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Quantum Algorithmic Advantage in Expectation Estimation

Mechanism: Quantum Amplitude Estimation (QAE) approximates expected values more efficiently than classical Monte Carlo methods. Readout: Readout: QAE achieves the same accuracy with quadratically fewer oracle queries, resulting in a significantly lower Root-Mean-Square Error (RMSE) for a given number of computations.

1. Introduction

Estimating the expected value of a random variable is a fundamental computational task across many scientific and industrial domains, including financial modeling, statistical physics, machine learning, and risk analysis. Classical approaches typically rely on Monte Carlo sampling, where the accuracy of the estimate improves with the number of samples drawn from the underlying distribution. However, Monte Carlo methods exhibit a well-known convergence rate proportional to 1 / 𝑁 1/ N ​

, where 𝑁 N is the number of samples.

Quantum computing offers a potential improvement to this process through quantum amplitude estimation (QAE), which leverages quantum interference to accelerate the estimation of probabilities and expectation values. Theoretical results indicate that amplitude estimation can achieve a quadratic improvement in sample complexity compared to classical Monte Carlo methods.

This work proposes a hypothesis regarding the measurable computational advantage achievable using quantum amplitude estimation for expectation estimation tasks.

1. Hypothesis

We hypothesize that:

A quantum algorithm based on amplitude estimation can estimate the expected value of a bounded function with lower query complexity than any classical Monte Carlo estimator achieving the same accuracy and confidence level.

Formally, let

𝜇

𝐸 [ 𝑓 ( 𝑋 ) ] μ=E[f(X)]

be the expected value of a bounded function 𝑓 f evaluated on samples drawn from a probability distribution 𝑋 X. Let 𝜖

0 ϵ>0 denote the desired additive error and 𝛼 α the failure probability.

The hypothesis states that:

A quantum algorithm implementing amplitude estimation can approximate 𝜇 μ within additive error 𝜖 ϵ with success probability at least 1 − 𝛼 1−α using

𝑂  ⁣ ( 1 𝜖 log ⁡ 1 𝛼 ) O( ϵ 1 ​

log α 1 ​

)

oracle queries.

Any classical Monte Carlo estimator requires at least

Ω  ⁣ ( 1 𝜖 2 log ⁡ 1 𝛼 ) Ω( ϵ 2 1 ​

log α 1 ​

)

samples to achieve the same accuracy and confidence.

Consequently, the quantum algorithm achieves a quadratic improvement in sample complexity relative to classical Monte Carlo methods.

1. Testable Prediction

The hypothesis implies a measurable difference in empirical scaling behavior between classical and quantum methods. Specifically:

The root-mean-square error (RMSE) of classical Monte Carlo estimators should scale proportionally to 1 / 𝑁 1/ N ​

, where 𝑁 N is the number of samples.

The RMSE of the quantum amplitude estimation algorithm should scale proportionally to 1 / 𝑁 1/N, where 𝑁 N represents oracle invocations.

If implemented on a sufficiently large and low-noise quantum processor, the quantum algorithm should reach a target accuracy 𝜖 ϵ using significantly fewer oracle calls than classical Monte Carlo methods.

1. Experimental Verification

To test the hypothesis, an experiment can be constructed as follows:

Select a benchmark expectation estimation problem (e.g., a probabilistic integral or option pricing payoff).

Implement a classical Monte Carlo estimator to compute the expectation.

Implement a quantum amplitude estimation algorithm encoding the same probability distribution and payoff function.

Measure estimation error as a function of the number of oracle queries or samples.

Compare the empirical scaling behavior and total runtime of the two methods.

Evidence supporting the hypothesis would consist of observing the predicted quadratic improvement in query complexity for the quantum algorithm relative to the classical baseline.

1. Implications

If validated, the hypothesis would demonstrate a practical instance of quantum algorithmic advantage, showing that quantum computers can outperform classical methods for a broadly applicable statistical task. Such results would have implications for fields heavily dependent on large-scale Monte Carlo simulations, including finance, physics simulations, and probabilistic machine learning.