BELT: Block Encoding of Linear Transformation on Density Matrices

TL;DR

BELT is a new quantum protocol that encodes arbitrary linear maps, including non-physical ones, into unitary operations, enabling their manipulation and analysis in quantum systems with improved efficiency.

Contribution

The paper introduces BELT, a systematic method for simulating any linear map on density matrices via block encoding, extending capabilities beyond existing quantum transformation techniques.

Findings

Enables simulation of non-completely positive maps like transpose.

Improves sample complexity over Hermitian-preserving map exponentiation.

Applications include entanglement detection and quantum channel inversion.

Abstract

Linear maps that are not completely positive play a crucial role in the study of quantum information, yet their non-completely positive nature renders them challenging to realize physically. The core difficulty lies in the fact that when acting such a map $\mathcal{N}$ on a state $\rho$, $\mathcal{N}(\rho)$ may not correspond to a valid density matrix, making it difficult to prepare directly in a physical system. We introduce Block Encoding of Linear Transformation (BELT), a systematic protocol that simulates arbitrary linear maps by embedding the output $\mathcal{N}(\rho)$ into a block of a unitary operator. BELT enables the manipulation and extraction of information about $\mathcal{N}(\rho)$ through coherent quantum evolution. Notably, BELT accommodates maps that fall outside the scope of quantum singular value transformation, such as the transpose map. BELT finds applications in entanglement detection, quantum channel inversion, and simulating pseudo-differential operators, and demonstrates improved sample complexity compared to protocols based on Hermitian-preserving map exponentiation.