Protocols and Trade-Offs of Quantum State Purification

TL;DR

This paper introduces a comprehensive framework for quantum state purification that optimizes fidelity and success probability, providing explicit protocols, proving their optimality, and demonstrating practical implementation methods for noisy quantum states.

Contribution

It presents a general, optimal quantum state purification framework with explicit protocols, optimality proofs, and practical implementation strategies for noisy quantum states.

Findings

Framework replicates and extends existing purification protocols.

Proves protocol optimality for two-copy states and numerically for higher copies.

Provides explicit circuits and estimates sample complexity for practical quantum purification.

Abstract

Quantum state purification is crucial in quantum communication and computation, aiming to recover a purified state from multiple copies of an unknown noisy state. This work introduces a general state purification framework designed to achieve the highest fidelity with a specified probability and characterize the associated trade-offs. For i.i.d. quantum states under depolarizing noise, our framework can replicate the purification protocol proposed by [Barenco et al., SIAM Journal on Computing, 26(5), 1997] and further provide exact formulas for the purification fidelity and probability with explicit trade-offs. We prove the protocols' optimality for two copies of noisy states with any dimension and confirm its optimality for higher numbers of copies and dimensions through numerical analysis. Our methodological approach paves the way for proving the protocol's optimality in more general scenarios and leads to optimal protocols for other noise models. Furthermore, we present a systematic implementation method via block encoding and parameterized quantum circuits, providing explicit circuits for purifying three-copy and four-copy states under depolarizing noise. Finally, we estimate the sample complexity and generalize the protocol to a recursive form, demonstrating its practicality for quantum computers with limited memory.