Analytical solution of thermal magnetization on memory stabilizer structures

TL;DR

This paper provides an analytical solution for the thermal magnetization of stabilizer-based memory structures, revealing that these systems do not exhibit a finite critical temperature in the thermodynamic limit, contrary to previous predictions.

Contribution

It introduces an analytical approach using controlled-not transformations to derive exact equations for magnetization and susceptibility in stabilizer memory models.

Findings

No finite critical temperature (Tc=0) in the thermodynamic limit.

Finite size systems show apparent finite critical temperatures due to finite-size effects.

Two-body interaction Hamiltonians have higher apparent critical temperatures than many-body Hamiltonians.

Abstract

We return to the question of how the choice of stabilizer generators affects the preservation of information on structures whose degenerate ground state encodes a classical redundancy code. Controlled-not gates are used to transform the stabilizer Hamiltonian into a Hamiltonian consisting of uncoupled single spins and/or pairs of spins. This transformation allows us to obtain an analytical partition function and derive closed form equations for the relative magnetization and susceptibility. These equations are in agreement with the numerical results presented in [arXiv:0907.0394v1] for finite size systems. Analytical solutions show that there is no finite critical temperature, Tc=0, for all of the memory structures in the thermodynamic limit. This is in contrast to the previously predicted finite critical temperatures based on extrapolation. The mismatch is a result of the infinite system being a poor approximation even for astronomically large finite size systems, where spontaneous magnetization still arises below an apparent finite critical temperature. We extend our analysis to the canonical stabilizer Hamiltonian. Interestingly, Hamiltonians with two-body interactions have a higher apparent critical temperature than the many-body Hamiltonian.