Quantum-Mechanical Variant of the Thouless–Anderson–Palmer Equation for Error-Correcting Codes

Abstract

Statistical mechanics of information has been applied to problems in various research topics of information science and technology [1],[2]. Among those research topics, error-correcting code Error-correcting code is one of the most developed subjects. In the research field of error-correcting codes, Nicolas Sourlas showed that the so-called convolutional codes can be constructed by spin glass Spin glass with infinite range p-body interactions and the decoded message should be corresponded to the ground state of the Hamiltonian [3]. Ruján pointed out that the bit error can be suppressed if one uses finite temperature equilibrium states as the decoding result, instead of the ground state [4], and the so-called Bayes-optimal decoding Bayes-optimal!decoding at some specific condition was proved by Nishimori [5] and Nishimori and Wong [6]. Kabashima and Saad succeeded in constructing more practical codes, namely low-density parity check (LDPC) codes by using the infinite range spin glass model with finite connectivities [7]. They used the so-called TAP (Thouless–Anderson–Palmer) Thouless–Anderson–Palmer (TAP)!equation equations to decode the original message for a given parity check.