How to Map Linear Differential Equations to Schr\"{o}dinger Equations
via Carleman and Koopman-von Neumann Embeddings for Quantum Algorithms
- URL: http://arxiv.org/abs/2311.15628v2
- Date: Sat, 23 Dec 2023 04:48:16 GMT
- Title: How to Map Linear Differential Equations to Schr\"{o}dinger Equations
via Carleman and Koopman-von Neumann Embeddings for Quantum Algorithms
- Authors: Yuki Ito, Yu Tanaka, Keisuke Fujii
- Abstract summary: We investigate the conditions for linear differential equations to be mapped to the Schr"odinger equation and solved on a quantum computer.
We compute the computational complexity associated with estimating the expected values of an observable.
These results are important in the construction of quantum algorithms for solving differential equations of large-degree-of-freedom.
- Score: 1.6003521378074745
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Solving linear and nonlinear differential equations with large degrees of
freedom is an important task for scientific and industrial applications. In
order to solve such differential equations on a quantum computer, it is
necessary to embed classical variables into a quantum state. While the Carleman
and Koopman-von Neumann embeddings have been investigated so far, the class of
problems that can be mapped to the Schr\"{o}dinger equation is not well
understood even for linear differential equations. In this work, we investigate
the conditions for linear differential equations to be mapped to the
Schr\"{o}dinger equation and solved on a quantum computer. Interestingly, we
find that these conditions are identical for both Carleman and Koopman-von
Neumann embeddings. We also compute the computational complexity associated
with estimating the expected values of an observable. This is done by assuming
a state preparation oracle, block encoding of the mapped Hamiltonian via either
Carleman or Koopman-von Neumann embedding, and block encoding of the observable
using $O(\log M)$ qubits with $M$ is the mapped system size. Furthermore, we
consider a general classical quadratic Hamiltonian dynamics and find a
sufficient condition to map it into the Schr\"{o}dinger equation. As a special
case, this includes the coupled harmonic oscillator model [Babbush et al.,
\cite{babbush_exponential_2023}]. We also find a concrete example that cannot
be described as the coupled harmonic oscillator but can be mapped to the
Schr\"{o}dinger equation in our framework. These results are important in the
construction of quantum algorithms for solving differential equations of
large-degree-of-freedom.
Related papers
- Quantum Algorithms for Stochastic Differential Equations: A Schrödingerisation Approach [29.662683446339194]
We propose quantum algorithms for linear differential equations.
The gate complexity of our algorithms exhibits an $mathcalO(dlog(Nd))$ dependence on the dimensions.
The algorithms are numerically verified for the Ornstein-Uhlenbeck processes, Brownian motions, and one-dimensional L'evy flights.
arXiv Detail & Related papers (2024-12-19T14:04:11Z) - Slow Mixing of Quantum Gibbs Samplers [47.373245682678515]
We present a quantum generalization of these tools through a generic bottleneck lemma.
This lemma focuses on quantum measures of distance, analogous to the classical Hamming distance but rooted in uniquely quantum principles.
We show how to lift classical slow mixing results in the presence of a transverse field using Poisson Feynman-Kac techniques.
arXiv Detail & Related papers (2024-11-06T22:51:27Z) - Calculating response functions of coupled oscillators using quantum phase estimation [40.31060267062305]
We study the problem of estimating frequency response functions of systems of coupled, classical harmonic oscillators using a quantum computer.
Our proposed quantum algorithm operates in the standard $s-sparse, oracle-based query access model.
We show that a simple adaptation of our algorithm solves the random glued-trees problem in time.
arXiv Detail & Related papers (2024-05-14T15:28:37Z) - 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) - Vectorization of the density matrix and quantum simulation of the von
Neumann equation of time-dependent Hamiltonians [65.268245109828]
We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations.
We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix.
A quantum algorithm to simulate the dynamics of the density matrix is proposed.
arXiv Detail & Related papers (2023-06-14T23:08:51Z) - Correspondence between open bosonic systems and stochastic differential
equations [77.34726150561087]
We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment.
A particular system with the form of a discrete nonlinear Schr"odinger equation is analyzed in more detail.
arXiv Detail & Related papers (2023-02-03T19:17:37Z) - Quantum Depth in the Random Oracle Model [57.663890114335736]
We give a comprehensive characterization of the computational power of shallow quantum circuits combined with classical computation.
For some problems, the ability to perform adaptive measurements in a single shallow quantum circuit is more useful than the ability to perform many shallow quantum circuits without adaptive measurements.
arXiv Detail & Related papers (2022-10-12T17:54:02Z) - 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) - On quantum algorithms for the Schr\"odinger equation in the
semi-classical regime [27.175719898694073]
We consider Schr"odinger's equation in the semi-classical regime.
Such a Schr"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics.
arXiv Detail & Related papers (2021-12-25T20:01:54Z) - Koopman-von Neumann Approach to Quantum Simulation of Nonlinear
Classical Dynamics [0.0]
Quantum computers can be used to simulate nonlinear non-Hamiltonian classical dynamics on phase space.
Koopman-von Neumann formulation implies that the conservation of the probability distribution function on phase space can be recast as an equivalent Schr"odinger equation on Hilbert space.
Quantum simulation of classical dynamics is exponentially more efficient than a deterministic Eulerian discretization of the Liouville equation.
arXiv Detail & Related papers (2020-03-22T19:47:19Z)
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.