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Are the Assumptions of Fault-Tolerant Quantum Error Correction Internally Consistent?

Robert Alicki, Daniel A. Lidar, Paolo Zanardi A

Published Jun 24, 2005• 7 citations
OpenAlex

Abstract

We critically examine the internal consistency of a set of minimal assumptions entering the theory of fault-tolerant quantum error correction for Markovian noise. We point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovian approximation. Namely, Markovian dynamics requires either the singular coupling limit (high temperature), or the weak coupling limit (weak system-bath interaction). The former is incompatible with the assumption of a constant and fresh supply of cold ancillas, while the latter is inconsistent with fast gates. We discuss ways to resolve these inconsistencies.

Related papers

Internal consistency of fault-tolerant quantum error correction in light of rigorous derivations of the quantum Markovian limit

Robert Alicki, Daniel A. Lidar, Paolo Zanardi A

We critically examine the internal consistency of a set of minimal assumptions entering the theory of fault-tolerant quantum error correction for Markovian noise. These assumptions are fast gates, a constant supply of fresh and cold ancillas, and a Markovian bath. We point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovian approximation. Namely, Markovian dynamics requires either the singular coupling limit (high temperature), or the weak coupling limit (weak system-bath interaction). The former is incompatible with the assumption of a constant and fresh supply of cold ancillas, while the latter is inconsistent with fast gates. We discuss ways to resolve these inconsistencies. As part of our discussion we derive, in the weak coupling limit, a new master equation for a system subject to periodic driving.

arXiv:quant-ph/0506201•Physical Review A•May 15, 2006•132 citations

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

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

Microscopic quantum dynamics study on the noise threshold of fault-tolerant quantum error correction

Yuan‐Chung Cheng, R. Silbey

Quantum circuits implementing fault-tolerant quantum error correction (QEC) for the three-qubit bit-flip code and five-qubit code are studied. To describe the effect of noise, we apply a model based on a generalized effective Hamiltonian where the system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. This noise model enables us to investigate the effect of noise in quantum circuits under realistic device conditions and avoid strong assumptions such as maximal parallelism and weak storage errors. Noise thresholds of the QEC codes are calculated. In addition, the effects of imprecision in projective measurements, collective bath, fault-tolerant repetition protocols, and level of parallelism in circuit constructions on the threshold values are also studied with emphasis on determining the optimal design for the fault-tolerant QEC circuit. These results provide insights into the fault-tolerant QEC process as well as useful information for designing the optimal fault-tolerant QEC circuit for particular physical implementation of quantum computer.

arXiv:quant-ph/0412168•Physical Review A•Jul 19, 2005•11 citations

Performance Limits of Fault-Tolerant Quantum Error Correction Schemes

Lorenzo Valentini, Diego Forlivesi, Marco Chiani

Quantum error correction (QEC) is essential for realizing scalable quantum computation. However, when evaluating its benefits, most analyses assume idealized components, overlooking the imperfections inherent in realistic fault-tolerant (FT) implementations. In this paper, we investigate the performance of QEC schemes taking into account that quantum gates and measurements are themselves error-prone. We derive bounds for the failure probability of Shor-style FT-QEC schemes using limited structural information, such as the number of flag qubits and quantum gates. Our analysis separates and quantifies two key contributors to the failure rate: decoding errors and residual errors arising from circuit-level faults. The derived bounds highlight fundamental limitations in Shor-style FT-QEC performance and quantify how circuit imperfections degrade error correction capabilities, under the assumption of depolarizing noise.

quant-pharXiv:2605.24501•IEEE Journal on Selected Areas in Communications•May 26, 2026•0 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
We critically examine the internal consistency of a set of minimal assumptions entering the theory of fault-tolerant quantum error correction for Markovian noise. We point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovian approximation. Namely, Markovian dynamics requires either the singular coupling limit (high temperature), or the weak coupling limit (weak system-bath interaction). The former is incompatible with the assumption of a constant and fresh supply of cold ancillas, while the latter is inconsistent with fast gates. We discuss ways to resolve these inconsistencies.
We critically examine the internal consistency of a set of minimal assumptions entering the theory of fault-tolerant quantum error correction for Markovian noise. These assumptions are fast gates, a constant supply of fresh and cold ancillas, and a Markovian bath. We point out that these assumptions may not be mutually consistent in light of rigorous formulations of the Markovian approximation. Namely, Markovian dynamics requires either the singular coupling limit (high temperature), or the weak coupling limit (weak system-bath interaction). The former is incompatible with the assumption of a constant and fresh supply of cold ancillas, while the latter is inconsistent with fast gates. We discuss ways to resolve these inconsistencies. As part of our discussion we derive, in the weak coupling limit, a new master equation for a system subject to periodic driving.
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.
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.
Quantum circuits implementing fault-tolerant quantum error correction (QEC) for the three-qubit bit-flip code and five-qubit code are studied. To describe the effect of noise, we apply a model based on a generalized effective Hamiltonian where the system-environment interactions are taken into account by including stochastic fluctuating terms in the system Hamiltonian. This noise model enables us to investigate the effect of noise in quantum circuits under realistic device conditions and avoid strong assumptions such as maximal parallelism and weak storage errors. Noise thresholds of the QEC codes are calculated. In addition, the effects of imprecision in projective measurements, collective bath, fault-tolerant repetition protocols, and level of parallelism in circuit constructions on the threshold values are also studied with emphasis on determining the optimal design for the fault-tolerant QEC circuit. These results provide insights into the fault-tolerant QEC process as well as useful information for designing the optimal fault-tolerant QEC circuit for particular physical implementation of quantum computer.
Quantum error correction (QEC) is essential for realizing scalable quantum computation. However, when evaluating its benefits, most analyses assume idealized components, overlooking the imperfections inherent in realistic fault-tolerant (FT) implementations. In this paper, we investigate the performance of QEC schemes taking into account that quantum gates and measurements are themselves error-prone. We derive bounds for the failure probability of Shor-style FT-QEC schemes using limited structural information, such as the number of flag qubits and quantum gates. Our analysis separates and quantifies two key contributors to the failure rate: decoding errors and residual errors arising from circuit-level faults. The derived bounds highlight fundamental limitations in Shor-style FT-QEC performance and quantify how circuit imperfections degrade error correction capabilities, under the assumption of depolarizing 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.