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:1111.1417

The Role of Correlated Noise in Quantum Computing

Daan Staudt

Published Nov 6, 2011• 3 citations
arXivHTMLPDF

Abstract

This paper aims to give an overview of the current state of fault-tolerant quantum computing, by surveying a number of results in the field. We show that thresholds can be obtained for a simple noise model as first proved in [AB97, Kit97, KLZ98], by presenting a proof for statistically independent noise, following the presentation of Aliferis, Gottesman and Preskill [AGP06]. We also present a result by Terhal and Burkard [TB05] and later improved upon by Aliferis, Gottesman and Preskill [AGP06] that shows a threshold can still be obtained for local non-Markovian noise, where we allow the noise to be weakly correlated in space and time. We then turn to negative results, presenting work by Ben-Aroya and Ta-Shma [BT11] who showed conditional errors cannot be perfectly corrected. We end our survey by briefly mentioning some more speculative objections, as put forth by Kalai [Kal08, Kal09, Kal11].

DOI
10.48550/arxiv.1111.1417

Related papers

A More Accurate Measurement Model for Fault-Tolerant Quantum Computing

Yingkai Ouyang

Aliferis, Gottesman and Preskill [1] reduce a non-Markovian noise model to a local noise model, under assumptions on the smallness of the norm of the system-bath interaction. They also prove constructively that given a local noise model, it is possible to simulate an ideal quantum circuit with size L and depth D up to any accuracy, using circuit constructed out of noisy gates from the Boykin set with size $L' = O(L (log L)^a)$ and depth $D'=O(D (log D)^b)$, where $a$ and $b$ are constants that depend on the error correction code that we choose and the design of the fault-tolerant architecture, in addition to more assumptions [1]. These two results combined give us a fault-tolerant threshold theorem for non-Markovian noise, provided that the strength of the effective local noise model is smaller than a positive number that depends on the fault-tolerant architecture we choose. However the ideal measurement process may involve a strong system-bath interaction which necessarily gives a local noise model of large strength. We refine the reduction of the non-Markovian noise model to the local noise model such that this need not be the case, provided that system-bath interactions from the non-ideal operations is sufficiently small. We make all assumptions that [1] has already made, in addition to a few more assumptions to obtain our result. We also give two specific instances where the norm of the fault gets suppressed by some paramater other than the norm of the system-bath interaction. These include the large ratio of the norm of the ideal Hamiltonian to the norm of the perturbation, and frequency of oscillation of the perturbation. We hence suggest finding specific phenomenological models of noise that exhibit these properties.

•Jan 30, 2009•0 citations

Comments on "Fault-Tolerant Quantum Computation for Local Non-Markovian Noise"

Robert Alicki

In the recent paper Terhal and Burkard derived a threshold result for fault-tolerant quantum computation under the assumption of the non-Markovian noise and claimed to rebut the objections rised by Alicki and Horodecki's. The purpose of this note is to show that the main condition used by Terhal and Burkard, although looking quite innocently, implies the assumption of the extremally low probability error per single quantum gate - p < 10^{-8} - 10^{-12} - i.e. the square of the expected threshold value for the case of Markovian noise.

arXiv:quant-ph/0402139•Feb 19, 2004•7 citations

Fault-tolerant quantum computation for local non-Markovian noise

Barbara M. Terhal, Guido Burkard

We derive a threshold result for fault-tolerant quantum computation for local non-Markovian noise models. The role of error amplitude in our analysis is played by the product of the elementary gate time ${t}_{0}$ and the spectral width of the interaction Hamiltonian between system and bath. We discuss extensions of our model and the applicability of our analysis.

arXiv:quant-ph/0402104•Physical Review A•Jan 31, 2005•180 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

