  Investigating principles for storage of quantum information at finite
temperature with minimal need for active error correction is an active area of
research. We bear upon this question in two-dimensional holographic conformal
field theories via the quantum null energy condition (QNEC) that we have shown
earlier to implement the restrictions imposed by quantum thermodynamics on such
many-body systems. We study an explicit encoding of a logical qubit into two
similar chirally propagating excitations of finite von-Neumann entropy on a
finite temperature background whose erasure can be implemented by an
appropriate inhomogeneous and instantaneous energy-momentum inflow from an
infinite energy memoryless bath due to which the system transits to a thermal
state. Holographically, these fast erasure processes can be depicted by
generalized AdS-Vaidya geometries described previously in which no assumption
of specific form of bulk matter is needed. We show that the quantum null energy
condition gives analytic results for the minimal finite temperature needed for
the deletion which is larger than the initial background temperature in
consistency with Landauer's principle. In particular, we find a simple
expression for the minimum final temperature needed for the erasure of a large
number of encoding qubits. We also find that if the encoding qubits are
localized over an interval shorter than a specific localization length, then
the fast erasure process is impossible, and furthermore this localization
length is the largest for an optimal amount of encoding qubits determined by
the central charge. We estimate the optimal encoding qubits for realistic
protection against fast erasure. We discuss possible generalizations of our
study for novel constructions of fault-tolerant quantum gates operating at
finite temperature.
