[Paper Review] Complexity of multivariate Feynman-Kac path integration in randomized and quantum settings

Marek Kwas|ArXiv.org|Oct 18, 2004

Coding theory and cryptography17 references9 citations

TL;DR

This paper establishes that the multivariate Feynman-Kac path integration problem, which suffers from the curse of dimensionality in the classical worst-case setting, achieves exponential speedup in both randomized and quantum settings. By leveraging uniform approximation with randomized and quantum algorithms, the complexity depends polynomially on ε⁻¹ and independently of dimension d, with exponents at most 2 (randomized) and 1 (quantum), thus vanquishing the curse of dimensionality.

ABSTRACT

The Feynman-Kac path integration problem was studied in the worst case setting by Plaskota et al. (J. Comp. Phys. 164 (2000) 335) for the univariate case and by Kwas and Li (J. Comp. 19 (2003) 730) for the multivariate case with d space variables. In this paper we consider the multivariate Feynman-Kac path integration problem in the randomized and quantum settings. For smooth multivariate functions, it was proven in Kwas and Li (2003) that the classical worst case complexity suffers from the curse of dimensionality in d. We show that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the number of function evaluations and/or quantum queries required to compute an e-approximation has a bound independent of d and depending polynomially on 1/e. The exponents of these polynomials are at most 2 in the randomized setting and at most 1 in the quantum setting. Hence we have exponential speedup over the classical worst case setting and quadratic speedup of the quantum setting over the randomized setting. However, both the randomized and quantum algorithms presented here still require extensive precomputing, similar to the algorithms of Plaskota et al. (2000) and Kwas and Li(2003).

Motivation & Objective

• To address the curse of dimensionality in multivariate Feynman-Kac path integration, which plagues classical worst-case complexity when the number of space variables d increases.
• To analyze the computational complexity of the multivariate Feynman-Kac path integration problem in randomized and quantum settings.
• To develop efficient algorithms based on uniform approximation that achieve improved complexity bounds compared to classical worst-case settings.
• To demonstrate that both randomized and quantum algorithms break the curse of dimensionality, with the quantum setting offering quadratic speedup over the randomized setting.

Proposed method

• The study employs uniform approximation as the core computational building block for solving the multivariate Feynman-Kac path integration problem.
• Randomized algorithms are constructed using Monte Carlo-type sampling with variance reduction techniques to improve convergence rates.
• Quantum algorithms are designed using amplitude amplification and quantum query models to achieve faster convergence than classical randomized methods.
• The complexity bounds are derived by relating the path integration problem to multivariate weighted integration, which is known to have tight complexity bounds in randomized and quantum settings.
• For smooth functions in the class F of r-times continuously differentiable functions, the uniform approximation complexity is shown to be ε⁻¹ with exponent depending on d and r.
• Theoretical analysis confirms that the randomized algorithm’s cost scales as O(ε⁻²/(1+2r/d)) and the quantum algorithm’s as O(ε⁻¹/(1+r/d)), both independent of d in asymptotic order.

Experimental results

Research questions

• RQ1 Can the curse of dimensionality in multivariate Feynman-Kac path integration be overcome in randomized and quantum settings?
• RQ2 What is the precise complexity of computing an ε-approximation to the multivariate Feynman-Kac path integral in the randomized and quantum settings?
• RQ3 How do the randomized and quantum algorithms compare in terms of query complexity and speedup relative to the classical worst-case setting?
• RQ4 Are the proposed randomized and quantum algorithms optimal for the multivariate Feynman-Kac path integration problem?
• RQ5 What role does the smoothness of the input functions (measured by r) play in determining the complexity in randomized and quantum settings?

Key findings

• The classical worst-case complexity of multivariate Feynman-Kac path integration suffers from the curse of dimensionality, with complexity growing as ε⁻¹ with exponent depending on d and r.
• In the randomized setting, the complexity is bounded by O(ε⁻²/(1+2r/d)), which is independent of d in asymptotic order and depends polynomially on ε⁻¹ with exponent at most 2.
• In the quantum setting, the complexity is bounded by O(ε⁻¹/(1+r/d)), with exponent at most 1, achieving quadratic speedup over the randomized setting for large d.
• For the class of r-times continuously differentiable functions, the randomized complexity scales as O(ε⁻²/(1+2r/d)) and the quantum complexity as O(ε⁻¹/(1+r/d)), both independent of d in dominant order.
• The proposed randomized and quantum algorithms are roughly optimal, as their complexity matches the known lower bounds for multivariate integration problems.
• For periodic functions with smoothness r, the same complexity bounds hold, and the algorithms are optimal due to matching lower bounds from integration complexity.