Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Open Systems & Information Dynamics

On Quasi-Inversion of Quantum Channels in 2 and in Higher Dimensions

Vahid Karimipour

Published Sep 1, 2022• 1 citation
Publisher

Abstract

We review the concept of the quasi-inverse of qubit channels and of higher dimensional channels. Quasi-inverse is a channel which when concatenaded to the original channel, increases its average fidelity in an optimal way. For qubit channels, we fully characterize the quasi-inverse, while for higher dimensional channels, we prove general theorems and provide bounds for the increased average fidelity. Nevertheless, explicit examples are given when exact quasi-inverses can be found.

DOI
10.1142/s1230161222500147

Related papers

Quasi-inversion of qubit channels

Vahid Karimipour, Fabio Benatti, Roberto Floreanini B

In general quantum operations, or quantum channels cannot be inverted by physical operations, i.e., by completely positive trace-preserving maps. An arbitrary state passing through a quantum channel loses its fidelity with the input. Given a quantum channel $\mathcal{E}$, we discuss the concept of its quasi-inverse as a completely positive trace-preserving map ${\mathcal{E}}^{qi}$ which when composed with $\mathcal{E}$ increases its average input-output fidelity in an optimal way. The channel ${\mathcal{E}}^{qi}$ comes as close as possible to the inverse of a quantum channel. We give a complete classification of such maps for qubit channels and provide quite a few illustrative examples.

arXiv:1909.06118•Physical Review A•Mar 16, 2020•12 citations

Quasi-inversion of quantum and classical channels in finite dimensions

Fereshte Shahbeigi, Koorosh Sadri, Morteza Moradi +2 more

Abstract We introduce the concept of quasi-inverse of quantum and classical channels, prove general properties of these inverses and determine them for a large class of channels acting in an arbitrary finite dimension. Therefore we extend the previous results of Karimipour et al (2020 Phys. Rev. A 101 032109) to arbitrary dimensional channels and to the classical domain. We demonstrate how application of the proposed scheme can increase on the average the fidelity between a given random pure state and its image transformed by the quantum channel followed by its quasi-inversion.

arXiv:2104.06062•Journal of Physics A Mathematical and Theoretical•Jul 13, 2021•9 citations

Quasi-inverse of qubit channels for mixed states

Muhammad Faizan, Muhammad Faryad

We found the quasi inverse of qubit channels as a unitary map, $\mathcal{E}^i$, 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 $\mathcal{E}$ and for arbitrary input states. The channel $\mathcal{E}$ 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.

arXiv:2410.02414•Physical Review A•Jun 24, 2025•0 citations

Quantum gate fidelity in terms of Choi matrices

Nathaniel Johnston, David W. Kribs

We provide new results for computing and comparing the quantum gate fidelity of quantum channels via their Choi matrices. We extend recent work that showed that there exist non-dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels. We use our characterization to show that when the dimension is 2 (or 3, under slightly stronger hypotheses), the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual—a fact that has been shown to be false when the dimension is 4 or larger. We also present a formula for the minimum gate fidelity of a channel in terms of a well-studied norm on a compression of its Choi matrix. As a consequence, several new ways of bounding and approximating the minimum gate fidelity follow, including a simple semidefinite program to compute it for qubit channels.

arXiv:1102.0948•Journal of Physics A Mathematical and Theoretical•Nov 18, 2011•21 citations

Physics-inspired neural networks as quasi inverse of quantum channels

Sameen Aziz, Muhammad Faryad

Quantum channels are not invertible in general. A quasi-inverse allows for a partial recovery of the input state, but its analytical results are found only in a restricted space of its parameters. This work explores the potential of neural networks to find the quasi-inverse of qubit channels for any values of the channel parameters while keeping the quasi-inverse as a physically realizable quantum operation. We introduce a physics-inspired loss function based on the mean of the square of the modified trace distance (MSMTD). The scaled trace distance is used so that the neural network does not increase the length of the Bloch vector of the quantum states, which ensures that the network behaves as a completely positive and trace-preserving (CPTP) quantum channel. The Kraus operators of the quasi-inverse channel were obtained by performing quantum process tomography on the trained neural network.

arXiv:2506.11716•Jun 13, 2025•0 citations

An exponential query advantage from access to Stinespring dilation unitaries in quantum channel learning

