Stratified Sampling for Quasi-Probability Decompositions

TL;DR

This paper introduces a stratified sampling framework for quasi-probability decompositions in quantum algorithms, significantly reducing variance and sampling costs without extra quantum resources.

Contribution

It develops a general variance reduction method using stratified sampling for QPDs and provides a classical algorithm for implementation, improving efficiency in quantum computations.

Findings

Achieves up to 80% variance reduction in simulations

Provides a classical dynamic programming approach for stratification

Demonstrates practical variance savings in quantum error mitigation

Abstract

Quasi-probability decompositions (QPDs) have proven essential in many quantum algorithms and protocols -- one replaces a ``difficult'' quantum circuit with an ensemble of ``easier'' circuit variants whose weighted outcomes reproduce any target observable. This, however, inevitably yields an increased configuration variance beyond Born-rule shot noise. We develop a broad framework for accounting for and reducing this variance and prove that stratified sampling -- under ideal proportional allocation -- results in an unbiased estimator with a variance that is never worse than na\"ive sampling (with equality only in degenerate cases). Furthermore, we provide a classical dynamic programme to enable stratification on arbitrary product-form QPDs. Numerical simulations of typical QPDs, such as Probabilistic Error Cancellation (PEC) and Probabilistic Angle Interpolation (PAI), demonstrate constant-factor reductions in overall variance (up to $\sim 60$--$80\%$ in an oracle model) and robust $\sim 10\%$ savings in the pessimistic single-shot regime. Our results can be applied immediately to reduce the net sampling cost of practically relevant QPDs that are commonly used in near term and early fault-tolerant algorithms without requiring additional quantum resources.