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

IEEE Information Theory Workshop (ITW)

Fidelity of a Quantum ARQ Protocol

Alexei Ashikhmin

Published May 25, 2006• 2 citations
Publisher

Abstract

We consider a generalization of a classical ARQ protocol for the case of quantum error correcting codes and the quantum depolarizing channel. We define the fidelity of the ARQ protocol as the probability that the outcome of a measurement of the received quantum state is collinear to the transmitted quantum state, under the condition that the measurement outcome belongs to the code space. Further, we derive tight upper and lower bounds on the exponent of the fidelity of the ARQ protocol. The obtained bounds show a threshold behavior of the fidelity. Namely, in asymptotics, as the code length tends to infinity, the fidelity tends to either 1 or 0 depending on the code rate and the probability of error of the quantum depolarizing channel.

DOI
10.1109/itw.2006.1633778

Related papers

Lower bounds on the quantum capacity and highest error exponent of general memoryless channels

M. Hamada

Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the author's earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.

arXiv:quant-ph/0112103•IEEE Transactions on Information Theory•Sep 1, 2002•24 citations

Exponential lower bound on the highest fidelity achievable by quantum error-correcting codes

Mitsuru Hamada

On a class of quantum channels which includes the depolarizing channel, the highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by $1\ensuremath{-}\mathrm{exp}[\ensuremath{-}\mathrm{nE}(R)+o(n)]$ for some function $E(R).$ The $E(R)$ is positive below some threshold ${R}_{0},$ which implies ${R}_{0}$ is a lower bound on the quantum capacity.

arXiv:quant-ph/0109114•Physical Review A•Apr 15, 2002•19 citations

Fidelity of Finite Length Quantum Codes in Qubit Erasure Channel

Alexei Ashikhmin

We consider the probability of decoding error, equivalently the fidelity, of finite length quantum codes assuming quantum information transmission via qubit erasure channel. First, we obtain an analytical expression on this probability via combinatorial invariants of quantum codes. Next, we derive a generalization of the MacWilliams identities for those invariants and use them to formulate a linear programming optimization problem, which allows us to obtain lower bounds on the number of unrecoverable sets of i erasures and the probability of decoding error. Our example shows that the obtain bounds can be very tight. Next, we derive upper (achievability) bounds on the probability of decoding error for finite length stabilizer and CSS codes. CSS codes have many advantages for practical applications. Our results show that CSS codes of small lengths, like 100 qubits, visibly lose in performance to generic stabilizer codes.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2020•2 citations

A Spectrum-Based Converse for Quantum State Discrimination and Its Applications to Classical-Quantum Channel Coding

Tam{á}s Havas, Hsuan-Yin Lin, Eirik Rosnes

We investigate converse bounds on the average decoding error probability in finite-blocklength classical-quantum channel coding. We first present a lower bound for multiple quantum hypothesis testing in terms of pairwise trace distances and derive a corresponding fidelity bound. We then obtain a spectrum-based converse that depends only on the a priori probabilities and spectra of the states. We show that the converse bound remains tight for the quantum depolarizing channel under suitable conditions. For codes with product-state outputs, this converse takes an explicit form involving products of output-state eigenvalues. We apply it to binary codes over the quantum amplitude damping channel using the input states $\vert+\rangle$ and $\vert-\rangle$. For this setting, we also discuss a normal approximation to the spectrum-based converse in the large blocklength regime. In all numerical examples considered, the spectrum-based converse is tighter than the other converse bounds at low noise levels.

quant-pharXiv:2610.06840•Oct 6, 2026•0 citations

An exponential lower bound on the highest fidelity achievable by quantum error-correcting codes

M. Hamada

On a quantum memoryless channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity of the channel.

IEEE International Symposium on Information Theory (ISIT)•Jun 25, 2003•1 citation

Upper and lower bounds on quantum codes

John P. Preskill, Graeme Smith

This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity. Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible. Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes. Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states. In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible. Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.

