optimal Hamiltonian simulations.
  arXiv  Detail & Related papers  (2021-04-01T19:06:00Z)
• Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra [53.46106569419296]
  We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression. We argue that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise.
  arXiv  Detail & Related papers  (2020-11-09T01:13:07Z)
• Iterative Algorithm Induced Deep-Unfolding Neural Networks: Precoding Design for Multiuser MIMO Systems [59.804810122136345]
  We propose a framework for deep-unfolding, where a general form of iterative algorithm induced deep-unfolding neural network (IAIDNN) is developed. An efficient IAIDNN based on the structure of the classic weighted minimum mean-square error (WMMSE) iterative algorithm is developed. We show that the proposed IAIDNN efficiently achieves the performance of the iterative WMMSE algorithm with reduced computational complexity.
  arXiv  Detail & Related papers  (2020-06-15T02:57:57Z)

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.