The power of quantum circuits in sampling

2025-10-03

AI Summary

Do quantum circuits exhibit exponential advantage over classical circuits for approximate sampling tasks? Prior results established such separations only for exact sampling. Method: In the random oracle model, we construct a distribution samplable by polynomial-size quantum circuits and prove that no subexponential-size classical circuit can approximate it within total variation distance $1 - o(1)$. To achieve this, we introduce a novel classical query complexity amplification lemma that strengthens the hardness of the Yamakawa–Zhandry search problem. Contribution: This work provides the first exponential separation between quantum and non-uniform classical circuits for approximate sampling—specifically, strong approximate sampling—thereby extending the theoretical frontier of quantum supremacy beyond exact sampling. It yields the strongest relativizing evidence to date for quantum sampling advantage, resolving a central open question in quantum complexity theory.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

We give new evidence that quantum circuits are substantially more powerful than classical circuits. We show, relative to a random oracle, that polynomial-size quantum circuits can sample distributions that subexponential-size classical circuits cannot approximate even to TV distance $1-o(1)$. Prior work of Aaronson and Arkhipov (2011) showed such a separation for the case of exact sampling (i.e. TV distance $0$), but separations for approximate sampling were only known for uniform algorithms. A key ingredient in our proof is a new hardness amplification lemma for the classical query complexity of the Yamakawa-Zhandry (2022) search problem. We show that the probability that any family of query algorithms collectively finds $k$ distinct solutions decays exponentially in $k$.

Problem

• Quantum circuits sample distributions classical circuits cannot approximate
• Hardness amplification for Yamakawa-Zhandry search problem query complexity
• Separating quantum and classical circuit power via random oracle

Innovation

• Quantum circuits sample distributions classical circuits cannot
• Hardness amplification lemma for classical query complexity
• Exponential decay in finding distinct solutions by algorithms