940 research outputs found
Optimal correction of concatenated fault-tolerant quantum codes
We present a method of concatenated quantum error correction in which
improved classical processing is used with existing quantum codes and
fault-tolerant circuits to more reliably correct errors. Rather than correcting
each level of a concatenated code independently, our method uses information
about the likelihood of errors having occurred at lower levels to maximize the
probability of correctly interpreting error syndromes. Results of simulations
of our method applied to the [[4,1,2]] subsystem code indicate that it can
correct a number of discrete errors up to half of the distance of the
concatenated code, which is optimal.Comment: 7 pages, 2 figures, published versio
Comparing the Overhead of Topological and Concatenated Quantum Error Correction
This work compares the overhead of quantum error correction with concatenated
and topological quantum error-correcting codes. To perform a numerical
analysis, we use the Quantum Resource Estimator Toolbox (QuRE) that we recently
developed. We use QuRE to estimate the number of qubits, quantum gates, and
amount of time needed to factor a 1024-bit number on several candidate quantum
technologies that differ in their clock speed and reliability. We make several
interesting observations. First, topological quantum error correction requires
fewer resources when physical gate error rates are high, white concatenated
codes have smaller overhead for physical gate error rates below approximately
10E-7. Consequently, we show that different error-correcting codes should be
chosen for two of the studied physical quantum technologies - ion traps and
superconducting qubits. Second, we observe that the composition of the
elementary gate types occurring in a typical logical circuit, a fault-tolerant
circuit protected by the surface code, and a fault-tolerant circuit protected
by a concatenated code all differ. This also suggests that choosing the most
appropriate error correction technique depends on the ability of the future
technology to perform specific gates efficiently
Error suppression via complementary gauge choices in Reed-Muller codes
Concatenation of two quantum error correcting codes with complementary sets
of transversal gates can provide a means towards universal fault-tolerant
computation. We first show that it is generally preferable to choose the inner
code with the higher pseudo-threshold in order to achieve lower logical failure
rates. We then explore the threshold properties of a wide range of
concatenation schemes. Notably, we demonstrate that the concatenation of
complementary sets of Reed-Muller codes can increase the code capacity
threshold under depolarizing noise when compared to extensions of previously
proposed concatenation models. We also analyze the properties of logical errors
under circuit level noise, showing that smaller codes perform better for all
sampled physical error rates. Our work provides new insights into the
performance of universal concatenated quantum codes for both code capacity and
circuit level noise.Comment: 11 pages + 4 appendices, 6 figures. In v2, Fig.1 was added to conform
to journal specification
Overhead and noise threshold of fault-tolerant quantum error correction
Fault tolerant quantum error correction (QEC) networks are studied by a
combination of numerical and approximate analytical treatments. The probability
of failure of the recovery operation is calculated for a variety of CSS codes,
including large block codes and concatenated codes. Recent insights into the
syndrome extraction process, which render the whole process more efficient and
more noise-tolerant, are incorporated. The average number of recoveries which
can be completed without failure is thus estimated as a function of various
parameters. The main parameters are the gate (gamma) and memory (epsilon)
failure rates, the physical scale-up of the computer size, and the time t_m
required for measurements and classical processing. The achievable computation
size is given as a surface in parameter space. This indicates the noise
threshold as well as other information. It is found that concatenated codes
based on the [[23,1,7]] Golay code give higher thresholds than those based on
the [[7,1,3]] Hamming code under most conditions. The threshold gate noise
gamma_0 is a function of epsilon/gamma and t_m; example values are
{epsilon/gamma, t_m, gamma_0} = {1, 1, 0.001}, {0.01, 1, 0.003}, {1, 100,
0.0001}, {0.01, 100, 0.002}, assuming zero cost for information transport. This
represents an order of magnitude increase in tolerated memory noise, compared
with previous calculations, which is made possible by recent insights into the
fault-tolerant QEC process.Comment: 21 pages, 12 figures, minor mistakes corrected and layout improved,
ref added; v4: clarification of assumption re logic gate
- …