•Jan 1, 2006•2 citations
We consider a generalization of a classical ARQ protocol for the case of quantum error correcting codes and the quantum depolarizing channel. We define the fidelity of the ARQ protocol as the probability that the outcome of a measurement of the received quantum state is collinear to the transmitted quantum state, under the condition that the measurement outcome belongs to the code space. Further, we derive tight upper and lower bounds on the exponent of the fidelity of the ARQ protocol. The obtained bounds show a threshold behavior of the fidelity. Namely, in asymptotics, as the code length tends to infinity, the fidelity tends to either 1 or 0 depending on the code rate and the probability of error of the quantum depolarizing channel.
Tradeoffs between the information rate and fidelity of quantum error-correcting codes are discussed. Quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive (CP) linear map, where the dimension of the underlying Hilbert space is a prime number. On such a quantum channel, the highest fidelity of a quantum error-correcting code of length n and rate R is proven to be lower-bounded by 1-exp[-nE(R)+o(n)] for some function E(R). The E(R) is positive below some threshold R/sub 0/, a direct consequence of which is that R/sub 0/ is a lower bound on the quantum capacity. This is an extension of the author's earlier result. While the earlier work states the result for the depolarizing channel and a slight generalization of it (Pauli channels), the result of this work applies to general discrete memoryless channels, including channel models derived from a physical law of time evolution.
On a class of quantum channels which includes the depolarizing channel, the highest fidelity of quantum error-correcting codes of length n and rate R is proven to be lower bounded by $1\ensuremath{-}\mathrm{exp}[\ensuremath{-}\mathrm{nE}(R)+o(n)]$ for some function E(R).E(R).E(R). The E(R)E(R)E(R) is positive below some threshold R0,{R}_{0},R0​, which implies R0{R}_{0}R0​ is a lower bound on the quantum capacity.
We consider the probability of decoding error, equivalently the fidelity, of finite length quantum codes assuming quantum information transmission via qubit erasure channel. First, we obtain an analytical expression on this probability via combinatorial invariants of quantum codes. Next, we derive a generalization of the MacWilliams identities for those invariants and use them to formulate a linear programming optimization problem, which allows us to obtain lower bounds on the number of unrecoverable sets of i erasures and the probability of decoding error. Our example shows that the obtain bounds can be very tight. Next, we derive upper (achievability) bounds on the probability of decoding error for finite length stabilizer and CSS codes. CSS codes have many advantages for practical applications. Our results show that CSS codes of small lengths, like 100 qubits, visibly lose in performance to generic stabilizer codes.
We investigate converse bounds on the average decoding error probability in finite-blocklength classical-quantum channel coding. We first present a lower bound for multiple quantum hypothesis testing in terms of pairwise trace distances and derive a corresponding fidelity bound. We then obtain a spectrum-based converse that depends only on the a priori probabilities and spectra of the states. We show that the converse bound remains tight for the quantum depolarizing channel under suitable conditions. For codes with product-state outputs, this converse takes an explicit form involving products of output-state eigenvalues. We apply it to binary codes over the quantum amplitude damping channel using the input states ∣+⟩\vert+\rangle∣+⟩ and ∣−⟩\vert-\rangle∣−⟩. For this setting, we also discuss a normal approximation to the spectrum-based converse in the large blocklength regime. In all numerical examples considered, the spectrum-based converse is tighter than the other converse bounds at low noise levels.
This thesis provides bounds on the performance of quantum error correcting codes when used for quantum communication and quantum key distribution. The first two chapters provide a bare-bones introduction to classical and quantum error correcting codes, respectively. The next four chapters present achievable rates for quantum codes in various scenarios. The final chapter is dedicated to an upper bound on the quantum channel capacity. Chapter 3 studies coding for adversarial noise using quantum list codes, showing there exist quantum codes with high rates and short lists. These can be used, together with a very short secret key, to communicate with high fidelity at noise levels for which perfect fidelity is, impossible. Chapter 4 explores the performance of a family of degenerate codes when used to communicate over Pauli channels, showing they can be used to communicate over almost any Pauli channel at rates that are impossible for a nondegenerate code and that exceed those of previously known degenerate codes. By studying the scaling of the optimal block length as a function of the channel's parameters, we develop a heuristic for designing even better codes. Chapter 5 describes an equivalence between a family of noisy preprocessing protocols for quantum key distribution and entanglement distillation protocols whose target state belongs to a class of private states called twisted states. In Chapter 6, the codes of Chapter 4 are combined with the protocols of Chapter 5 to provide higher key rates for one-way quantum key distribution than were previously thought possible. Finally, Chapter 7 presents a new upper bound on the quantum channel that is both additive and convex, and which can be interpreted as the of the channel for communication given access to side channels from a class of zero cloning channels. This clone assisted capacity is equal to the unassisted for channels that are degradable, which we use to find new upper bounds on the of a depolarizing channel.