Semiclassical grounds of the Calderbank-Shor-Steane quantum error correction codes

Abstract

A valid Calderbank-Shor-Steane (CSS) error correction code requires two classical linear codes for the preparation of the initial state (codewords). This code allow to correct for certain errors caused by an unwanted interaction which produces a degraded quantum state. However, this initial seven qubits encoding can be obtained from a maximally entangled Bell state (0000000>+|1111111)/√ through an operation Hint whose explicit expression is derived in the present work. The price the CSS syndrome has to pay due to its classical grounds is that the operator Hint is not unitary. In other words, Hint is not a valid quantum gate i. e. this does not represent a logical operation. Consequently, the final state is not completely robust for the standard cryptography of Quantum Computation. Besides to be a non unitary operator, Hint, is not reversible introducing with this dissipative effects that destroy the coherence in the quantum computer. Additionally, this operator is not invariant under rotations of the protector qubits inducing then preferred directions of the propagation of the logical information. These are indeed the reasons that prompt us for extending the semi classical CSS quantum error correction codes formalism to a pure quantum Hamming codes.