Symmetric dilations of Pauli channels and semigroups

TL;DR

This paper investigates the symmetry properties of Stinespring dilations of single-qubit Pauli channels, providing explicit constructions and constraints relevant for quantum simulation and quantum computing implementations.

Contribution

It offers new insights into the symmetry constraints on dilations of Pauli channels, including explicit constructions for time-dependent dilations and collision models.

Findings

Derived the representation of the Pauli group on environment Hilbert space.

Showed covariance constraints strongly restrict dilation Hamiltonians and initial states.

Provided explicit constructions of dilations for various Pauli channels and semigroups.

Abstract

We explore the symmetry properties of Stinespring dilations of single-qubit Pauli channels, addressing both the generic case and the specific examples of phase damping and depolarizing channels. For each scenario, we derive the representation of the Pauli group acting on the Hilbert space of the environment. We then focus on dilations that are continuous in time and driven by a time-independent Hamiltonian, and on collision models that generate a Pauli dynamical semigroup in the limit of fast collisions. First, we complement some recent general results on these types of dilations (M. Cattaneo, Phys. Rev. A 111, 022209 (2025)) with some additions and clarifications, including the case of covariant channels with strongly conserved quantities. Next, we show that the covariance property of Pauli channels impose strong constraints on both the dilation Hamiltonian and the initial state of the environment, and demonstrate how these constraints can be exploited to explicitly construct the time-dependent dilations in all considered cases. Our results are relevant for the quantum simulation of Pauli channels via unitary dilations and of Pauli semigroups via collision models, both in the laboratory and on quantum computers.