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:quant-ph/0509203

Fault-tolerance threshold for a distance-three quantum code

Ben W. Reichardt

Published Sep 28, 2005• 8 citations
arXivHTMLPDF

Abstract

The quantum error threshold is the highest (model-dependent) noise rate which we can tolerate and still quantum-compute to arbitrary accuracy. Although noise thresholds are frequently estimated for the Steane seven-qubit, distance-three quantum code, there has been no proof that a constant threshold even exists for distance-three codes. We prove the existence of a constant threshold. The proven threshold is well below estimates, based on simulations and analytic models, of the true threshold, but at least it is now known to be positive.

DOI
10.48550/arxiv.quant-ph/0509203

Related papers

Threshold for the distance three Steane quantum code

Ben W. Reichardt

The error threshold is the model-dependent noise rate which we can tolerate and still compute to arbitrary accuracy. Although noise thresholds are frequently estimated for the Steane seven-qubit distance three quantum code, there has been no proof that a constant threshold even exists for distance three codes. We prove the existence of a constant threshold. The proven threshold is well below estimates of the true threshold, but at least it is now known to be positive.

•Sep 28, 2005•4 citations

Quantum accuracy threshold for concatenated distance-3 code

Panos Aliferis, Daniel Gottesman, John P. Preskill

We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold $\varepsilon_0$. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, $\varepsilon_0 \ge 2.73\times 10^{-5}$ for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.

Quantum Information and Computation•Mar 1, 2006•398 citations

Quantum accuracy threshold for concatenated distance-3 codes

Panos Aliferis, Daniel Gottesman, John P. Preskill

We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold epsilon_0. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, epsilon_0 > 2.73 \times 10^{-5} for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.

arXiv:quant-ph/0504218•Apr 28, 2005•84 citations

Fault-Tolerant Quantum Computation With Constant Error Rate

Dorit Aharonov, Michael Ben-Or

This paper proves the threshold result, which asserts that quantum computation can be made robust against errors and inaccuracies, when the error rate, $η$, is smaller than a constant threshold, $η_c$. The result holds for a very general, not necessarily probabilistic noise model, for quantum particles with any number of states, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation. The proceeding version was very succinct, and here we fill all the missing details, and elaborate on many parts of the proof. In particular, we devote a section for a discussion of universality issues and proofs that the sets of gates that we use are universal. Another section is devoted to a rigorous proof that fault tolerance can be achieved in the presence of general non probabilistic noise. The systematic structure of the fault tolerant procedures for polynomial codes is explained in length. The proof that the concatenation scheme works is written in a clearer way. The paper also contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.

arXiv:quant-ph/9906129•Jun 30, 1999•18 citations

Fault-tolerant thresholds for quantum error correction with the surface code

Ashley M. Stephens

The surface code is a promising candidate for fault-tolerant quantum computation, achieving a high threshold error rate with nearest-neighbor gates in two spatial dimensions. Here, through a series of numerical simulations, we investigate how the precise value of the threshold depends on the noise model, measurement circuits, and decoding algorithm. We observe thresholds between 0.502(1)% and 1.140(1)% per gate, values which are generally lower than previous estimates.

arXiv:1311.5003•Physical Review A•Feb 14, 2014•131 citations

Quantum accuracy threshold for distance-5 codes

Omar. Khalid

A quantum computer can only solve classically intractable problems like factoring large integers if it can perform quantum computations scalably. The minimum accuracy required among the components of a quantum computer to perform scalable computation is called the quantum accuracy threshold, epsilon 0. We explore the accuracy threshold for Calderbank-Shor-Steane codes of distance 5. Our accuracy thresholds are based on the threshold theorem proven by Aliferis et al. [3]. Their threshold theorem is applicable to concatenated codes and can be used to derive a rigorous lower bound on the threshold. In this thesis we consider degenerate codes with parameters [[17,1,5]] and [[19,1,5]], and a non-degenerate [[21,1,5]] code. For each code we design a set of logical operations sufficient to perform universal computation. We estimate the thresholds using simulation tools for an independent stochastic error model. Our simulations incorporate randomized sampling techniques to estimate the number of ways our largest logical operation can fail. A lower bound of epsilon0 ≥ 3.624 x 10-5 is derived for the [[17,1,5]] code.

•Jan 1, 2006•0 citations
The quantum error threshold is the highest (model-dependent) noise rate which we can tolerate and still quantum-compute to arbitrary accuracy. Although noise thresholds are frequently estimated for the Steane seven-qubit, distance-three quantum code, there has been no proof that a constant threshold even exists for distance-three codes. We prove the existence of a constant threshold. The proven threshold is well below estimates, based on simulations and analytic models, of the true threshold, but at least it is now known to be positive.
We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold ε0\varepsilon_0ε0​. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, ε0≥2.73×10−5\varepsilon_0 \ge 2.73\times 10^{-5}ε0​≥2.73×10−5 for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.
We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold epsilon_0. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, epsilon_0 > 2.73 \times 10^{-5} for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.
This paper proves the threshold result, which asserts that quantum computation can be made robust against errors and inaccuracies, when the error rate, ηηη, is smaller than a constant threshold, ηcη_cηc​. The result holds for a very general, not necessarily probabilistic noise model, for quantum particles with any number of states, and is also generalized to one dimensional quantum computers with only nearest neighbor interactions. No measurements, or classical operations, are required during the quantum computation. The proceeding version was very succinct, and here we fill all the missing details, and elaborate on many parts of the proof. In particular, we devote a section for a discussion of universality issues and proofs that the sets of gates that we use are universal. Another section is devoted to a rigorous proof that fault tolerance can be achieved in the presence of general non probabilistic noise. The systematic structure of the fault tolerant procedures for polynomial codes is explained in length. The proof that the concatenation scheme works is written in a clearer way. The paper also contains new and significantly simpler proofs for most of the known results which we use. For example, we give a simple proof that it suffices to correct bit and phase flips, we significantly simplify Calderbank and Shor's original proof of the correctness of CSS codes. We also give a simple proof of the fact that two-qubit gates are universal. The paper thus provides a self contained and complete proof for universal fault tolerant quantum computation.
A quantum computer can only solve classically intractable problems like factoring large integers if it can perform quantum computations scalably. The minimum accuracy required among the components of a quantum computer to perform scalable computation is called the quantum accuracy threshold, epsilon 0. We explore the accuracy threshold for Calderbank-Shor-Steane codes of distance 5. Our accuracy thresholds are based on the threshold theorem proven by Aliferis et al. [3]. Their threshold theorem is applicable to concatenated codes and can be used to derive a rigorous lower bound on the threshold. In this thesis we consider degenerate codes with parameters [[17,1,5]] and [[19,1,5]], and a non-degenerate [[21,1,5]] code. For each code we design a set of logical operations sufficient to perform universal computation. We estimate the thresholds using simulation tools for an independent stochastic error model. Our simulations incorporate randomized sampling techniques to estimate the number of ways our largest logical operation can fail. A lower bound of epsilon0 ≥ 3.624 x 10-5 is derived for the [[17,1,5]] code.