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@article{Fowler-2013,
title = {Analytic asymptotic performance of topological
codes},
author = {Fowler, Austin G.},
journal = {Phys. Rev. A},
volume = 87,
pages = 040301,
year = 2013,
doi = {10.1103/PhysRevA.87.040301},
url = {https://link.aps.org/doi/10.1103/PhysRevA.87.040301},
eprint = {arXiv:1208.1334},
annote = {Topological quantum error-correction codes are
extremely practical, typically requiring only a
two-dimensional lattice of qubits with tunable
nearest-neighbor interactions yet tolerating high
physical error rates p . It is computationally
expensive to simulate the performance of such codes
at low p , yet this is a regime we wish to study as
low physical error rates lead to low qubit
overhead. We present a very general method of
analytically estimating the low- p performance of
the most promising class of topological codes. Our
method can handle arbitrary periodic quantum
circuits implementing the error detection associated
with this class of codes, and arbitrary Pauli error
models for each type of quantum gate. Our analytic
expressions take only seconds to obtain, versus
hundreds of hours to perform equivalent low- p
simulations.}
}
@article{Bravyi-Vargo-2013,
title = {Simulation of rare events in quantum error
correction},
author = {Bravyi, Sergey and Vargo, Alexander},
journal = {Phys. Rev. A},
volume = 88,
pages = 062308,
numpages = 14,
year = 2013,
doi = {10.1103/PhysRevA.88.062308},
url = {https://link.aps.org/doi/10.1103/PhysRevA.88.062308},
eprint = {arXiv:1308.6270},
annote = {We consider the problem of calculating the logical
error probability for a stabilizer quantum code
subject to random Pauli errors. To access the regime
of large code distances where logical errors are
extremely unlikely we adopt the splitting method
widely used in Monte Carlo simulations of rare
events and Bennett's acceptance ratio method for
estimating the free energy difference between two
canonical ensembles. To illustrate the power of
these methods in the context of error correction, we
calculate the logical error probability P L for the
two-dimensional surface code on a square lattice
with a pair of holes for all code distances d ≤ 20
and all error rates p below the fault-tolerance
threshold. Our numerical results confirm the
expected exponential decay P L ∼ exp [ − α ( p ) d ]
and provide a simple fitting formula for the decay
rate α ( p ) . Both noiseless and noisy syndrome
readout circuits are considered.}
}
@Article{Pryadko-2020,
author = {Pryadko, L. P.},
title = {On maximum-likelihood decoding with circuit-level
errors},
journal = {Quantum},
year = 2020,
volume = 4,
pages = 304,
doi = {10.22331/q-2020-08-06-304},
eprint = {arXiv:1909.06732},
annote = {Error probability distribution associated with a
given Clifford measurement circuit is described
exactly in terms of the circuit error-equivalence
group, or the circuit subsystem code previously
introduced by Bacon, Flammia, Harrow, and Shi. This
gives a prescription for maximum-likelihood decoding
with a given measurement circuit. Marginal
distributions for subsets of circuit errors are also
analyzed; these generate a family of related
asymmetric LDPC codes of varying degeneracy. More
generally, such a family is associated with any
quantum code. Implications for decoding
highly-degenerate quantum codes are discussed.}
}