Belief propagation algorithm for computing correlation functions in finite-temperature quantum many-body systems on loopy graphs

We address the problem of decoding sparse quantum error correction codes. For Pauli channels, this task can be accomplished by a version of the belief propagation algorithm used for decoding sparse classical codes. Quantum codes pose two new challenges however. Firstly, their Tanner graph unavoidably contain small loops which typically undermines the performance of belief propagation. Secondly, sparse quantum codes are by definition highly degenerate. The standard belief propagation algorithm does not exploit this feature, but rather it is impaired by it. We propose heuristic methods to improve belief propagation decoding, specifically targeted at these two problems. While our results exhibit a clear improvement due to the proposed heuristic methods, they also indicate that the main source of errors in the quantum coding scheme remains in the decoding.

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Accelerating Q-ary Sliding-Window Belief Propagation Algorithm with GPU

Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering - IoT as a Service ◽

10.1007/978-3-030-44751-9_1 ◽

2020 ◽

pp. 3-8 ◽

Cited By ~ 1

Author(s):

Bowei Shan ◽

Sihua Chen ◽

Yong Fang

Keyword(s):

Belief Propagation ◽

Sliding Window ◽

Propagation Algorithm ◽

Belief Propagation Algorithm

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On the iterative decoding of sparse quantum codes

Quantum Information and Computation ◽

10.26421/qic8.10-8 ◽

2008 ◽

Vol 8 (10) ◽

pp. 986-1000

Author(s):

D. Poulin ◽

Y. Chung

Keyword(s):

Iterative Decoding ◽

Belief Propagation ◽

Quantum Codes ◽

Error Correction Codes ◽

Heuristic Methods ◽

Tanner Graph ◽

Coding Scheme ◽

Propagation Algorithm ◽

Belief Propagation Algorithm ◽

Quantum Error

We address the problem of decoding sparse quantum error correction codes. For Pauli channels, this task can be accomplished by a version of the belief propagation algorithm used for decoding sparse classical codes. Quantum codes pose two new challenges however. Firstly, their Tanner graph unavoidably contain small loops which typically undermines the performance of belief propagation. Secondly, sparse quantum codes are by definition highly degenerate. The standard belief propagation algorithm does not exploit this feature, but rather it is impaired by it. We propose heuristic methods to improve belief propagation decoding, specifically targeted at these two problems. While our results exhibit a clear improvement due to the proposed heuristic methods, they also indicate that the main source of errors in the quantum coding scheme remains in the decoding.

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Coarse-Graining, Scaling, and Hierarchies

Nonextensive Entropy ◽

10.1093/oso/9780195159769.003.0026 ◽

2004 ◽

Author(s):

Juan Pérez-Mercade

Keyword(s):

Differential Equations ◽

Fixed Points ◽

Correlation Functions ◽

Nonlinear Partial Differential Equations ◽

Coarse Graining ◽

Power Laws ◽

Effective Parameters ◽

Many Body ◽

Anomalous Dimensions ◽

Many Body Systems

We present a scenario that is useful for describing hierarchies within classes of many-component systems. Although this scenario may be quite general, it will be illustrated in the case of many-body systems whose space-time evolution can be described by a class of stochastic parabolic nonlinear partial differential equations. The stochastic component we will consider is in the form of additive noise, but other forms of noise such as multiplicative noise may also be incorporated. It will turn out that hierarchical behavior is only one of a class of asymptotic behaviors that can emerge when an out-of-equilibrium system is coarse grained. This phenomenology can be analyzed and described using the renormalization group (RG) [6, 15]. It corresponds to the existence of complex fixed points for the parameters characterizing the system. As is well known (see, for example, Hochberg and Perez-Mercader [8] and Onuki [12] and the references cited there), parameters such as viscosities, noise couplings, and masses evolve with scale. In other words, their values depend on the scale of resolution at which the system is observed (examined). These scaledependent parameters are called effective parameters. The evolutionary changes due to coarse graining or, equivalently, changes in system size, are analyzed using the RG and translate into differential equations for the probability distribution function [8] of the many-body system, or the n-point correlation functions and the effective parameters. Under certain conditions and for systems away from equilibrium, some of the fixed points of the equations describing the scale dependence of the effective parameters can be complex; this translates into complex anomalous dimensions for the stochastic fields and, therefore, the correlation functions of the field develop a complex piece. We will see that basic requirements such as reality of probabilities and maximal correlation lead, in the case of complex fixed points, to hierarchical behavior. This is a first step for the generalization of extensive behavior as described by real power laws to the case of complex exponents and the study of hierarchical behavior.

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