Errors and pseudothresholds for incoherent and coherent noise

Gutiérrez, Mauricio; Smith, Conor; Lulushi, Livia; Janardan, Smitha; Brown, Kenneth R.

doi:10.1103/physreva.94.042338

Cited by 56 publications

(63 citation statements)

References 61 publications

(85 reference statements)

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“…This leads to a diamond-distance logical error rate of order  1 greater than would be obtained by replacing the physical error by its Pauli twirl. The same result was also obtained numerically recently [14].…”

Section: Introductionsupporting

confidence: 87%

“…It is now understood that despite the projective nature of QEC stabilizer measurements, coherent physical errors give rise to a logical error that is also coherent to some extent [12,14,15]. Several strategies for mitigating coherent errors have been proposed, including averaging over random gate sequences [17][18][19][20][21][22], and optimization of the decoding algorithm [28].…”

Section: Resultsmentioning

confidence: 99%

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Modeling coherent errors in quantum error correction

Greenbaum

Dutton

2017

Quantum Sci. Technol.

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Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Pauli model. Here we examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ò) under the repetition code. We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n, in the small error limit. We find that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than  --error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failure more quickly than the Pauli model would predict, and this may need to be combated with coherent suppression methods at the physical level or larger codes.

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Section: Introductionsupporting

confidence: 87%

Section: Resultsmentioning

confidence: 99%

Modeling coherent errors in quantum error correction

Greenbaum

Dutton

2017

Quantum Sci. Technol.

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“…Using such methods, small surface codes (up to distance 3) have been simulated under nonClifford noise [4]. In another study, brute-force simulation of the seven-qubit Steane code was performed without concatenation [5]. Simulation of such low distance codes allows comparison of noise at the logical level to the noise on the physical level; however, it is difficult to infer quantities of interest such as thresholds or overheads from such small simulations.…”

mentioning

confidence: 99%

Tensor-Network Simulations of the Surface Code under Realistic Noise

2017

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The surface code is a many-body quantum system, and simulating it in generic conditions is computationally hard. While the surface code is believed to have a high threshold, the numerical simulations used to establish this threshold are based on simplified noise models. We present a tensor-network algorithm for simulating error correction with the surface code under arbitrary local noise. We use this algorithm to study the threshold and the subthreshold behavior of the amplitudedamping and systematic rotation channels. We also compare these results to those obtained by making standard approximations to the noise models.Introduction. -The working principle behind quantum error correction is to "fight entanglement with entanglement," i.e., protect the data against local interaction with the environment by encoding them into delocalized degrees of freedom of a many-body system. Thus, characterizing a fault-tolerant scheme is ultimately a problem of quantum many-body physics.While simulating quantum many-body systems is generically hard, particular systems have additional structure that can be taken advantage of. For example, free-fermion Hamiltonians have algebraic properties that make them exactly solvable. An analogy in stabilizer quantum error correction is Pauli noise, where errors are Pauli operators drawn from some fixed distribution. Because of their algebraic structure, Pauli noise models can be simulated efficiently using the stabilizer formalism [1]. Beyond Pauli noise, noise composed of Clifford gates and projections onto Pauli eigenstates can also be simulated efficiently using the same methods [2].While such efficiently simulable noise models can be useful to benchmark fault-tolerant schemes, they do not represent most models of practical interest. For instance, qubits that are built out of nondegenerate energy eigenstates are often subject to relaxation, a.k.a. amplitude damping. Miscalibrations often result in systematic errors corresponding to small unitary rotations [3]. Given that these processes do not have efficient descriptions within the stabilizer formalism, understanding how a given fault-tolerant scheme will respond to them is a difficult and important problem.The simplest approach to such many-body problems is brute-force simulation, where an arbitrary state in Hilbert space is represented as an exponentially large vector of coefficients. Using such methods, small surface codes (up to distance 3) have been simulated under nonClifford noise [4]. In another study, brute-force simulation of the seven-qubit Steane code was performed without concatenation [5]. Simulation of such low distance codes allows comparison of noise at the logical level to the noise on the physical level; however, it is difficult to infer quantities of interest such as thresholds or overheads from such small simulations. Another approach, akin to the use of tight-binding approximations in solid-state physics, is to approximate these noise processes with efficiently simulable ones [2,6]. However, the accuracy of these ap...

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“…We report the level-1 pseudothreshold of the code under this error model. In the Appendix we present two alternative Monte Carlo error sampling schemes that we employed in the simulations, and we report the level-1 pseudothreshold for two well studied distance-3 quantum error correcting codes: the Steane code [25][26][27] and the 5-qubit code [28,29], for comparison.…”

Section: Arxiv:170201155v3 [Quant-ph] 3 Oct 2017mentioning

confidence: 99%

Fault tolerance with bare ancillary qubits for a [[7,1,3]] code

Gutiérrez

David

et al. 2017

Phys. Rev. A

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We present a [[7, 1, 3]] quantum error-correcting code that is able to achieve fault-tolerant syndrome measurement using one ancillary qubit per stabilizer for an error model of independent single-qubit Pauli errors. All single-qubit Pauli errors on the ancillary qubits propagate to form exclusively correctable errors on the data qubits. The situation changes for error models with two-qubit Pauli errors. We compare the level-1 logical error rates under two noise models: the standard Pauli symmetric depolarizing error model and an anisotropic error model. The anisotropic model is motivated by control errors on two-qubit gates commonly applied to trapped ion qubits. We find that one ancillary qubit per syndrome measurement is sufficient for fault-tolerance for the anisotropic error, but is not sufficient for the standard depolarizing errors. We then show how to achieve fault tolerance for the standard depolarizing errors by adding flag qubits to check for errors on select ancilary qubits. Our results on this [[7, 1, 3]] code demonstrates how physically motivated noise models may simplify fault-tolerent protocols.

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Errors and pseudothresholds for incoherent and coherent noise

Cited by 56 publications

References 61 publications

Modeling coherent errors in quantum error correction

Modeling coherent errors in quantum error correction

Tensor-Network Simulations of the Surface Code under Realistic Noise

Fault tolerance with bare ancillary qubits for a [[7,1,3]] code

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