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

arXiv:2308.15884

Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry

Yeow Meng Chee, Hoang Ta, Van Khu Vu

Published Aug 30, 2023• 0 citations
arXivHTMLPDF

Abstract

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of $ε$ in $\mathrm{poly}(1/ε, \text{input dimension})$ time.

DOI
10.48550/arxiv.2308.15884

Related papers

An Efficient Parameterized Algorithm for Computing Quantum Channel Fidelity via Symmetries Exploitation

Yeow Meng Chee, Hoang Ta, Van Khu Vu

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of$\epsilon $in$\mathrm {poly}(1/\epsilon, \text {input dimension})$time, compared to the$\exp (1/\epsilon, \text {input dimension})$running time required for direct computation.

IEEE Transactions on Information Theory•Jul 16, 2025•0 citations

A Hierarchy of Efficient Bounds on Quantum Capacities Exploiting Symmetry

Omar Fawzi, Ala Shayeghi, Hoang Ta

Optimal rates for achieving an information processing task are often characterized in terms of regularized information measures. In many cases of quantum tasks, we do not know how to compute such quantities. Here, we exploit the symmetries in the recently introduced$\mathrm {D}^{\#}$in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As applications, we give a general procedure to give efficient bounds on the regularized Umegaki channel divergence as well as the classical capacity and two-way assisted quantum capacity of quantum channels. In particular, we obtain slight improvements for the capacity of the amplitude damping channel. We also prove that for fixed input and output dimensions, the regularized sandwiched Rényi divergence between any two quantum channels can be approximated up to an$\epsilon $accuracy in time that is polynomial in$1/\epsilon $.

arXiv:2203.02127•IEEE Transactions on Information Theory•Jun 10, 2022•10 citations

Semidefinite Programming for Quantum Channel Learning

Mikhail Gennadievich Belov, Victor Victorovich Dubov, Vadim Konstantinovich Ivanov +3 more

The problem of reconstructing a quantum channel from a sample of classical data is considered. When the total fidelity can be represented as a ratio of two quadratic forms (e.g., in the case of mapping a mixed state to a pure state, projective operators, unitary learning, and others), Semidefinite Programming (SDP) can be applied to solve the fidelity optimization problem with respect to the Choi matrix. A remarkable feature of SDP is that the optimization is convex, which allows the problem to be efficiently solved by a variety of numerical algorithms. We have tested several commercially available SDP solvers, all of which allowed for the reconstruction of quantum channels of different forms. A notable feature is that the Kraus rank of the obtained quantum channel typically comprises less than a few percent of its maximal possible value. This suggests that a relatively small Kraus rank quantum channel is typically sufficient to describe experimentally observed classical data. The theory was also applied to the problem of reconstructing projective operators from data. Finally, we discuss a classical computational model based on quantum channel transformation, performed and calculated on a classical computer, possibly hardware-optimized.

cs.LGCross-listedarXiv:2601.12502•Physical review. E•Jan 20, 2026•0 citations

On approximate quantum error correction for symmetric noise

Gereon Koßmann, Julius A. Zeiss, Omar Fawzi +1 more

We revisit the extendability-based semi-definite programming hierarchy introduced by Berta et al. [Mathematical Programming, 1 - 49 (2021)], which provides converging outer bounds on the optimal fidelity of approximate quantum error correction (AQEC). As our first contribution, we introduce a measurement-based rounding scheme that extracts inner sequences of certifiably good encoder-decoder pairs from this outer hierarchy. To address the computational complexity of evaluating fixed levels of the hierarchy, we investigate the use of symmetry-based dimension reduction. In particular, we combine noise symmetries - such as those present in multiple copies of the qubit depolarizing channel - with the permutational symmetry arising from the extendability of the optimization variable. This framework is illustrated through basic, but already challenging numerical examples that showcase its practical effectiveness. Our results contribute to narrowing the gap between theoretical developments in quantum information theory and their practical applications in the analysis of small-scale quantum error-correcting codes.

arXiv:2507.12326•Jul 16, 2025•0 citations

Semidefinite programming hierarchies for constrained bilinear optimization

Mario Berta, Francesco Borderi, Omar Fawzi +1 more

Abstract We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form $${\mathrm {Tr}}\big [H(D\otimes E)\big ]$$ Tr [ H ( D ⊗ E ) ] , maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from $$O(d^{k/2})$$ O ( d k / 2 ) to $${\mathrm {poly}}(d,k)$$ poly ( d , k ) , where d denotes local dimension and k the number of copies.

