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arXiv:quant-ph/0611058 · Physical Review A

Simulating a single-qubit channel using a mixed-state environment

Geetu Narang, Arvind

Published Mar 5, 2007• 17 citations
arXivHTMLPDF

Abstract

We analyze the class of single-qubit channels with the environment modeled by a one-qubit mixed state. The set of affine transformations for this class of channels is computed analytically, employing the canonical form for the two-qubit unitary operator. We demonstrate that $\frac{3}{8}$ of the generalized depolarizing channels can be simulated by the one-qubit mixed-state environment by explicitly obtaining the shape of the volume occupied by this class of channels within the tetrahedron representing the generalized depolarizing channels. Further, as a special case, we show that the two-Pauli channel cannot be simulated by a one-qubit mixed-state environment.

DOI
10.1103/physreva.75.032305

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We analyze the class of single-qubit channels with the environment modeled by a one-qubit mixed state. The set of affine transformations for this class of channels is computed analytically, employing the canonical form for the two-qubit unitary operator. We demonstrate that 38\frac{3}{8}83​ of the generalized depolarizing channels can be simulated by the one-qubit mixed-state environment by explicitly obtaining the shape of the volume occupied by this class of channels within the tetrahedron representing the generalized depolarizing channels. Further, as a special case, we show that the two-Pauli channel cannot be simulated by a one-qubit mixed-state environment.
We study a generalized amplitude damping channel when environment is initially in the single-qubit mixed state. Representing the affine transformation of the generalized amplitude damping by a three-dimensional volume, we plot explicitly the volume occupied by the channels simulatable by a single-qubit mixed-state environment. As expected, this volume is embedded in the total volume by the channels which is simulated by two-qubit enviroment. The volume ratio is approximately 0.08 which is much smaller than 3/8, the volume ratio for generalized depolarizing channels.
We found the quasi inverse of qubit channels as a unitary map, Ei\mathcal{E}^iEi, by minimizing the average trace distance between the input state to the channel and the output of the quasi inverse channel for arbitrary qubit channel E\mathcal{E}E and for arbitrary input states. The channel E\mathcal{E}E was assumed completely positive and trace-preserving. To find the quasi inverse for mixed states, we proposed an alternative definition of the quasi inverse based on the mean square of the trace distance (MSTD) of a channel. The definition based on the trace distance allowed easy generalization of the quasi inverse to mixed input states. The quasi inverse of the Pauli, generalized amplitude damping, mixed unitary, and tetrahedron channels calculated based on the MSTD agreed with the one computed using average fidelity in the special case of input states being pure.
Pauli channels are fundamental in the context of quantum computing as they model the simplest kind of noise in quantum devices. We propose a quantum algorithm for simulating Pauli channels and extend it to encompass Pauli dynamical maps (parametrized Pauli channels). A parametrized quantum circuit is employed to accommodate for dynamical maps. We also establish the mathematical conditions for an N-qubit transformation to be achievable using a parametrized circuit where only one single-qubit operation depends on the parameter. The implementation of the proposed circuit is demonstrated using IBM's quantum computers for the case of one qubit, and the fidelity of this implementation is reported.
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.
The development of classical ergodic theory has had a significant impact on the areas of mathematics, physics, and, in general, applied sciences. The quantum ergodic theory of Hamiltonian dynamics has its motivations in understanding thermodynamics and statistical mechanics. The quantum channel, a completely positive trace-preserving map, represents the most general representation of quantum dynamics and is an essential aspect of quantum information theory and quantum computation. In this work, we study the ergodic theory of quantum channels by characterizing different levels of ergodic hierarchy from integrable to mixing. The quantum channels on single systems are constructed from the unitary operators acting on bipartite states and tracing out the environment. The interaction strength of these unitary operators measured in terms of operator entanglement provides sufficient conditions for the channel to be mixing. By using block-diagonal unitary operators, we construct a set of nonergodic channels. By using the canonical form of the two-qubit unitary operator, we analytically construct channels on a single qubit ranging from integrable to mixing. Moreover, we also study interacting many-body quantum systems that include the famous Sachdev-Ye-Kitaev model and show that they display mixing within the framework of the quantum channel.
Dephasing decoherence induced by interaction of one qubit with a quantum bath can be simulated classically by random unitary evolution without the need for a bath and this random unitary evolution is equivalent to the quantum case. For a general dephasing model and a single-qubit system, we explicitly construct the noise functional and completely specify the random unitary evolution. To demonstrate the technique, we applied our results to three paradigmatic models: spin-boson, central spin, and quantum impurity. For multiple qubits, we identify a class of generalized quantum dephasing models that can be simulated classically. Finally we show that depolarizing quantum models can be simulated classically for all dimensionalities of the principal system.