Quantum

An Exponential Sample-Complexity Advantage for Coherent Quantum Inference

Zhaoyi Li, Elias Theil, Aram W. Harrow, Isaac Chuang

Featured May 25, 2026

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Simply

By keeping quantum information coherent throughout processing, new methods achieve a massive sample complexity advantage, needing far fewer input copies than older measurement-based techniques to perform tasks like amplifying quantum purity or simulating quantum evolution.

In depth

The paper introduces a theory of coherent quantum inference (CQI), demonstrating that quantum protocols which preserve coherence can achieve exponentially lower sample complexity compared to incoherent, measurement-mediated methods. This advantage is rigorously proven for three key tasks: quantum purity amplification (QPA), density matrix exponentiation (DME), and random purification (RP), where coherent approaches often show a sample complexity independent of or only weakly dependent on the Hilbert-space dimension

d

, while incoherent methods scale linearly with

d

. The work establishes a framework for understanding the fundamental limits and benefits of processing quantum information directly without classical bottlenecks.

Key Takeaways

1
Coherent quantum inference protocols can achieve an exponential sample-complexity advantage over incoherent, measurement-mediated protocols for tasks requiring quantum outputs.
2
For quantum purity amplification (QPA), coherent processing requires $O (1/ ε)$ copies, significantly outperforming incoherent methods that need $Ω (d / ε)$ copies, where $d$ is the Hilbert-space dimension.
3
The framework of coherent quantum inference is formalized and applied to random purification (RP) and density matrix exponentiation (DME), consistently demonstrating dimension-dependent separations in sample complexity.

Conceptual Flow

HIGH LEVEL

Methodology: Coherent vs. Incoherent Processing

Instead of measuring quantum data and losing its special properties, the new method processes it directly, keeping all its quantum magic.

Quantum Data

Process Directly

Quantum Output

Results: Exponential Sample Advantage

This direct processing means the new method needs much less starting data to get a good result, especially for bigger problems.

Many Old Samples

Few New Samples

Achieve Same Result

Good Output