Exact Evaluation of Bit- and Symbol-Error Rates for Arbitrary Two-Dimensional Modulation and Nonuniform Signaling in AWGN Channel

2006 ◽  
Vol 54 (5) ◽  
pp. 956-956
Author(s):  
L. Szczecinski ◽  
S. Aissa ◽  
C. Gonzalez ◽  
M. Bacic
Keyword(s):  
Error Rates ◽  
Two Dimensional ◽  
Awgn Channel ◽  
Exact Evaluation

scholarly journals Exponential suppression of bit or phase errors with cyclic error correction

Nature ◽  
2021 ◽  
Vol 595 (7867) ◽  
pp. 383-387
Author(s):  
◽  
Zijun Chen ◽  
Kevin J. Satzinger ◽  
Juan Atalaya ◽  
Alexander N. Korotkov ◽  
...  
Keyword(s):  
Error Correction ◽  
Error Detection ◽  
Fault Tolerant ◽  
Error Rates ◽  
Quantum Error Correction ◽  
Superconducting Qubits ◽  
Two Dimensional ◽  
Logical Error ◽  
Quantum Error ◽  
Logical Qubit

AbstractRealizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

On the correlation structure of unilateral AR processes on the plane

2000 ◽  
Vol 32 (02) ◽  
pp. 408-425
Author(s):  
F. Champagnat ◽  
J. Idier
Keyword(s):  
Markov Chain ◽  
Maximum Entropy ◽  
Random Fields ◽  
Sampling Scheme ◽  
Two Dimensional ◽  
Chain Conditions ◽  
Finite Lattices ◽  
Exact Evaluation ◽  
Type Property ◽  
Correlation Matching

In, Tory and Pickard show that a simple subclass of unilateral AR processes identifies with Gaussian Pickard random fields on Z 2. First, we extend this result to the whole class of unilateral AR processes, by showing that they all satisfy a Pickard-type property, under which correlation matching and maximum entropy properties are assessed. Then, it is established that the Pickard property provides the ‘missing’ equations that complement the two-dimensional Yule-Walker equations, in the sense that the conjunction defines a one-to-one mapping between the set of AR parameters and a set of correlations. It also implies Markov chain conditions that allow exact evaluation of the likelihood and an exact sampling scheme on finite lattices.

scholarly journals A Study of Ising Formulations for Minimizing Setup Cost in the Two-Dimensional Cutting Stock Problem

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 182
Author(s):  
Hiroshi Arai ◽  
Harumi Haraguchi
Keyword(s):  
Ising Model ◽  
Optimal Solution ◽  
Computer Experiments ◽  
Error Rates ◽  
Cutting Stock ◽  
Two Dimensional ◽  
Setup Cost ◽  
Proposed Model ◽  
Acceptance Rates ◽  
Future Work

We proposed the method that translates the two-dimensional CSP for minimizing the number of cuts to the Ising model. After that, we conducted computer experiments of the proposed model using the benchmark problem. From the above, the following results are obtained. (1) The proposed Ising model adequately represents the target problem. (2) Acceptance rates were as low as 0.2% to 9.8% and from 21.8% to 49.4%. (3) Error rates from optimal solution were as broad as 0% to 25.9%. For future work, we propose the following changes: (1) Improve the Hamiltonian for constraints. (2) Improve the proposed model to adjust more complex two-dimensional CSP and reduce the number of spins when it deals with large materials and components. (3) Conduct experiments using a quantum annealer.

scholarly journals DeLTA 2.0: A deep learning pipeline for quantifying single-cell spatial and temporal dynamics

2021 ◽  
Author(s):  
Owen M. O'Connor ◽  
Razan N. Alnahhas ◽  
Jean-Baptiste Lugagne ◽  
Mary Dunlop
Keyword(s):  
Single Cell ◽  
Temporal Dynamics ◽  
Single Cells ◽  
Time Lapse ◽  
Error Rates ◽  
Two Dimensions ◽  
Age And Growth ◽  
Two Dimensional ◽  
Spatial Effects ◽  
Deep Convolutional Neural Networks

Improvements in microscopy software and hardware have dramatically increased the pace of image acquisition, making analysis a major bottleneck in generating quantitative, single-cell data. Although tools for segmenting and tracking bacteria within time-lapse images exist, most require human input, are specialized to the experimental set up, or lack accuracy. Here, we introduce DeLTA 2.0, a purely Python workflow that can rapidly and accurately analyze single cells on two-dimensional surfaces to quantify gene expression and cell growth. The algorithm uses deep convolutional neural networks to extract single-cell information from time-lapse images, requiring no human input after training. DeLTA 2.0 retains all the functionality of the original version, which was optimized for bacteria growing in the mother machine microfluidic device, but extends results to two-dimensional growth environments. Two-dimensional environments represent an important class of data because they are more straightforward to implement experimentally, they offer the potential for studies using co-cultures of cells, and they can be used to quantify spatial effects and multi-generational phenomena. However, segmentation and tracking are significantly more challenging tasks in two-dimensions due to exponential increases in the number of cells that must be tracked. To showcase this new functionality, we analyze mixed populations of antibiotic resistant and susceptible cells, and also track pole age and growth rate across generations. In addition to the two-dimensional capabilities, we also introduce several major improvements to the code that increase accessibility, including the ability to accept many standard microscopy file formats and arbitrary image sizes as inputs. DeLTA 2.0 is rapid, with run times of less than 10 minutes for complete movies with hundreds of cells, and is highly accurate, with error rates around 1%, making it a powerful tool for analyzing time-lapse microscopy data.

scholarly journals Performance and error analysis of Knill's postselection scheme in a two-dimensional architecture

2014 ◽  
Vol 14 (9&10) ◽  
pp. 807-822
Author(s):  
Ching-Yi Lai ◽  
Gerardo Paz ◽  
Martin Suchara ◽  
Todd A. Brun
Keyword(s):  
Fault Tolerant ◽  
Error Rates ◽  
Error Threshold ◽  
Noise Model ◽  
Concatenated Codes ◽  
Two Dimensional ◽  
Memory Error ◽  
Computation Scheme ◽  
Quantum Architecture ◽  
With Memory

Knill demonstrated a fault-tolerant quantum computation scheme based on concatenated error-detecting codes and postselection with a simulated error threshold of $3\%$ over the depolarizing channel. We show how to use Knill's postselection scheme in a practical two-dimensional quantum architecture that we designed with the goal to optimize the error correction properties, while satisfying important architectural constraints. In our 2D architecture, one logical qubit is embedded in a tile consisting of $5\times 5$ physical qubits. The movement of these qubits is modeled as noisy SWAP gates and the only physical operations that are allowed are local one- and two-qubit gates. We evaluate the practical properties of our design, such as its error threshold, and compare it to the concatenated Bacon-Shor code and the concatenated Steane code. Assuming that all gates have the same error rates, we obtain a threshold of $3.06\times 10^{-4}$ in a local adversarial stochastic noise model, which is the highest known error threshold for concatenated codes in 2D. We also present a Monte Carlo simulation of the 2D architecture with depolarizing noise and we calculate a pseudo-threshold of about $0.1\%$. With memory error rates one-tenth of the worst gate error rates, the threshold for the adversarial noise model, and the pseudo-threshold over depolarizing noise, are $4.06\times 10^{-4}$ and $0.2\%$, respectively. In a hypothetical technology where memory error rates are negligible, these thresholds can be further increased by shrinking the tiles into a $4\times 4$ layout.

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