A linearized stabilizer formalism for systems of finite dimension

TL;DR

This paper introduces a linearized stabilizer formalism for finite-dimensional systems, simplifying the simulation of stabilizer circuits on qudits by representing operators in discrete phase space and analyzing their evolution.

Contribution

It presents a new formalism using displacement operators in phase space, enabling simpler proofs and efficient simulation of stabilizer circuits for qudits of any fixed dimension.

Findings

Simulation of stabilizer circuits is complete for coMod_{d}L.

Simulation can be performed with O(log(n)^2) depth boolean circuits.

Formalism applies to systems of any fixed dimension d >= 2.

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.