Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quantum Topological Data Analysis with Linear Depth and Exponential
  Speedup   Quantum computing offers the potential of exponential speedups for certain
classical computations. Over the last decade, many quantum machine learning
(QML) algorithms have been proposed as candidates for such exponential
improvements. However, two issues unravel the hope of exponential speedup for
some of these QML algorithms: the data-loading problem and, more recently, the
stunning dequantization results of Tang et al. A third issue, namely the
fault-tolerance requirements of most QML algorithms, has further hindered their
practical realization. The quantum topological data analysis (QTDA) algorithm
of Lloyd, Garnerone and Zanardi was one of the first QML algorithms that
convincingly offered an expected exponential speedup. From the outset, it did
not suffer from the data-loading problem. A recent result has also shown that
the generalized problem solved by this algorithm is likely classically
intractable, and would therefore be immune to any dequantization efforts.
However, the QTDA algorithm of Lloyd et~al. has a time complexity of
$O(n^4/(\epsilon^2 \delta))$ (where $n$ is the number of data points,
$\epsilon$ is the error tolerance, and $\delta$ is the smallest nonzero
eigenvalue of the restricted Laplacian) and requires fault-tolerant quantum
computing, which has not yet been achieved. In this paper, we completely
overhaul the QTDA algorithm to achieve an improved exponential speedup and
depth complexity of $O(n\log(1/(\delta\epsilon)))$. Our approach includes three
key innovations: (a) an efficient realization of the combinatorial Laplacian as
a sum of Pauli operators; (b) a quantum rejection sampling approach to restrict
the superposition to the simplices in the complex; and (c) a stochastic rank
estimation method to estimate the Betti numbers. We present a theoretical error
analysis, and the circuit and computational time and depth complexities for
Betti number estimation.

31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an