[Paper Review] A Survey of Quantum Alternatives to Randomized Algorithms: Monte Carlo Integration and Beyond

Philip Intallura, Georgios Korpas+5 more|arXiv (Cornell University)|Mar 8, 2023

Quantum Computing Algorithms and Architecture8 citations

TL;DR

This survey analyzes quantum approaches to replace or augment classical Monte Carlo, focusing on amplitude estimation and variants to speed up sampling-based uncertainty quantification.

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.

Motivation & Objective

• Motivate the need for faster Monte Carlo methods in applications like computational finance and uncertainty quantification.
• Survey existing and emerging quantum algorithms that can replace or augment classical Monte Carlo procedures.
• Assess adaptive and hardware-aware variants of quantum Monte Carlo techniques.
• Provide a framework to compare classical mixing times with quantum-speedup promises.
• Highlight practical considerations for implementing these quantum approaches on near-term devices.

Proposed method

• Review quantum amplitude estimation (QAE) and its role in replacing Monte Carlo sampling by achieving O(1/ε) query complexity.
• Describe how quantum amplitude amplification generalizes Grover to estimate integrals and expectations.
• Present quantum approximate counting as a means to estimate averages of Booleanized function values.
• Discuss Montanaro’s algorithm for near-quadratic speedups in general Monte Carlo settings with bounded variance.
• Summarize practical variants that reduce circuit depth (MLE-QAE, Iterative QAE, robust amplitude estimation) and parallelization strategies for QPE/QAE.

Experimental results

Research questions

• RQ1 Can quantum algorithms provide a quadratic or near-quadratic speedup for Monte Carlo integration under realistic assumptions?
• RQ2 What are the trade-offs between fault-tolerant and near-term quantum devices for quantum Monte Carlo (depth, qubit count, and noise resilience)?
• RQ3 How do adaptive and variant schemes (MLE-QAE, IQAE, RAE) compare in terms of accuracy, depth, and practicality on NISQ devices?
• RQ4 What is the impact of quantum-enhanced mixing times on applications such as derivative pricing and risk assessment?
• RQ5 How can parallelization of QPE/QAE reduce gate depth while preserving or enhancing accuracy?

Key findings

• Quantum amplitude estimation can yield a quadratic speedup over classical Monte Carlo under certain assumptions.
• Several variants (MLE-QAE, Iterative QAE, RAE) reduce circuit depth and hardware requirements, aiding NISQ usability.
• Montanaro’s framework extends quantum speedups to a broad class of randomized quantum algorithms with bounded variance.
• Parallelized QPE/QAE strategies can lower gate depth and improve robustness to decoherence, enabling deeper use on near-term devices.
• The survey emphasizes that quantum advantages are context-dependent and not universally guaranteed across all Monte Carlo tasks.