High-precision quantum algorithms for partial differential equations

• URL: http://arxiv.org/abs/2002.07868v2
• Date: Thu, 4 Nov 2021 19:48:10 GMT
• Title: High-precision quantum algorithms for partial differential equations
• Authors: Andrew M. Childs, Jin-Peng Liu, Aaron Ostrander
• Abstract summary: Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm. We develop quantum algorithms based on adaptive-order finite difference methods and spectral methods. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound.
• Score: 1.4050836886292872
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity $\mathrm{poly}(1/\epsilon)$, where $\epsilon$ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be $\mathrm{poly}(d, \log(1/\epsilon))$, where $d$ is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.

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)
• Quantum and classical algorithms for nonlinear unitary dynamics [0.5729426778193399]
  We present a quantum algorithm for a non-linear differential equation of the form $fracd|urangledt. We also introduce a classical algorithm based on the Euler method allowing comparably scaling to the quantum algorithm in a restricted case.
  arXiv  Detail & Related papers  (2024-07-10T14:08:58Z)
• Quantum algorithms for Hopcroft's problem [45.45456673484445]
  We study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n4/3)$ time. Our results are two different quantum algorithms with time complexity $widetilde O(n5/6)$.
  arXiv  Detail & Related papers  (2024-05-02T10:29:06Z)
• 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)
• Quantum simulation of Maxwell's equations via Schr\"odingersation [27.193565893837356]
  We present quantum algorithms for electromagnetic fields governed by Maxwell's equations. The algorithms are based on the Schr"odingersation approach. Instead of qubits, the quantum algorithms can also be formulated in the continuous variable quantum framework.
  arXiv  Detail & Related papers  (2023-08-16T14:52:35Z)
• Fast quantum algorithm for differential equations [0.5895819801677125]
  We present a quantum algorithm with numerical complexity that is polylogarithmic in $N$ but is independent of $kappa$ for a large class of PDEs. Our algorithm generates a quantum state that enables extracting features of the solution.
  arXiv  Detail & Related papers  (2023-06-20T18:01:07Z)
• Efficient quantum linear solver algorithm with detailed running costs [0.0]
  We introduce a quantum linear solver algorithm combining ideasdiabatic quantum computing with filtering techniques based on quantum signal processing. Our protocol reduces the cost of quantum linear solvers over state-of-the-art close to an order of magnitude for early implementations.
  arXiv  Detail & Related papers  (2023-05-19T00:07:32Z)
• Alternatives to a nonhomogeneous partial differential equation quantum algorithm [52.77024349608834]
  We propose a quantum algorithm for solving nonhomogeneous linear partial differential equations of the form $Apsi(textbfr)=f(textbfr)$. These achievements enable easier experimental implementation of the quantum algorithm based on nowadays technology.
  arXiv  Detail & Related papers  (2022-05-11T14:29:39Z)
• Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations [0.25782420501870296]
  We propose a quantum algorithm for solving $n$-dimensional nonlinear dissipative ordinary differential equations (ODEs) Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in $n$ or $epsilon$.
  arXiv  Detail & Related papers  (2021-11-15T01:34:43Z)
• Provably Faster Algorithms for Bilevel Optimization [54.83583213812667]
  Bilevel optimization has been widely applied in many important machine learning applications. We propose two new algorithms for bilevel optimization. We show that both algorithms achieve the complexity of $mathcalO(epsilon-1.5)$, which outperforms all existing algorithms by the order of magnitude.
  arXiv  Detail & Related papers  (2021-06-08T21:05:30Z)
• Efficient quantum algorithm for dissipative nonlinear differential equations [1.1988695717766686]
  We develop a quantum algorithm for dissipative quadratic $n$-dimensional ordinary differential equations. Our algorithm has complexity $T2 qmathrmpoly(log T, log n, log 1/epsilon)/epsilon$, where $T$ is the evolution time, $epsilon$ is the allowed error, and $q$ measures decay of the solution.
  arXiv  Detail & Related papers  (2020-11-06T04:27:00Z)

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.