Quantum element-wise transforms
Abstract
Quantum algorithms for basic numerical linear algebraic tasks have proven essential for translating diverse problems to a unified quantum computational context. Many of these tasks---e.g., applying a function to the spectrum of a matrix embedded in a unitary process (a so-called block encoding), or taking linear combinations of block encodings---are well-addressed by techniques like quantum singular value transformation or linear combination of unitaries. However, there exist useful matrix transforms whose realization is unclear or inefficient. In this work we construct improved quantum algorithms for some of these transforms, the simplest of which is a polynomial function applied element-wise. We show the space required to compute quantum element-wise transforms (QEWTs) can be reduced exponentially in the degree of the applied function compared to prior work, and rectify errors in previous constructions. We prove novel space and query lower bounds for QEWTs, construct useful function families nearly saturating these bounds, and establish the hardness of QEWT dequantization. This represents the first sound theoretical treatment of QEWTs, and brings their resource requirements on par with other practical block encoding techniques. We present our algorithms alongside concrete applications to machine learning inference and signal processing.
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BibTeX
@misc{rossi2026quantum,
title = {{Quantum element-wise transforms}},
author = {Zane M. Rossi and Rahul Sarkar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.06456},
eprint = {2606.06456},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.06456},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum element-wise transforms
AU - Zane M. Rossi
AU - Rahul Sarkar
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2606.06456
UR - https://arxiv.org/abs/2606.06456
AB - Quantum algorithms for basic numerical linear algebraic tasks have proven essential for translating diverse problems to a unified quantum computational context. Many of these tasks---e.g., applying a function to the spectrum of a matrix embedded in a unitary process (a so-called block encoding), or taking linear combinations of block encodings---are well-addressed by techniques like quantum singular value transformation or linear combination of unitaries. However, there exist useful matrix transforms whose realization is unclear or inefficient. In this work we construct improved quantum algorithms for some of these transforms, the simplest of which is a polynomial function applied element-wise. We show the space required to compute quantum element-wise transforms (QEWTs) can be reduced exponentially in the degree of the applied function compared to prior work, and rectify errors in previous constructions. We prove novel space and query lower bounds for QEWTs, construct useful function families nearly saturating these bounds, and establish the hardness of QEWT dequantization. This represents the first sound theoretical treatment of QEWTs, and brings their resource requirements on par with other practical block encoding techniques. We present our algorithms alongside concrete applications to machine learning inference and signal processing.
ER -