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Analysis and Applications

Complexity of quantum integration on generalized Hölder classes

Wan Li, Peixin Ye

Published Feb 27, 2026• 0 citations
Publisher

Abstract

In this paper, we study the numerical integration problem of functions from the generalized Hölder class [Formula: see text], which is determined by a generalized modulus of smoothness [Formula: see text], in the quantum model of computation. We obtain the exact order of complexity of this problem. The results show that quantum algorithms have faster convergence rates than those of deterministic and randomized ones. Quantitatively, the improvement amounts to the factors [Formula: see text] for the deterministic case and [Formula: see text] for the randomized case, where [Formula: see text] is the number of function evaluations involved in the computational process. Furthermore, we apply our general results to the case [Formula: see text] with [Formula: see text] and [Formula: see text] and obtain the corresponding complexity.

DOI
10.1142/s0219530526500375

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In this paper, we study the numerical integration problem of functions from the generalized Hölder class [Formula: see text], which is determined by a generalized modulus of smoothness [Formula: see text], in the quantum model of computation. We obtain the exact order of complexity of this problem. The results show that quantum algorithms have faster convergence rates than those of deterministic and randomized ones. Quantitatively, the improvement amounts to the factors [Formula: see text] for the deterministic case and [Formula: see text] for the randomized case, where [Formula: see text] is the number of function evaluations involved in the computational process. Furthermore, we apply our general results to the case [Formula: see text] with [Formula: see text] and [Formula: see text] and obtain the corresponding complexity.
We obtain the optimal order of high-dimensional integration complexity in the quantum computation model in anisotropic Sobolev classes W∞r([0,1]d)W_{\infty}^{\bf r}([0,1]^d)W∞r​([0,1]d) and Hölder Nikolskii classes H∞r([0,1]d)H_{\infty}^{\bf r}([0,1]^d)H∞r​([0,1]d). It is proved that for these classes of functions there is a speed-up of quantum algorithms over deterministic classical algorithms due to factor n−1n^{-1}n−1 and over randomized classical methods due to factor n−1/2n^{-1/2}n−1/2. Moreover, we give an estimation for optimal query complexity in the class H∞Λ(D)H_{\infty}^{\Lambda}(D)H∞Λ​(D) whose smoothness index is the boundary of some complete set in Z+d\mathbb{Z}_+^dZ+d​.
We obtain the optimal order of high-dimensional integration complexity in the quantumcomputation model in anisotropic Sobolev classes W∞^r ([0, 1]^d) and Hǒlder Nikolskii classes H∞^r([0, 1]^d). It is proved that for these classes of functions there is a speed-up of quantum algorithms over deterministic classical algorithms due to factor n^-1 and over randomized classical methods due to factor n^-1/2. Moreover, we give an estimation for optimal query complexity in the class H∞^∧ (D) whose smoothness index is the boundary of some complete set in Z+^d.
Quantum computing has attracted a lot of attention in different research fields, such as mathematics, physics and computer science. Quantum algorithms can solve certain problems significantly faster than classical algorithms. There are many numerical problems, especially those arising from quantum systems, which are notoriously difficult to solve using classical computers, since the computational time required often scales exponentially with the size of the problem. However, quantum computers have the potential to solve these problems efficiently, which is also one of the founding ideas of the field of quantum computing. In this thesis, we explore five computational problems, designing innovative quantum algorithms and studying their computational complexity. First, we design an adiabatic quantum algorithm based on the Berry phases for the counting problem. For the running time, it is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result should be robust to decoherence and resilient to certain kinds of noise. Since the counting problem is the foundation of many other numerical problems, such as high-dimensional integration and path integration, our adiabatic algorithms can be directly generalized to these kinds of problems. In addition, we study the quantum PAC learning model, offering an improved lower bound on the query complexity. The lower bound is very close to the best lower bound on query complexity known for the classical PAC learning model. We also study the algorithms and the cost of simulating a system evolving under a given Hamiltonian. We consider high order splitting methods that are particularly applicable in quantum simulation and obtain bounds on the number of exponentials required. Moreover, we derive the optimal order of convergence given the required error bound. We compare our complexity estimates to previously known ones and show the resulting speedup. Furthermore, we consider randomized algorithms for Hamiltonian simulation. The evolution is simulated by a product of exponentials in a random sequence and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. We provide a scheme to bound the error of the final quantum state in a randomized algorithm, and obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement. We also apply Hamiltonian simulation in estimating the ground state energy of a multiparticle system, which is also known as the multivariate Sturm-Liouville eigenvalue problem. Since the cost of this problem grows exponentially with the number of particles using deterministic classical algorithms, it suffers from the curse of dimensionality. Quantum computers can vanquish the curse, and we exhibit a quantum algorithm whose total cost are linear in the number of particles.
We study the complexity of high-dimensional integration of periodic functions from Besov spaces Sp,qrB(Td)S^r_{p,q}B(\mathbb{T}^d)Sp,qr​B(Td) with mixed smoothness in the quantum model of computation. We construct new quantum algorithms based on the multivariate Faber-Schauder decomposition of such functions and analyze their convergence rates. We show a gain of one convergence order compared to the well-known deterministic rates of convergence. We additionally obtain lower bounds showing the optimality of our results up to logarithms.