arXiv:1810.12197•Mathematical Programming•Apr 15, 2021•32 citations

Optimising the relative entropy under semidefinite constraints

Gereon Koßmann, René Schwonnek

Abstract Finding the minimal relative entropy of two quantum states under semidefinite constraints is a pivotal problem located at the mathematical core of various applications in quantum information theory. An efficient method for providing provable upper and lower bounds is the central result of this work. Our primordial motivation stems from the essential task of estimating secret key rates for QKD from the measurement statistics of a real device. Further applications include the computation of channel capacities, the estimation of entanglement measures and many more. We build on a recently introduced integral representation of quantum relative entropy by [Frenkel, Quantum 7, 1102 (2023)] and provide reliable bounds as a sequence of semidefinite programs (SDPs). Our approach ensures provable sublinear convergence in the discretization, while also maintaining resource efficiency in terms of SDP matrix dimensions. Additionally, we can provide gap estimates to the optimum at each iteration stage.

arXiv:2404.17016•npj Quantum Information•Jan 23, 2026•6 citations
Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of εεε in poly(1/ε,input dimension)\mathrm{poly}(1/ε, \text{input dimension})poly(1/ε,input dimension) time.
Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of$\epsilon ininin\mathrm {poly}(1/\epsilon, \text {input dimension})time,comparedtothetime, compared to thetime,comparedtothe\exp (1/\epsilon, \text {input dimension})$running time required for direct computation.
Optimal rates for achieving an information processing task are often characterized in terms of regularized information measures. In many cases of quantum tasks, we do not know how to compute such quantities. Here, we exploit the symmetries in the recently introducedD#\mathrm {D}^{\#}D#in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As applications, we give a general procedure to give efficient bounds on the regularized Umegaki channel divergence as well as the classical capacity and two-way assisted quantum capacity of quantum channels. In particular, we obtain slight improvements for the capacity of the amplitude damping channel. We also prove that for fixed input and output dimensions, the regularized sandwiched Rényi divergence between any two quantum channels can be approximated up to an$\epsilon $accuracy in time that is polynomial in$1/\epsilon $.
The problem of reconstructing a quantum channel from a sample of classical data is considered. When the total fidelity can be represented as a ratio of two quadratic forms (e.g., in the case of mapping a mixed state to a pure state, projective operators, unitary learning, and others), Semidefinite Programming (SDP) can be applied to solve the fidelity optimization problem with respect to the Choi matrix. A remarkable feature of SDP is that the optimization is convex, which allows the problem to be efficiently solved by a variety of numerical algorithms. We have tested several commercially available SDP solvers, all of which allowed for the reconstruction of quantum channels of different forms. A notable feature is that the Kraus rank of the obtained quantum channel typically comprises less than a few percent of its maximal possible value. This suggests that a relatively small Kraus rank quantum channel is typically sufficient to describe experimentally observed classical data. The theory was also applied to the problem of reconstructing projective operators from data. Finally, we discuss a classical computational model based on quantum channel transformation, performed and calculated on a classical computer, possibly hardware-optimized.
We revisit the extendability-based semi-definite programming hierarchy introduced by Berta et al. [Mathematical Programming, 1 - 49 (2021)], which provides converging outer bounds on the optimal fidelity of approximate quantum error correction (AQEC). As our first contribution, we introduce a measurement-based rounding scheme that extracts inner sequences of certifiably good encoder-decoder pairs from this outer hierarchy. To address the computational complexity of evaluating fixed levels of the hierarchy, we investigate the use of symmetry-based dimension reduction. In particular, we combine noise symmetries - such as those present in multiple copies of the qubit depolarizing channel - with the permutational symmetry arising from the extendability of the optimization variable. This framework is illustrated through basic, but already challenging numerical examples that showcase its practical effectiveness. Our results contribute to narrowing the gap between theoretical developments in quantum information theory and their practical applications in the analysis of small-scale quantum error-correcting codes.
Abstract We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[H(D⊗E)]{\mathrm {Tr}}\big [H(D\otimes E)\big ]Tr[H(D⊗E)] Tr [ H ( D ⊗ E ) ] , maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(dk/2)O(d^{k/2})O(dk/2) O ( d k / 2 ) to poly(d,k){\mathrm {poly}}(d,k)poly(d,k) poly ( d , k ) , where d denotes local dimension and k the number of copies.
Abstract Finding the minimal relative entropy of two quantum states under semidefinite constraints is a pivotal problem located at the mathematical core of various applications in quantum information theory. An efficient method for providing provable upper and lower bounds is the central result of this work. Our primordial motivation stems from the essential task of estimating secret key rates for QKD from the measurement statistics of a real device. Further applications include the computation of channel capacities, the estimation of entanglement measures and many more. We build on a recently introduced integral representation of quantum relative entropy by [Frenkel, Quantum 7, 1102 (2023)] and provide reliable bounds as a sequence of semidefinite programs (SDPs). Our approach ensures provable sublinear convergence in the discretization, while also maintaining resource efficiency in terms of SDP matrix dimensions. Additionally, we can provide gap estimates to the optimum at each iteration stage.