Quantum computing is a very active and diverse area, with immense research and development occurring in quantum hardware design and algorithm creation for applications in various scientific fields. The key motivation behind such efforts is the potential for significant speed-up relative to simulations on classical computers. Quantum Monte Carlo Integration (QMCI) is a quantum algorithm that promises a quadratic advantage in terms of the required number of samples for a required accuracy over classical Monte Carlo Integration (MCI), which is a key statistical resource for integrating multi-dimensional space. In this study, we explore the reality of this potential, specifically for applications in Computational Fluid Dynamics (CFD). We perform an eigenvalue stability analysis for a 1D convection-diffusion equation, in which the coefficients depend on a random number. The quantum algorithm utilized uses quantum amplitude estimation (QAE) to accelerate the integral estimation process. Numerical values obtained from QMCI results are compared against numerical integration techniques, and acceptably low errors are observed. The performance of QMCI is compared against MCI by equating the number of oracle calls in QMCI with the number of samples required in MCI. No significant difference in this metric is observed, with QMCI requiring more oracle calls. This suggests that the expected quantum supremacy is not attainable for the considered numerical integration problem and its quantum-circuit implementation. Extension to available Noisy-Intermediate-Scale-Quantum (NISQ) hardware is expected to reveal more about the algorithm implementation and the simulator emulation. Potential future directions that can be explored in algorithmic development based on this preliminary research are outlined.
Quantum computing is a very active and diverse area, with immense research and development occurring in quantum hardware design and algorithm creation for applications in various scientific fields. The key motivation behind such efforts is the potential for significant speed-up relative to simulations on classical computers. Quantum Monte Carlo Integration (QMCI) is a quantum algorithm that promises a quadratic advantage in terms of the required number of samples for a required accuracy over classical Monte Carlo Integration (MCI), which is a key statistical resource for integrating multi-dimensional space. In this study, we explore the reality of this potential, specifically for applications in Computational Fluid Dynamics (CFD). We perform an eigenvalue stability analysis for a 1D convection-diffusion equation, in which the coefficients depend on a random number. The quantum algorithm utilized uses quantum amplitude estimation (QAE) to accelerate the integral estimation process. Numerical values obtained from QMCI results are compared against numerical integration techniques, and acceptably low errors are observed. The performance of QMCI is compared against MCI by equating the number of oracle calls in QMCI with the number of samples required in MCI. No significant difference in this metric is observed, with QMCI requiring more oracle calls. This suggests that the expected quantum supremacy is not attainable for the considered numerical integration problem and its quantum-circuit implementation. Extension to available Noisy-Intermediate-Scale-Quantum (NISQ) hardware is expected to reveal more about the algorithm implementation and the simulator emulation. Potential future directions that can be explored in algorithmic development based on this preliminary research are outlined.
Efficient and robust uncertainty quantification (UQ) is essential in aerospace systems, as emphasized by both NASA’s CFD Vision 2030 and Certification by Analysis (CbA) 2040 initiatives. However, the extensive computational resources required by traditional UQ methods, especially when coupled with high-fidelity computational fluid dynamics (CFD) simulations, pose significant barriers. Quantum computing (QC) has been identified by the CFD Vision 2030 study as a promising technology to address these challenges, but has not yet been extensively explored in the context of UQ. In this study, a Quantum Computing-based Monte Carlo (QCMC) algorithm is implemented using a Qiskit simulator. This method utilizes Quantum Amplitude Estimation (QAE), specifically tailored for near-term Noisy Intermediate-Scale Quantum (NISQ) computers. The QCMC method demonstrates comparable accuracy to classical Monte Carlo (MC) methods on a two-dimensional heat diffusion problem while reducing the required number of samples (oracle calls) by ∼8X. By substantially lowering the computational overhead of UQ, this quantum computing-based algorithm could be a promising tool required to enable robust uncertainty-informed aerospace design.
This paper presents a novel approach to simulating complex fluid dynamics, specifically turbulent flows, utilizing quantum-inspired algorithms. Traditional numerical methods, such as Navier-Stokes solvers, often struggle with the computational demands of accurately representing turbulence, leading to significant processing times and limitations in practical applications. We introduce a simulation strategy based on quantum amplitude estimation (QAE), a technique inspired by quantum computing principles, to accelerate the solution process. The core idea is to approximate the probability amplitude associated with different fluid states, thereby reducing the number of iterations required to reach a stable solution. We demonstrate, through theoretical analysis and a simplified numerical framework, that QAE offers a promising path toward more efficient and accurate simulations of turbulent flows, potentially bridging the gap between classical computational fluid dynamics (CFD) and the computational benefits anticipated from full-scale quantum simulation. This work lays the groundwork for future research exploring the full potential of quantum-inspired algorithms in tackling complex fluid dynamics problems.
