Towards quantum advantage via topological data analysis

Abstract

A particularly promising line of quantum machine leaning (QML) algorithms with the potential to exhibit exponential speedups over their classical counterparts has recently been set back by a series of "dequantization" results, that is, quantum-inspired classical algorithms which perform equally well in essence. This raises the important question whether other QML algorithms are susceptible to such dequantization, or whether it can be formally argued that they are out of reach of classical computers. In this paper, we study the quantum algorithm for topological data analysis by Lloyd, Garnerone and Zanardi (LGZ). We provide evidence that certain crucial steps in this algorithm solve problems that are classically intractable by closely relating them to the one clean qubit model, a restricted model of quantum computation whose power is strongly believed to lie beyond that of classical computation. While our results do not imply that the topological data analysis problem solved by the LGZ algorithm (i.e., Betti number estimation) is itself DQC1-hard, our work does provide the first steps towards answering the question of whether it is out of reach of classical computers. Additionally, we discuss how to extend the applicability of this algorithm beyond its original aim of estimating Betti numbers and demonstrate this by looking into quantum algorithms for spectral entropy estimation. Finally, we briefly consider the suitability of the LGZ algorithm for near-term implementations.

Quantum Topological Data Analysis with Linear Depth and Exponential Speedup

“…Remark 1 (Time Complexity Discrepancy). We note that there is a discrepancy in the total time complexity of the QTDA algorithm reported in [17] and in the subsequent articles [18,19], primarily due to differences in the underlying assumptions. This relates to simulation of the matrix B or ∆ k , where Lloyd et al [17] suggest the requirement of constructing and applying the projector Pk at each round (possibly using Grover's search algorithm, although some implementation details are missing).…”

“…A k-simplex is then added for every subset of k + 1 data-points that are pair-wise connnected (i.e., for every k-clique, the associated k-simplex is added). The resulting simplicial complex is related to the clique-complex from graph theory [19].…”

“…The problem of Betti number estimation (BNE) can be defined as follows [19]: Given a set of n points, its corresponding Vietoris-Rips complex Γ, an integer 0 ≤ k ≤ n − 1, and the parameters (ǫ, η) ∈ (0, 1), find the random value χ k ∈ [0, 1] that satisfies with probability 1 − η the condition…”

“…where dim Hk is the dimension of the Hilbert space spanned by the set of k-simplices in the complex (i.e., the number of k-simplices |S k | in Γ). Gyurik et al [19] showed that this problem is likely classically intractable; specifically, it is shown that a generalization of the problem is as hard as simulating the one clean qubit model (DQC1-hard), which is believed to take super-polynomial time to compute on a classical computer. They also argued that BNE is likely to be immune to dequantization.…”

“…The seminal approach of Lloyd et al [17] to estimate the Betti numbers using quantum computers, which was further analyzed by Gunn and Kornerup [18] and Gyurik et al [19], comprises two main steps. The first step of the algorithm is to create a mixed state ρ k over the states |s k of k-simplices (over Hk ) that are in the complex Γ.…”