An Exponential Sample-Complexity Advantage for Coherent Quantum Inference

TL;DR

This paper demonstrates that coherent quantum inference protocols can exponentially reduce sample complexity compared to traditional incoherent, measurement-based methods, especially in tasks like quantum purity amplification.

Contribution

It introduces a new theoretical framework showing exponential sample complexity advantages of coherent over incoherent quantum inference protocols.

Findings

Coherent protocols achieve error ε with O(1/ε) copies for quantum purity amplification.

Incoherent protocols require Ω(d/ε) copies, showing a linear dependence on input dimension d.

The paper establishes a sharp separation between coherent and incoherent quantum inference methods.

Abstract

Standard quantum inference converts quantum data into classical outputs. We study an alternative inference setting in which the desired output is quantum, preserving coherence. Such settings include quantum purity amplification (QPA), mixed-state approximate purification or cloning, and density matrix exponentiation. We show that such protocols can achieve exponentially lower sample complexity than incoherent, measurement-mediated protocols. For QPA with principal eigenstate targets and $d$-dimensional inputs, coherent processing achieves error $\varepsilon$ using $O(1/\varepsilon)$ copies, versus the $\Omega(d/\varepsilon)$ copies required by any incoherent protocol. Together, these sharp coherent-incoherent separations seed a theory of coherent quantum inference, with an entanglement-breaking limit identifying the optimal incoherent counterpart of each coherent protocol.