Quantum algorithm for the Navier Stokes equations by using the streamfunction vorticity formulation and the lattice Boltzmann method

Budinski, Ljubomir

doi:10.48550/arxiv.2103.03804

Cited by 4 publications

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“…Ω. Then we construct the linear system defined in equation (22) and solve the linear system with the algorithm proposed in [2]. The normalized solution of equation ( 22) is represented as…”

Section: Resultsmentioning

confidence: 99%

“…However, because of the linearity of quantum mechanics, solving nonlinear equations with quantum computing is challenging, some related algorithms are proposed [17][18][19][20][21][22][23][24][25]. An early quantum algorithm for solving nonlinear ODEs is proposed in [17], but the complexity of the algorithm increases exponentially with the evolution time.…”

Section: Introductionmentioning

confidence: 99%

“…O A is constructed by querying O F1 O(c) times and O F2 O(1) times. By lemma 1, |0 |y in can be prepared by querying O u O(c) times.Substituting equations (127)-(129) and considering the choice of all parameters, the query complexity of solving equation(22) for O F1 , O F2 and O u isO sT( F 1 + F 2 )poly log gηsT F 1 F 2 (1 − 2K 2 ) u in . (130)Using amplitude amplification algorithm[38], we repeat the above process O(1/ √ p) times and get |u out (T) = |x l,0 (T) which satisfies |u out (T) − u(T)/ u(T)with Ω(1) success probability.…”

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confidence: 99%

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Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Xue

Guo

2021

New J. Phys.

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While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving n-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state ε-close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is O(gηTpoly(log(nT/ε))), where η, g measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in n or ε.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Xue

Guo

2021

New J. Phys.

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show abstract

“…However, because of the linearity of quantum mechanics, solving nonlinear equations with quantum computing is challenging, some related algorithms are proposed [17][18][19][20][21][22][23][24]. An early quantum algorithm for solving nonlinear ordinary differential equations (ODEs) is proposed in [17], but the complexity of the algorithm increases exponentially with the evolution time.…”

Section: Introductionmentioning

confidence: 99%

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Xue,

Wu,

Guo

2021

Preprint

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While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving n-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into the other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state ǫ-close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is O(gηT poly(log(nT /ǫ))), where η, g measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in n or ǫ.

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“…There have been some quantum algorithms for solving nonlinear differential equations [21][22][23][24][25][26][27][28]. However, quantum algorithms for solving system of nonlinear equations are relatively few.…”

Section: Introductionmentioning

confidence: 99%

Quantum Algorithm for Solving Quadratic Nonlinear System of Equations

Xue,

Xu,

et al. 2021

Preprint

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High-dimensional nonlinear system of equations that appears in all kinds of fields is difficult to be solved on a classical computer, we present an efficient quantum algorithm for solving ndimensional quadratic nonlinear system of equations. Our algorithm embeds the equations into a finite-dimensional system of linear equations with homotopy perturbation method and a linearization technique, then we solve the linear equations with quantum linear system solver and obtain a state which is ǫ-close to the normalized exact solution of the original nonlinear equations with success probability Ω(1). The complexity of our algorithm is O(poly(log (n/ǫ))), which provides an exponential improvement over the optimal classical algorithm in dimension n.

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Quantum algorithm for the Navier Stokes equations by using the streamfunction vorticity formulation and the lattice Boltzmann method

Cited by 4 publications

References 0 publications

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Quantum Algorithm for Solving Quadratic Nonlinear System of Equations

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