New methods in quantum error correction and fault-tolerant quantum computing

Christopher Chamberland

Quantum computers have the potential to solve several interesting problems in polynomial \ntime for which no polynomial time classical algorithms have been found. However, \none of the major challenges in building quantum devices is that quantum systems are very \nsensitive to noise arising from undesired interactions with the environment. Noise can lead \nto errors which can corrupt the results of the computation. Quantum error correction is \none way to mitigate the effects of noise arising in quantum devices. \nWith a plethora of quantum error correcting codes that can be used in various settings, \none of the main challenges of quantum error correction is understanding how well various \ncodes perform under more realistic noise models that can be observed in experiments. \nThis thesis proposes a new decoding algorithm which can optimize threshold values of \nerror correcting codes under different noise models. The algorithm can be applied to \nany Markovian noise model. Further, it is shown that for certain noise models, logical \nClifford corrections can further improve a code's threshold value if the code obeys certain \nsymmetries. \n \nSince gates and measurements cannot in general be performed with perfect precision, \nthe operations required to perform quantum error correction can introduce more errors \ninto the system thus negating the benefits of error correction. Fault-tolerant quantum \ncomputing is a way to perform quantum error correction with imperfect operations while \nretaining the ability to suppress errors as long as the noise is below a code's threshold. \nOne of the main challenges in performing fault-tolerant error correction is the high resource \nrequirements that are needed to obtain very low logical noise rates. With the use of \nflag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable \nto arbitrary distance codes. Various code families are shown to satisfy the requirements \nof flag fault-tolerant error correction. We also provide circuits using a constant number of \nqubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer \nqubits than previous fault-tolerant error correction protocols. \n \nIt is often the case that the noise afflicting a quantum device cannot be fully characterized. \nFurther, even with some knowledge of the noise, it can be very challenging to use \nanalytic decoding methods to improve the performance of a fault-tolerant scheme. This \nthesis presents decoding schemes using several state of the art machine learning techniques \nwith a focus on fault-tolerant quantum error correction in regimes that are relevant to near \nterm experiments. It is shown that even in low noise rate regimes and with no knowledge \nof the noise, noise can be further suppressed for small distance codes. Limitations of machine \nlearning decoders as well as the classical resources required to perform active error \ncorrection are discussed. \n \nIn many cases, gate times can be much shorter than typical measurement times of \nquantum states. Further, classical decoding of the syndrome information used in quantum \nerror correction to compute recovery operators can also be much slower than gate times. \nFor these reasons, schemes where error correction can be implemented in a frame (known \nas the Pauli frame) have been developed to avoid active error correction. In this thesis, we \ngeneralize previous Pauli frame schemes and show how Clifford frame error correction can \nbe implemented with minimal overhead. Clifford frame error correction is necessary if the \nlogical component of recovery operators were chosen from the Clifford group, but could \nalso be used in randomized benchmarking schemes.

•Oct 22, 2018•0 citations

Fault-Tolerant Quantum Computation with Long-Range Correlated Noise

Dorit Aharonov, Alexei Kitaev, John P. Preskill

We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1/r(D).