Quantum algorithms for Monte Carlo integration (QMCI), which are based on quantum amplitude estimation (QAE), provide quadratic speed-up compared with classical counterparts, and are therefore widely investigated, along with applications to industries such as finance. One typical feature of problems in finance is high dimensionality, or, in other words, the necessity to create numerous random numbers (RNs) to calculate the integrand. In such a situation, the original implementation of QMCI, in which states encoding the probability distributions of the RNs are generated on different quantum registers, requires a large number of qubits. In this poster, we propose an implementation of QMCI, in which pseudo-random numbers (PRNs) are sequentially generated on one register, and therefore the qubit number is tremendously reduced. Moreover, we show that, when the integrand has the form such that contributions from different RNs are separable into different terms, we can also achieve time complexity reduction with respect to the number of dimensions by combining the nested QAE and use of PRNs.
Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. The existing proposals for fault-tolerant quantum algorithms for CFD have almost exclusively been based on the Carleman embedding method, used to encode nonlinearities on a quantum computer. Here, we first identify several key challenges that must be addressed in order to establish quantum advantage using this method, including Carleman convergence, time-stepping requirements, condition-number scaling, and data extraction. Building on this analysis, we develop a novel algorithm for the incompressible lattice Boltzmann equation that addresses them, and provide its detailed analysis, including all potential sources of algorithmic complexity, as well as gate count estimates. We find that for an end-to-end problem, a modest quantum advantage may be preserved for selected observables in the high-error-tolerance regime. We establish a lower bound for the Reynolds number scaling of our quantum algorithm in dimension D at Kolmogorov microscale resolution with O(Re43(1+2D)×qM), where qM is a multiplicative overhead for data extraction with qM=O(Re83) for the drag force. This upper bounds the scaling improvement over classical algorithms by O(Re83D). However, our numerical investigations suggest a lower speedup, with a scaling estimate of O(Re1.936×qM) for D=2. Our results give robust evidence that small, but nontrivial, quantum advantages can be achieved in the context of CFD, and motivate the need for additional rigorous end-to-end quantum algorithm development.
This paper explores the application of quantum-inspired algorithms to accelerate the simulation of fluid dynamics. Traditional computational fluid dynamics (CFD) methods, particularly those solving the Navier-Stokes equations, are often computationally expensive, limiting their applicability in real-time simulations and complex scenarios. This research investigates the potential of mimicking quantum phenomena, specifically wave interference and amplitude estimation, through the design and implementation of novel algorithms. The core idea is to leverage techniques inspired by quantum computing to achieve a significant speedup in solving these equations. We present a framework utilizing amplitude estimation to efficiently approximate solutions, offering a potential pathway to overcome the computational bottlenecks inherent in conventional CFD. The presented approach provides a novel methodology for simulating fluid dynamics with reduced computational cost, opening opportunities for advancements in areas such as aerospace, automotive, and weather forecasting.
The approximation of an integral with numerical integration methods is a relevant problem in many fields, from finance to chemistry. Monte Carlo methods are a family of methods widely used for numerical integration but which are not always applicable to industrial tasks. Instead, there are quantum computing methods such as Quantum Amplitude Estimation able to offer a speed-up compared to the Monte Carlo methods for numerical integration which promise a significant impact, especially in finance. The Quantum Amplitude Estimation algorithm is a fundamental quantum algorithm with the potential to achieve a quadratic speed-up. Unfortunately, being competitive over classical Monte Carlo methods would require hardware performances currently not available. Therefore, alternative methods have been developed in order to exploit fewer resources while maintaining an advantageous theoretical scaling. We present a comparison between the standard QAE algorithm and two NISQ-friendly versions of the QAE on a numerical integration task. The Monte Carlo technique of Metropolis-Hastings instead is used as a classical benchmark. The scaling of the different algorithms are evaluated in terms of estimation error as a function of the number of samples, computational time, and length of the quantum circuits required by the quantum solutions. The conclusion highlights that an alternative approach is preferable with respect to employng the phase estimation routine. Indeed, the maximum likelihood estimation guarantees the best trade-off between the length of the quantum circuits and the precision in the integral estimation.