Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Xue, Cheng; Wu, Yu-Chun; Guo, Guo-Ping

doi:10.1088/1367-2630/ac3eff

Cited by 17 publications

(11 citation statements)

References 45 publications

(86 reference statements)

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“…First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by [3], but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse, invertible matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas [2] and [3] additionally require normality.…”

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confidence: 90%

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Improved quantum algorithms for linear and nonlinear differential equations

Krovi¹

2022

Preprint

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We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). In [1], a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices. The algorithm here can also be exponentially faster for certain classes of diagonalizable matrices.Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in [2]). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by [3], but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse, invertible matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas [2] and [3] additionally require normality.

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mentioning

confidence: 90%

“…However, the ratio of nonlinearity to dissipation is required to be smaller than in [2]. The algorithm of [3] is also only for homogeneous linear differential equations with the assumption of normality.…”

Section: Introductionmentioning

confidence: 99%

Improved quantum algorithms for linear and nonlinear differential equations

Krovi¹

2022

Preprint

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“…The purpose of the |z| −2n factor included in (41) is to simplify this projection step. Specifically, this cancels the |z| 2n factor in (15), which makes the projection step independent of n:…”

Section: B Mapping Of Evolution Termsmentioning

confidence: 99%

“…These include quantum linear systems algorithms [4,5], quantum algorithms for linear differential equations [6,7], and quantum algorithms for linear simulations of classical waves, fluids, and plasmas [8][9][10][11][12]. Applying quantum computers to solve nonlinear problems is less straightforward, but a variety of approaches have been proposed and studied [11][12][13][14][15][16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Correspondence between open bosonic systems and stochastic differential equations

Engel¹,

Parker²

2023

Preprint

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Bosonic mean-field theories can approximate the dynamics of systems of n bosons provided that n 1. 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 and the mean-field theory is replaced by a stochastic differential equation. When the n → ∞ limit is taken, the stochastic terms in this differential equation vanish, and a mean-field theory is recovered. Besides providing insight into the differences between the behavior of finite quantum systems and their classical limits given by n → ∞, the developed mathematics can provide a basis for quantum algorithms that solve some stochastic nonlinear differential equations. We discuss conditions on the efficiency of these quantum algorithms, with a focus on the possibility for the complexity to be polynomial in the log of the stochastic system size. A particular system with the form of a stochastic discrete nonlinear Schrödinger equation is analyzed in more detail.

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“…However, the computational complexity of this work involves exponential dependency on the time interval used in the time integration. A small number of more recent works have addressed nonlinear differential equations, e.g., Lloyd et al (2020), Liu et al (2021), and Xue et al (2021). An example of an application to a nonlinear fluid dynamics problem is the work of Gaitan (2020), where for a very specific one-dimensional problem the complexity analysis showed potential for quantum speed-up.…”

Section: Introductionmentioning

confidence: 99%

Investigating hardware acceleration for simulation of CFD quantum circuits

2022

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Among the many computational models for quantum computing, the Quantum Circuit Model is the most well-known and used model for interacting with current quantum hardware. The practical implementation of quantum computers is a very active research field. Despite this progress, access to physical quantum computers remains relatively limited. Furthermore, the existing machines are susceptible to random errors due to quantum decoherence, as well as being limited in number of qubits, connectivity and built-in error correction. Simulation on classical hardware is therefore essential to allow quantum algorithm researchers to test and validate new algorithms in a simulated-error environment. Computing systems are becoming increasingly heterogeneous, using a variety of hardware accelerators to speed up computational tasks. One such type of accelerators, Field Programmable Gate Arrays (FPGAs), are reconfigurable circuits that can be programmed using standardized high-level programming models such as OpenCL and SYCL. FPGAs allow to create specialized highly-parallel circuits capable of mimicking the quantum parallelism properties of quantum gates, in particular for the class of quantum algorithms where many different computations can be performed concurrently or as part of a deep pipeline. They also benefit from very high internal memory bandwidth. This paper focuses on the analysis of quantum algorithms for applications in computational fluid dynamics. In this work we introduce novel quantum-circuit implementations of model lattice-based formulations for fluid dynamics, specifically the D1Q3 model using quantum computational basis encoding, as well as, efficient simulation of the circuits using FPGAs. This work forms a step toward quantum circuit formulation of the Lattice Boltzmann Method (LBM). For the quantum circuits implementing the nonlinear equilibrium distribution function in the D1Q3 lattice model, it is shown how circuit transformations can be introduced that facilitate the efficient simulation of the circuits on FPGAs, exploiting their fine-grained parallelism. We show that these transformations allow us to exploit more parallelism on the FPGA and improve memory locality. Preliminary results show that for this class of circuits the introduced transformations improve circuit execution time. We show that FPGA simulation of the reduced circuits results in more than 3× improvement in performance per Watt compared to the CPU simulation. We also present results from evaluating the same kernels on a GPU.

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Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

Cited by 17 publications

References 45 publications

Improved quantum algorithms for linear and nonlinear differential equations

Improved quantum algorithms for linear and nonlinear differential equations

Correspondence between open bosonic systems and stochastic differential equations

Investigating hardware acceleration for simulation of CFD quantum circuits

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