Manaki Arihara, Satoshi Yoshida, Philip Taranto +1 more

We present a learning task for quantum channels that exhibits an exponential separation in query complexity (in terms of the number of qubits) between two access models: i) black-box channel access and ii) access to a Stinespring dilation unitary of the channel along with its inverse. We consider the task of learning how close the unknown channel is to the set of unitary channels. This is characterized by the \textit{optimal average gate fidelity (OAGF)}, defined as the average gate fidelity maximized over all unitary channels. For constant Kraus rank and a dilation environment of dimension $\operatorname{polylog}(d)$, we provide an algorithm that estimates the OAGF to fixed additive accuracy using $\operatorname{polylog}(d)$ queries to the Stinespring dilation unitary and its inverse. In contradistinction, under the black-box access model, this estimation requires at least $Ω(\sqrt{d})$ queries, even for channels of constant Kraus rank and constant additive accuracy. As a corollary, we rule out a universal simulator that uses only $\operatorname{poly}(\log d,q)$ queries to the unknown channel to approximate the output of arbitrary $q$-query algorithms with access to a randomly-sampled Stinespring dilation unitary and its inverse, even to sufficiently small constant error. These results demonstrate that access to the system-environment joint evolution and its inverse can provide an exponential advantage in estimating an intrinsic property of a quantum channel.

quant-pharXiv:2609.40001•Oct 1, 2026•0 citations
We review the concept of the quasi-inverse of qubit channels and of higher dimensional channels. Quasi-inverse is a channel which when concatenaded to the original channel, increases its average fidelity in an optimal way. For qubit channels, we fully characterize the quasi-inverse, while for higher dimensional channels, we prove general theorems and provide bounds for the increased average fidelity. Nevertheless, explicit examples are given when exact quasi-inverses can be found.
In general quantum operations, or quantum channels cannot be inverted by physical operations, i.e., by completely positive trace-preserving maps. An arbitrary state passing through a quantum channel loses its fidelity with the input. Given a quantum channel E\mathcal{E}E, we discuss the concept of its quasi-inverse as a completely positive trace-preserving map Eqi{\mathcal{E}}^{qi}Eqi which when composed with E\mathcal{E}E increases its average input-output fidelity in an optimal way. The channel Eqi{\mathcal{E}}^{qi}Eqi comes as close as possible to the inverse of a quantum channel. We give a complete classification of such maps for qubit channels and provide quite a few illustrative examples.
Abstract We introduce the concept of quasi-inverse of quantum and classical channels, prove general properties of these inverses and determine them for a large class of channels acting in an arbitrary finite dimension. Therefore we extend the previous results of Karimipour et al (2020 Phys. Rev. A 101 032109) to arbitrary dimensional channels and to the classical domain. We demonstrate how application of the proposed scheme can increase on the average the fidelity between a given random pure state and its image transformed by the quantum channel followed by its quasi-inversion.
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.
We provide new results for computing and comparing the quantum gate fidelity of quantum channels via their Choi matrices. We extend recent work that showed that there exist non-dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels. We use our characterization to show that when the dimension is 2 (or 3, under slightly stronger hypotheses), the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual—a fact that has been shown to be false when the dimension is 4 or larger. We also present a formula for the minimum gate fidelity of a channel in terms of a well-studied norm on a compression of its Choi matrix. As a consequence, several new ways of bounding and approximating the minimum gate fidelity follow, including a simple semidefinite program to compute it for qubit channels.
Quantum channels are not invertible in general. A quasi-inverse allows for a partial recovery of the input state, but its analytical results are found only in a restricted space of its parameters. This work explores the potential of neural networks to find the quasi-inverse of qubit channels for any values of the channel parameters while keeping the quasi-inverse as a physically realizable quantum operation. We introduce a physics-inspired loss function based on the mean of the square of the modified trace distance (MSMTD). The scaled trace distance is used so that the neural network does not increase the length of the Bloch vector of the quantum states, which ensures that the network behaves as a completely positive and trace-preserving (CPTP) quantum channel. The Kraus operators of the quasi-inverse channel were obtained by performing quantum process tomography on the trained neural network.
We present a learning task for quantum channels that exhibits an exponential separation in query complexity (in terms of the number of qubits) between two access models: i) black-box channel access and ii) access to a Stinespring dilation unitary of the channel along with its inverse. We consider the task of learning how close the unknown channel is to the set of unitary channels. This is characterized by the \textit{optimal average gate fidelity (OAGF)}, defined as the average gate fidelity maximized over all unitary channels. For constant Kraus rank and a dilation environment of dimension polylog⁡(d)\operatorname{polylog}(d)polylog(d), we provide an algorithm that estimates the OAGF to fixed additive accuracy using polylog⁡(d)\operatorname{polylog}(d)polylog(d) queries to the Stinespring dilation unitary and its inverse. In contradistinction, under the black-box access model, this estimation requires at least Ω(d)Ω(\sqrt{d})Ω(d​) queries, even for channels of constant Kraus rank and constant additive accuracy. As a corollary, we rule out a universal simulator that uses only poly⁡(log⁡d,q)\operatorname{poly}(\log d,q)poly(logd,q) queries to the unknown channel to approximate the output of arbitrary qqq-query algorithms with access to a randomly-sampled Stinespring dilation unitary and its inverse, even to sufficiently small constant error. These results demonstrate that access to the system-environment joint evolution and its inverse can provide an exponential advantage in estimating an intrinsic property of a quantum channel.