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A Survey of Quantum Alternatives to Randomized Algorithms: Monte Carlo Integration and Beyond

Philip Intallura, Georgios Korpas, Sudeepto Chakraborty, Vyacheslav Kungurtsev, Rufus Lawrence, Ales Wodecki, Jakub Marecek

TL;DR

The quantum algorithms that could replace classical Monte Carlo and then consider both the existing quantum algorithms and the potential quantum realizations that include adaptive enhancements as alternatives to the classical procedure are considered.

Abstract

Monte Carlo sampling is a powerful toolbox of algorithmic techniques widely used for a number of applications wherein some noisy quantity, or summary statistic thereof, is sought to be estimated. In this paper, we survey the literature for implementing Monte Carlo procedures using quantum circuits, focusing on the potential to obtain a quantum advantage in the computational speed of these procedures. We revisit the quantum algorithms that could replace classical Monte Carlo and then consider both the existing quantum algorithms and the potential quantum realizations that include adaptive enhancements as alternatives to the classical procedure.

A Survey of Quantum Alternatives to Randomized Algorithms: Monte Carlo Integration and Beyond

TL;DR

The quantum algorithms that could replace classical Monte Carlo and then consider both the existing quantum algorithms and the potential quantum realizations that include adaptive enhancements as alternatives to the classical procedure are considered.

Abstract

Monte Carlo sampling is a powerful toolbox of algorithmic techniques widely used for a number of applications wherein some noisy quantity, or summary statistic thereof, is sought to be estimated. In this paper, we survey the literature for implementing Monte Carlo procedures using quantum circuits, focusing on the potential to obtain a quantum advantage in the computational speed of these procedures. We revisit the quantum algorithms that could replace classical Monte Carlo and then consider both the existing quantum algorithms and the potential quantum realizations that include adaptive enhancements as alternatives to the classical procedure.
Paper Structure (36 sections, 63 equations, 3 figures, 4 tables)

This paper contains 36 sections, 63 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Basic principles of Monte Carlo methods
  3. Quantum Alternatives to Classical Monte Carlo
  4. Quantum Amplitude Estimation
  5. Summation of Boolean functions
  6. Integration of real-valued functions
  7. Regularity and query complexity
  8. Quantum Approximate Counting
  9. Loading probability distributions
  10. Opportunities with Quantum Alternatives to Classical Monte Carlo
  11. Montanaro's Algorithm
  12. MLE-QAE
  13. Iterative QAE
  14. Full advantage with minimal depth
  15. Robust Amplitude Estimation
  16. ...and 21 more sections

Figures (3)

  • Figure 1: The evolution of algorithms for quantum replacement of Monte Carlo integration. Note that certain variations of QAE might not explicitly appear on this diagram.
  • Figure 2: A unitary $f(U)$ prepares an eigenstate of $U$ on the $\ell$ registers whose eigenvalue is computed at 3-bit precision using QPE.
  • Figure 3: An extra entangling qubit $p_j$ achieves a horizontal compression in the gate execution, as compared to the standard and directly parallelized QPE.