Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra via Schrodingerisation

• URL: http://arxiv.org/abs/2304.02865v1
• Date: Thu, 6 Apr 2023 04:55:15 GMT
• Title: Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra via Schrodingerisation
• Authors: Shi Jin and Nana Liu
• Abstract summary: We introduce the quantum Jacobi and quantum power methods for solving the quantum linear systems of equations. The proposed quantum simulation can be performed on either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems.
• Score: 32.104513049339936
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Quantum simulation is known to be capable of simulating certain dynamical systems in continuous time -- Schrodinger's equations being the most direct and well-known -- more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrodinger's equations via a method called Schrodingerisation. Building on the observation that iterative methods in linear algebra, and more generally discrete linear dynamical systems, can be viewed as discrete time approximations of dynamical systems which evolve continuously in time, we can apply the Schrodingerisation technique. Thus quantum simulation can be directly applied to the continuous-time limits of some of the simplest iterative methods. This applies to general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we introduce the quantum Jacobi and quantum power methods for solving the quantum linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix respectively. The proposed quantum simulation can be performed on either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative method to solve linear algebra problems using quantum simulation.

Related papers

• Variational quantum simulation using non-Gaussian continuous-variable systems [39.58317527488534]
  We present a continuous-variable variational quantum eigensolver compatible with state-of-the-art photonic technology. The framework we introduce allows us to compare discrete and continuous variable systems without introducing a truncation of the Hilbert space.
  arXiv  Detail & Related papers  (2023-10-24T15:20:07Z)
• Variational Quantum Algorithms for Simulation of Lindblad Dynamics [0.0]
  We introduce a variational hybrid classical-quantum algorithm to simulate the Lindblad master equation and its adjoint for time-evolving Markovian open quantum systems and quantum observables. We design and optimize low-depth variational quantum circuits that efficiently capture the unitary and non-unitary dynamics of the solutions.
  arXiv  Detail & Related papers  (2023-05-04T13:25:44Z)
• 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 simulation of partial differential equations via Schrodingerisation [31.986350313948435]
  We present a simple new way to simulate general linear partial differential equations via quantum simulation. Using a simple new transform, referred to as the warped phase transformation, any linear partial differential equation can be recast into a system of Schrodinger's equations. This can be seen directly on the level of the dynamical equations without more sophisticated methods.
  arXiv  Detail & Related papers  (2022-12-28T17:32:38Z)
• Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations [31.986350313948435]
  We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations.
  arXiv  Detail & Related papers  (2022-09-18T05:50:23Z)
• Quantum algorithms for matrix operations and linear systems of equations [65.62256987706128]
  We propose quantum algorithms for matrix operations using the "Sender-Receiver" model. These quantum protocols can be used as subroutines in other quantum schemes.
  arXiv  Detail & Related papers  (2022-02-10T08:12:20Z)
• QOptCraft: A Python package for the design and study of linear optical quantum systems [0.0]
  The package QOptCraft gives a collection of methods to solve some of the most usual problems when designing quantum experiments with linear interferometers. The routines are chosen to avoid usual numerical problems when dealing with the unitary matrices that appear in the description of linear systems.
  arXiv  Detail & Related papers  (2021-08-13T12:05:21Z)
• Quantum algorithms for quantum dynamics: A performance study on the spin-boson model [68.8204255655161]
  Quantum algorithms for quantum dynamics simulations are traditionally based on implementing a Trotter-approximation of the time-evolution operator. variational quantum algorithms have become an indispensable alternative, enabling small-scale simulations on present-day hardware. We show that, despite providing a clear reduction of quantum gate cost, the variational method in its current implementation is unlikely to lead to a quantum advantage.
  arXiv  Detail & Related papers  (2021-08-09T18:00:05Z)
• Bernstein-Greene-Kruskal approach for the quantum Vlasov equation [91.3755431537592]
  The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed.
  arXiv  Detail & Related papers  (2021-02-18T20:55:04Z)
• Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms [74.17312533172291]
  We describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) We then explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation)
  arXiv  Detail & Related papers  (2020-12-12T00:01:10Z)
• Efficient classical simulation of random shallow 2D quantum circuits [104.50546079040298]
  Random quantum circuits are commonly viewed as hard to simulate classically. We show that approximate simulation of typical instances is almost as hard as exact simulation. We also conjecture that sufficiently shallow random circuits are efficiently simulable more generally.
  arXiv  Detail & Related papers  (2019-12-31T19:00:00Z)

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.