Pauli Decomposition by Character Theory: A Memory-Bounded Algorithm for Qubits and Qudits
Abstract
Pauli decomposition of an n-qubit operator is expensive if each of the 4n Pauli strings is treated as an unstructured object. Labeling those strings by the configuration space (ℤ2)n makes them displacement operators X⊗ xZ⊗ z on the computational space: shifts by~x and multiplications by characters Z⊗ z in the computational basis. That labeling organizes the operator into 2n fibers of length 2n, one per shift~x, each collecting the computational-basis amplitudes that hop by~x. Expanding each fiber function in the character basis of (ℤ2)n --- a length-2n Walsh--Hadamard transform --- recovers the Pauli coefficients at total cost O(n·4n). The same fiber / shift--character picture extends to (ℤp)n (Heisenberg--Weyl qudits). \texttt{paulikit} is a memory-bounded realization for qubits: for matrix-sparse inputs, peak memory need not densify to a full 2n×2n operator. Performance is checked on a stated low-end GNU/Linux platform for a coupled-oscillator sweep beyond a billion Pauli terms.
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BibTeX
@misc{khellat2026pauli,
title = {{Pauli Decomposition by Character Theory: A Memory-Bounded Algorithm for Qubits and Qudits}},
author = {Mohammadreza Khellat and Mohammad Masoumi and Saman Nasoori and Soroush Nasoori},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08299},
eprint = {2610.08299},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08299},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Pauli Decomposition by Character Theory: A Memory-Bounded Algorithm for Qubits and Qudits
AU - Mohammadreza Khellat
AU - Mohammad Masoumi
AU - Saman Nasoori
AU - Soroush Nasoori
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.08299
UR - https://arxiv.org/abs/2610.08299
AB - Pauli decomposition of an n-qubit operator is expensive if each of the 4^n Pauli strings is treated as an unstructured object. Labeling those strings by the configuration space (ℤ_2)^n makes them displacement operators X^⊗ xZ^⊗ z on the computational space: shifts by~x and multiplications by characters Z^⊗ z in the computational basis. That labeling organizes the operator into 2^n fibers of length 2^n, one per shift~x, each collecting the computational-basis amplitudes that hop by~x. Expanding each fiber function in the character basis of (ℤ_2)^n --- a length-2^n Walsh--Hadamard transform --- recovers the Pauli coefficients at total cost O(n·4^n). The same fiber / shift--character picture extends to (ℤ_p)^n (Heisenberg--Weyl qudits). \texttt{paulikit} is a memory-bounded realization for qubits: for matrix-sparse inputs, peak memory need not densify to a full 2^n×2^n operator. Performance is checked on a stated low-end GNU/Linux platform for a coupled-oscillator sweep beyond a billion Pauli terms.
ER -