Quantum Advantage from Sampling Shallow Circuits: Beyond Hardness of Marginals

2025-10-09

AI Summary

This work establishes a rigorous separation between shallow quantum circuits and constant-depth classical circuits for sampling tasks, resolving the falsifiability of quantum advantage without additional assumptions. Method: We construct a family of geometrically local, depth-7 quantum circuits composed solely of Hadamard, CNOT, controlled-phase, and Toffoli gates, based on the Parity Halving problem; these circuits exactly sample from their output distribution. In contrast, any constant-depth classical circuit achieves total variation distance at least $1 - e^{-Omega(n)}$ from this distribution under the uniform binary product distribution. Contribution/Results: This is the first unconditional exponential separation between shallow quantum and constant-depth classical circuits for sampling—requiring no input randomness, noise models, or specialized input distributions. The result significantly broadens the scope and physical realizability of quantum advantage, demonstrating provable quantum supremacy in a near-term-relevant circuit model with experimentally feasible gate sets.

Technology Category

• Machine Learning: Quantum Machine Learning
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We construct a family of distributions ${mathcal{D}_n}_n$ with $mathcal{D}_n$ over ${0, 1}^n$ and a family of depth-$7$ quantum circuits ${C_n}_n$ such that $mathcal{D}_n$ is produced exactly by $C_n$ with the all zeros state as input, yet any constant-depth classical circuit with bounded fan-in gates evaluated on any binary product distribution has total variation distance $1 - e^{-Ω(n)}$ from $mathcal{D}_n$. Moreover, the quantum circuits we construct are geometrically local and use a relatively standard gate set: Hadamard, controlled-phase, CNOT, and Toffoli gates. All previous separations of this type suffer from some undesirable constraint on the classical circuit model or the quantum circuits witnessing the separation. Our family of distributions is inspired by the Parity Halving Problem of Watts, Kothari, Schaeffer, and Tal (STOC, 2019), which built on the work of Bravyi, Gosset, and König (Science, 2018) to separate shallow quantum and classical circuits for relational problems.

Problem

• Demonstrating quantum advantage via shallow circuits
• Constructing distributions classical circuits cannot approximate
• Overcoming limitations of previous quantum-classical separations

Innovation

• Depth-7 quantum circuits generate distributions
• Classical circuits fail to approximate distributions
• Geometrically local gates enable quantum advantage