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arXiv:2205.05808 · Physical Review A

Pauli component erasing quantum channels

Jose Alfredo de Leon, Alejandro Fonseca, F. Leyvraz, David Davalos, Carlos Pineda

Published Oct 5, 2022• 3 citations
arXivHTMLPDF

Abstract

Decoherence of quantum systems is described by quantum channels. However, a complete understanding of such channels, especially in the multiparticle setting, is still an ongoing difficult task. We propose the family of quantum maps that preserve or completely erase the components of a multiqubit system in the basis of Pauli strings, which we call Pauli component erasing maps. For the corresponding channels, it is shown that the preserved components can be interpreted as a finite vector subspace, from which we derive several properties and complete the characterization. Moreover, we show that the obtained family of channels forms a semigroup and derive its generators. We use this simple structure to determine physical implementations and connect the obtained family of channels with Markovian processes.

DOI
10.1103/physreva.106.042604

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Decoherence of quantum systems is described by quantum channels. However, a complete understanding of such channels, especially in the multiparticle setting, is still an ongoing difficult task. We propose the family of quantum maps that preserve or completely erase the components of a multiqubit system in the basis of Pauli strings, which we call Pauli component erasing maps. For the corresponding channels, it is shown that the preserved components can be interpreted as a finite vector subspace, from which we derive several properties and complete the characterization. Moreover, we show that the obtained family of channels forms a semigroup and derive its generators. We use this simple structure to determine physical implementations and connect the obtained family of channels with Markovian processes.
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We present a noise deconvolution technique for obtaining noiseless expectation values of noisy observables at the output of multiqubit quantum channels. For any number of qubits or in the presence of correlations, our protocol applies to any mathematically invertible noise model, even when its inverse map is not physically implementable, i.e., when it is neither completely positive nor trace preserving. For a generic observable affected by Pauli noise it provides a quadratic speedup, always producing a rescaling of its Pauli basis components. We show that it is still possible to achieve the deconvolution while experimentally estimating the noise parameters, whenever these are unknown (bypassing resource-heavy techniques such as quantum process tomography). We provide a simulation, with examples for both Pauli and non-Pauli channels.
Quantum channels define key objects in quantum information theory. They are represented by completely positive trace-preserving linear maps in matrix algebras. We analyze a family of quantum channels defined through the use of the Weyl operators. Such channels provide generalization of the celebrated qubit Pauli channels. Moreover, they are covariant with respective to the finite group generated by Weyl operators. In what follows, we study self-adjoint Weyl channels by providing a special Hermitian representation. For a prime dimension of the corresponding Hilbert space, the self-adjoint Weyl channels contain well-known generalized Pauli channels as a special case. We propose multipartite generalization of Weyl channels. In particular, we analyze the power of prime dimensions using finite fields and study the covariance properties of these objects.
In matrix algebras, we introduce a class of linear maps that are irreducibly covariant with respect to the finite group generated by the Weyl operators. In particular, we analyze the irreducibly covariant quantum channels, that is, the completely positive and trace-preserving linear maps. Interestingly, imposing additional symmetries leads to the so-called generalized Pauli channels, which were recently considered in the context of the non-Markovian quantum evolution. Finally, we provide examples of irreducibly covariant positive but not necessarily completely positive maps.
Abstract Analysis of quantum processes, especially in the context of noise, errors, and decoherence is essential for the improvement of quantum devices. An intuitive representation of those processes modeled by quantum channels are Pauli transfer matrices. They display the action of a linear map in the n-qubit Pauli basis in a way, that is more intuitive, since Pauli strings are more tangible objects than the standard basis matrices. We set out to investigate classical algorithms that convert the various representations into Pauli transfer matrices. We propose new algorithms that make explicit use of the tensor product structure of the Pauli basis. They convert a quantum channel in a given representation (Chi or process matrix, Choi matrix, superoperator, or Kraus operators) to the corresponding Pauli transfer matrix. Moreover, the underlying principle can also be used to calculate the Pauli transfer matrix of other linear operations over n-qubit matrices such as left-, right-, and sandwich multiplication as well as forming the (anti-)commutator with a given operator. Finally, we investigate the runtime of these algorithms, derive their asymptotic scaling and demonstrate improved performance using instances with up to seven qubits.
We investigate the possibility of dividing quantum channels into concatenations of other channels, thereby studying the semigroup structure of the set of completely-positive trace-preserving maps. We show the existence of ‘ indivisible ’ channels which can not be written as non-trivial products of other channels and study the set of ‘ infinitesimal divisible ’ channels which are elements of continuous completely positive evolutions. For qubit channels we obtain a complete characterization of the sets of indivisible and infinitesimal divisible channels. Moreover, we identify those channels which are solutions of time-dependent master equations for both positive and completely positive evolutions. For arbitrary finite dimension we prove a representation theorem for elements of continuous completely positive evolutions based on new results on determinants of quantum channels and Markovian approximations.