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Comparing quantum and classical Monte Carlo algorithms for estimating Betti numbers of clique complexes

Ismail Yunus Akhalwaya, Ahmed Bhayat, Adam Connolly, Steven Herbert, Lior Horesh, Julien Sorci, Shashanka Ubaru

TL;DR

This work assesses quantum and classical Monte Carlo algorithms for estimating normalised Betti numbers on clique complexes, focusing on a modular framework that exposes shared trace-estimation structures. It analyzes existing QBNE-Chebyshev and CBNE-Power/Chebyshev approaches, deriving upper bounds on sampling costs and revealing an exponential dependence on the spectral gap in the quantum Chebyshev method. To overcome this, the authors introduce QBNE-Power, a new quantum algorithm leveraging the reflected Laplacian that achieves polynomial scaling in problem size and gap, while preserving quantum speedups in Monte Carlo sampling. Numerical experiments on dense, structured graphs verify the theoretical bounds and show that QBNE-Power offers superior empirical convergence and reduced sample complexity compared with classical counterparts. Overall, the paper highlights a path to practical quantum advantages in topological data analysis via tailored block-encodings and polynomial-filter techniques, with potential impact for near-term quantum devices.

Abstract

Several quantum and classical Monte Carlo algorithms for Betti Number Estimation (BNE) on clique complexes have recently been proposed, though it is unclear how their performances compare. We review these algorithms, emphasising their common Monte Carlo structure within a new modular framework. We derive upper bounds for the number of samples needed to reach a given level of precision, and use them to compare these algorithms. By recombining the different modules, we create a new quantum algorithm with an exponentially-improved dependence in the sample complexity. We run classical simulations to verify convergence within the theoretical bounds and observe the predicted exponential separation, even though empirical convergence occurs substantially earlier than the conservative theoretical bounds.

Comparing quantum and classical Monte Carlo algorithms for estimating Betti numbers of clique complexes

TL;DR

This work assesses quantum and classical Monte Carlo algorithms for estimating normalised Betti numbers on clique complexes, focusing on a modular framework that exposes shared trace-estimation structures. It analyzes existing QBNE-Chebyshev and CBNE-Power/Chebyshev approaches, deriving upper bounds on sampling costs and revealing an exponential dependence on the spectral gap in the quantum Chebyshev method. To overcome this, the authors introduce QBNE-Power, a new quantum algorithm leveraging the reflected Laplacian that achieves polynomial scaling in problem size and gap, while preserving quantum speedups in Monte Carlo sampling. Numerical experiments on dense, structured graphs verify the theoretical bounds and show that QBNE-Power offers superior empirical convergence and reduced sample complexity compared with classical counterparts. Overall, the paper highlights a path to practical quantum advantages in topological data analysis via tailored block-encodings and polynomial-filter techniques, with potential impact for near-term quantum devices.

Abstract

Several quantum and classical Monte Carlo algorithms for Betti Number Estimation (BNE) on clique complexes have recently been proposed, though it is unclear how their performances compare. We review these algorithms, emphasising their common Monte Carlo structure within a new modular framework. We derive upper bounds for the number of samples needed to reach a given level of precision, and use them to compare these algorithms. By recombining the different modules, we create a new quantum algorithm with an exponentially-improved dependence in the sample complexity. We run classical simulations to verify convergence within the theoretical bounds and observe the predicted exponential separation, even though empirical convergence occurs substantially earlier than the conservative theoretical bounds.
Paper Structure (27 sections, 22 theorems, 85 equations, 1 figure, 4 tables, 5 algorithms)

This paper contains 27 sections, 22 theorems, 85 equations, 1 figure, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Simplicial complexes, Laplacians and Betti numbers
  3. Stochastic trace estimation
  4. Classical stochastic trace estimation for powers of sparse matrices
  5. Quantum stochastic trace estimation for positive semi-definite matrices
  6. The QBNE-Chebyshev algorithm
  7. Outline
  8. Quantum circuits for the Laplacian
  9. Polynomial constructions
  10. Complexity analysis
  11. The classical BNE algorithms
  12. Outline
  13. CBNE-Power algorithm
  14. CBNE-Chebyshev algorithm
  15. The QBNE-Power algorithm
  16. ...and 12 more sections

Key Result

Lemma 3.1

Let $X_1,..., X_q$ be independent random variables such that $a_i \leq X_i \leq b_i$ almost surely, and write $s_q = \sum_{i=1}^q X_i$. Then

Figures (1)

  • Figure 1: The circuit $U_{I-P_\Gamma}$ which block-encodes the projection $I-P_{\Gamma}$ given a circuit $U_{P_\Gamma}$ that block-encodes the projection $P_\Gamma$. The multi-controlled $X$ operation is controlled on the $0$ of each of the auxiliary qubits of $U_{P_\Gamma}$. See akhalwaya2024topological for a circuit implementation of $U_{P_\Gamma}$ using $\mathcal{O}(n^2)$ auxiliary qubits.

Theorems & Definitions (44)

  • Definition
  • Lemma 3.1: Hoeffding's Inequality Hoeffding1963
  • Definition
  • Lemma 3.2
  • proof
  • Definition : Block-encoding
  • Lemma 3.3
  • Definition
  • Lemma 4.1
  • proof
  • ...and 34 more