A linearized stabilizer formalism for systems of finite dimension
Abstract
The stabilizer formalism is a scheme, generalizing well-known techniques developed by Gottesman [quant-ph/9705052] in the case of qubits, to efficiently simulate a class of transformations ("stabilizer circuits", which include the quantum Fourier transform and highly entangling operations) on standard basis states of d-dimensional qudits. To determine the state of a simulated system, existing treatments involve the computation of cumulative phase factors which involve quadratic dependencies. We present a simple formalism in which Pauli operators are represented using displacement operators in discrete phase space, expressing the evolution of the state via linear transformations modulo D <= 2d. We thus obtain a simple proof that simulating stabilizer circuits on n qudits, involving any constant number of measurement rounds, is complete for the complexity class coMod_{d}L and may be simulated by O(log(n)^2)-depth boolean circuits for any constant d >= 2.
Keywords
Cite
@article{arxiv.1102.3354,
title = {A linearized stabilizer formalism for systems of finite dimension},
author = {Niel de Beaudrap},
journal= {arXiv preprint arXiv:1102.3354},
year = {2023}
}
Comments
25 pages, 3 figures. Reorganized to collect complexity results; some corrections and elaborations of technical results. Differs slightly from the version to be published (fixed typos, changes of wording to accommodate page breaks for a different article format). To appear as QIC vol 13 (2013), pp.73--115