arXiv:quant-ph/0510231•Physical Review Letters•Feb 7, 2006•243 citations
This paper aims to give an overview of the current state of fault-tolerant quantum computing, by surveying a number of results in the field. We show that thresholds can be obtained for a simple noise model as first proved in [AB97, Kit97, KLZ98], by presenting a proof for statistically independent noise, following the presentation of Aliferis, Gottesman and Preskill [AGP06]. We also present a result by Terhal and Burkard [TB05] and later improved upon by Aliferis, Gottesman and Preskill [AGP06] that shows a threshold can still be obtained for local non-Markovian noise, where we allow the noise to be weakly correlated in space and time. We then turn to negative results, presenting work by Ben-Aroya and Ta-Shma [BT11] who showed conditional errors cannot be perfectly corrected. We end our survey by briefly mentioning some more speculative objections, as put forth by Kalai [Kal08, Kal09, Kal11].
Aliferis, Gottesman and Preskill [1] reduce a non-Markovian noise model to a local noise model, under assumptions on the smallness of the norm of the system-bath interaction. They also prove constructively that given a local noise model, it is possible to simulate an ideal quantum circuit with size L and depth D up to any accuracy, using circuit constructed out of noisy gates from the Boykin set with size L′=O(L(logL)a)L' = O(L (log L)^a)L′=O(L(logL)a) and depth D′=O(D(logD)b)D'=O(D (log D)^b)D′=O(D(logD)b), where aaa and bbb are constants that depend on the error correction code that we choose and the design of the fault-tolerant architecture, in addition to more assumptions [1]. These two results combined give us a fault-tolerant threshold theorem for non-Markovian noise, provided that the strength of the effective local noise model is smaller than a positive number that depends on the fault-tolerant architecture we choose. However the ideal measurement process may involve a strong system-bath interaction which necessarily gives a local noise model of large strength. We refine the reduction of the non-Markovian noise model to the local noise model such that this need not be the case, provided that system-bath interactions from the non-ideal operations is sufficiently small. We make all assumptions that [1] has already made, in addition to a few more assumptions to obtain our result. We also give two specific instances where the norm of the fault gets suppressed by some paramater other than the norm of the system-bath interaction. These include the large ratio of the norm of the ideal Hamiltonian to the norm of the perturbation, and frequency of oscillation of the perturbation. We hence suggest finding specific phenomenological models of noise that exhibit these properties.
In the recent paper Terhal and Burkard derived a threshold result for fault-tolerant quantum computation under the assumption of the non-Markovian noise and claimed to rebut the objections rised by Alicki and Horodecki's. The purpose of this note is to show that the main condition used by Terhal and Burkard, although looking quite innocently, implies the assumption of the extremally low probability error per single quantum gate - p < 10^{-8} - 10^{-12} - i.e. the square of the expected threshold value for the case of Markovian noise.
We derive a threshold result for fault-tolerant quantum computation for local non-Markovian noise models. The role of error amplitude in our analysis is played by the product of the elementary gate time t0{t}_{0}t0​ and the spectral width of the interaction Hamiltonian between system and bath. We discuss extensions of our model and the applicability of our analysis.
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.
Quantum computers have the potential to solve several interesting problems in polynomial \ntime for which no polynomial time classical algorithms have been found. However, \none of the major challenges in building quantum devices is that quantum systems are very \nsensitive to noise arising from undesired interactions with the environment. Noise can lead \nto errors which can corrupt the results of the computation. Quantum error correction is \none way to mitigate the effects of noise arising in quantum devices. \nWith a plethora of quantum error correcting codes that can be used in various settings, \none of the main challenges of quantum error correction is understanding how well various \ncodes perform under more realistic noise models that can be observed in experiments. \nThis thesis proposes a new decoding algorithm which can optimize threshold values of \nerror correcting codes under different noise models. The algorithm can be applied to \nany Markovian noise model. Further, it is shown that for certain noise models, logical \nClifford corrections can further improve a code's threshold value if the code obeys certain \nsymmetries. \n \nSince gates and measurements cannot in general be performed with perfect precision, \nthe operations required to perform quantum error correction can introduce more errors \ninto the system thus negating the benefits of error correction. Fault-tolerant quantum \ncomputing is a way to perform quantum error correction with imperfect operations while \nretaining the ability to suppress errors as long as the noise is below a code's threshold. \nOne of the main challenges in performing fault-tolerant error correction is the high resource \nrequirements that are needed to obtain very low logical noise rates. With the use of \nflag qubits, this thesis develops new fault-tolerant error correction protocols that are applicable \nto arbitrary distance codes. Various code families are shown to satisfy the requirements \nof flag fault-tolerant error correction. We also provide circuits using a constant number of \nqubits for these codes. It is shown that the proposed flag fault-tolerant method uses fewer \nqubits than previous fault-tolerant error correction protocols. \n \nIt is often the case that the noise afflicting a quantum device cannot be fully characterized. \nFurther, even with some knowledge of the noise, it can be very challenging to use \nanalytic decoding methods to improve the performance of a fault-tolerant scheme. This \nthesis presents decoding schemes using several state of the art machine learning techniques \nwith a focus on fault-tolerant quantum error correction in regimes that are relevant to near \nterm experiments. It is shown that even in low noise rate regimes and with no knowledge \nof the noise, noise can be further suppressed for small distance codes. Limitations of machine \nlearning decoders as well as the classical resources required to perform active error \ncorrection are discussed. \n \nIn many cases, gate times can be much shorter than typical measurement times of \nquantum states. Further, classical decoding of the syndrome information used in quantum \nerror correction to compute recovery operators can also be much slower than gate times. \nFor these reasons, schemes where error correction can be implemented in a frame (known \nas the Pauli frame) have been developed to avoid active error correction. In this thesis, we \ngeneralize previous Pauli frame schemes and show how Clifford frame error correction can \nbe implemented with minimal overhead. Clifford frame error correction is necessary if the \nlogical component of recovery operators were chosen from the Clifford group, but could \nalso be used in randomized benchmarking schemes.
We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1/r(D).