Characterizing unitaries via quasi-process functions
Abstract
It is well-known that every n-qubit unitary can be decomposed into a product of controlled singlequbit unitaries. We show that the control conditions in such a decomposition naturally define a quasi-process function, an object that originally came up in the study of indefinite causal order. Here quasi-process functions yield a representation of every n-qubit unitary in a canonical form, namely, as a quasi-process function unitary (qPFU). We use this representation to show how quantum nonlocality without entanglement (QNLWE) can be understood in terms of unitaries that encode the trade-off between causal order and locality noted in R. Kunjwal and Ä. Baumeler, Phys. Rev. Lett. 131, 120201 (2023). Specifically, we obtain two sufficient (but not necessary) criteria-mutual exclusivity of control conditions (MECC) and Unambiguity-for a unitary to be a process function unitary (PFU), i.e., it admits a qPFU representation where the quasi-process function is a process function, an object that can model paradox-free classical causal loops. We also show the inequivalence of these criteria, neither implying the other. We then show that the circuit depth of MECC unitaries admits a characterization in terms of clique-partitioning problems on Hamming graphs and use this correspondence to provide bounds on the circuit depth of n-qubit MECC unitaries. Finally, we show how MECC unitaries provide a recipe for constructing a unitary purification of the associated process function, thereby providing a large class of unitary processes of interest in quantum causality. The generality of this causality-inspired representation of unitaries makes it a versatile tool for analyzing the structure of quantum circuits in other domains.
Related
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BibTeX
@misc{ahmed2026characterizing,
title = {{Characterizing unitaries via quasi-process functions}},
author = {Nasra Daher Ahmed and Ravi Kunjwal},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2610.00579},
eprint = {2610.00579},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05791425v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Characterizing unitaries via quasi-process functions
AU - Nasra Daher Ahmed
AU - Ravi Kunjwal
PY - 2026
DA - 2026/09/30
PB - HAL
DO - 10.48550/arxiv.2610.00579
UR - https://hal.science/hal-05791425v1
AB - It is well-known that every n-qubit unitary can be decomposed into a product of controlled singlequbit unitaries. We show that the control conditions in such a decomposition naturally define a quasi-process function, an object that originally came up in the study of indefinite causal order. Here quasi-process functions yield a representation of every n-qubit unitary in a canonical form, namely, as a quasi-process function unitary (qPFU). We use this representation to show how quantum nonlocality without entanglement (QNLWE) can be understood in terms of unitaries that encode the trade-off between causal order and locality noted in R. Kunjwal and Ä. Baumeler, Phys. Rev. Lett. 131, 120201 (2023). Specifically, we obtain two sufficient (but not necessary) criteria-mutual exclusivity of control conditions (MECC) and Unambiguity-for a unitary to be a process function unitary (PFU), i.e., it admits a qPFU representation where the quasi-process function is a process function, an object that can model paradox-free classical causal loops. We also show the inequivalence of these criteria, neither implying the other. We then show that the circuit depth of MECC unitaries admits a characterization in terms of clique-partitioning problems on Hamming graphs and use this correspondence to provide bounds on the circuit depth of n-qubit MECC unitaries. Finally, we show how MECC unitaries provide a recipe for constructing a unitary purification of the associated process function, thereby providing a large class of unitary processes of interest in quantum causality. The generality of this causality-inspired representation of unitaries makes it a versatile tool for analyzing the structure of quantum circuits in other domains.
ER -