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

Incompleteness of relational simulations in the blocking paradigm

Refinement is the notion of development between formal specifications For specifications given in a relational formalism downward and upward simulations are the standard method to verify that a refinement holds their usefulness based upon their soundness and joint completeness This is known to be true for total relational specifications and has been claimed to hold for partial relational specifications in both the non-blocking and blocking interpretations In this paper we show that downward and upward simulations in the blocking interpretation where domains are guards are not Jointly complete This contradicts earlier claims in the literature We illustrate this with an example (based on one recently constructed by Reeves and Streader) and then construct a proof to show why Joint completeness fails in general (C) 2010 Elsevier B V All rights reserve

Refinement and verification of concurrent systems specified in Object-Z and CSP

The formal development of large or complex systems can often be facilitated by the use of more than one formal specification language. Such a combination of languages is particularly suited to the specification of concurrent or distributed systems, where both the modelling of processes and state is necessary. This paper presents an approach to refinement and verification of specifications written using a combination of Object-Z and CSP. A common semantic basis for the two languages enables a unified method of refinement to be used, based upon CSP refinement. To enable state-based techniques to be used for the Object-Z components of a specification we develop state-based refinement relations which are sound and complete with respect to CSP refinement. In addition, a verification method for static and dynamic properties is presented. The method allows us to verify properties of the CSP system specification in terms of its component Object-Z classes by using the laws of the CSP operators together with the logic for Object-Z

Rapid Recovery for Systems with Scarce Faults

Our goal is to achieve a high degree of fault tolerance through the control of a safety critical systems. This reduces to solving a game between a malicious environment that injects failures and a controller who tries to establish a correct behavior. We suggest a new control objective for such systems that offers a better balance between complexity and precision: we seek systems that are k-resilient. In order to be k-resilient, a system needs to be able to rapidly recover from a small number, up to k, of local faults infinitely many times, provided that blocks of up to k faults are separated by short recovery periods in which no fault occurs. k-resilience is a simple but powerful abstraction from the precise distribution of local faults, but much more refined than the traditional objective to maximize the number of local faults. We argue why we believe this to be the right level of abstraction for safety critical systems when local faults are few and far between. We show that the computational complexity of constructing optimal control with respect to resilience is low and demonstrate the feasibility through an implementation and experimental results.Comment: In Proceedings GandALF 2012, arXiv:1210.202

Computer Science and Game Theory: A Brief Survey

There has been a remarkable increase in work at the interface of computer science and game theory in the past decade. In this article I survey some of the main themes of work in the area, with a focus on the work in computer science. Given the length constraints, I make no attempt at being comprehensive, especially since other surveys are also available, and a comprehensive survey book will appear shortly.Comment: To appear; Palgrave Dictionary of Economic