Method for sending classical data in quantum information processing systems and corresponding system

1. Method for sending first data as quantum information in qubits (|φ ) and classical second data (S i ) over a quantum channel, in particular in quantum information communication systems, which includes applying quantum error correction (QECC) encoding to said qubits (|φ ) obtaining quantum information codewords (|ψ ),

wherein said method includes

applying intentional errors (P i ) having error syndromes (S i ) representing said second classical data to said quantum information codewords (|ψ ) obtaining quantum information codewords with intentional errors (P i ) applied upon (P i |ψ i ), and

transmitting from a transmitting side said quantum information codewords with intentional errors applied upon (P i |ψ i ) over said quantum channel which outputs received codewords (P i |ψ i ; E i P i |ψ i ) at a receiving side,

computing error syndromes (Ŝ i ; R i ) from said received codewords (P i |ψ i ; E i P i |ψ i ),

performing a QECC error correction operation on said received codewords (P i |ψ i ; E i P i |ψ i ) by applying a correction operator ({circumflex over (P)} i † , {circumflex over (P)} i † Ê i † ) obtained at least by said computed syndromes (Ŝ i ; R i ) to obtain corrected codewords (|{circumflex over (ψ)} i ),

outputting said corrected codewords (|{circumflex over (ψ)} i ) and said computed syndromes (Ŝ i ).

2. The method of claim 1 , wherein the method further includes

encoding information data (B k ) with a classical error code correction encoder to obtain encoded error syndromes (S i ) which are applied as said error syndromes to the quantum information codewords (|ψ ),

said computing error syndromes (S i; ; R i ) from said received codewords (E i P i |ψ i ) including

a step of computing channel affected syndromes (R i ) from the received codewords (E i P i |ψ i ) and a step of classical syndrome error correction on said channel affected syndromes (R i ) to obtain classical corrected error syndromes (S i ),

said applying a correction operator ({circumflex over (P)} i † ,{circumflex over (P)} i † Ê i † ) obtained at least by said computed syndromes (S i; ; R i ) to obtain corrected codewords (|{circumflex over (ψ)} i includes

obtaining said correction operator ({circumflex over (P)} i † ,{circumflex over (P)} i † Ê i † ) by

performing an operation of computation of the intentional error ({circumflex over (P)} i ) and of a channel error of the quantum channel to which the computed syndrome is associated (S(Ê i )) on the basis of said channel affected syndromes (R i ) and said classical corrected error syndromes (S i ),

using said computed intentional error ({circumflex over (P)} i ) and channel error (Ê i ) to obtain said correction operator ({circumflex over (P)} i † ,{circumflex over (P)} i † Ê i † ), in particular as inverse of the computed intentional error ({circumflex over (P)} i ) and channel error S(Ê i ).

3. The method of claim 1 , wherein said applying a correction operator (P i |ψ i ; E i P i |ψ i ) obtained at least by said computed syndromes (S i; ; R i ) to obtain corrected codewords (|{circumflex over (ψ)} i includes obtaining the intentional error from said computed syndromes and computing the correction on the basis of said intentional error, in particular as inverse of the intentional error ({circumflex over (P)} i † ).

4. The method of claim 1 , wherein applying intentional errors (P i ) having error syndromes (S i ) representing said second classical data to said quantum information codewords (|ψ ) obtaining quantum information codewords with intentional errors (P i ) applied upon (P i |ψ i ) includes introducing in the quantum information codewords intentional errors (P i ) determined by corresponding syndromes (S i ).

5. The method of claim 1 , wherein said second classical data (S i ) represent communication control data.

6. The method of claim 1 , wherein said second classical data (S i ) represent a synchronization word which is attached to selected codewords.

7. A quantum communication system configured to send first data as quantum information in qubits (|φ ) and classical second data (S i ) over a quantum channel, comprising a quantum information transmission module which includes a quantum error correction (QECC) encoder configured to apply QECC encoding to said qubits (|φ ) obtaining quantum information codewords (|ψ ),

wherein said quantum information transmission module is configured to

apply intentional errors (P i ) having error syndromes (S i ) representing said second classical data to said quantum information codewords (|ψ ) obtaining quantum information codewords with intentional errors (P i ) applied upon (P i |ψ i ), and

transmit from a transmitting side said quantum information codewords with intentional errors applied upon (P i |ψ i ) over said quantum channel which outputs received codewords (P i |ψ i ; E i P i |ψ i ) at a receiver module comprised in said system,

said receiver module being configured to

compute error syndromes (S i; ; R i ) from said received codewords (P i |ψ i ; E i P i |ψ i ),

perform a QECC error correction operation on said received codewords (P i |ψ i ; E i P i |ψ i ) by applying a correction operator ({circumflex over (P)} i † ,{circumflex over (P)} i † Ê i † ) obtained at least by said computed syndromes (S i; ; R i ) to obtain corrected codewords (|{circumflex over (ψ)} i ),

output said corrected codewords (|{circumflex over (ψ)} i ) and said computed syndromes (S i; ).

8. The system of claim 7 , wherein

said transmitter module includes a classical error code correction encoder configured to encode information data (B k ) to obtain encoded error syndromes (S i ) which are applied as said error syndromes to the quantum information codewords (|ψ ),

said receiver module configured to compute error syndromes (S i; ; R i ) from said received codewords (E i P i |ψ i ) including one or more modules configured to

compute channel affected syndromes (R i ) from the received codewords (E i P i |ψ i ), and

perform a classical syndrome error correction on said channel affected syndromes (R i ) to obtain classical corrected error syndromes (S i ),

an application of a correction operator ({circumflex over (P)} i † ,{circumflex over (P)} i † Ê i † ) obtained at least by said computed syndromes (S i; ; R i ) to obtain corrected codewords (|{circumflex over (ψ)} i including

obtaining said correction operator ({circumflex over (P)} i † ,{circumflex over (P)} i † Ê i † ) by

performing an operation of computation of the intentional error ({circumflex over (P)} i ) and of a channel error (Ê i ) of the quantum channel on the basis of said channel affected syndromes (R i ) and said classical corrected error syndromes (S i ),

using said computed intentional error ({circumflex over (P)} i ) and channel error (Ê i ) to obtain said correction operator ({circumflex over (P)} i † ;{circumflex over (P)} i † Ê i † ), in particular as inverse of the computed intentional error ({circumflex over (P)} i ) and channel error ({right arrow over (E)} i ).

9. The system of claim 7 , wherein the receiver module is configured to apply a correction operator ({circumflex over (P)} i † ,{circumflex over (P)} i † Ê i † ) obtained at least by said computed syndromes (S i; ; R i ) to obtain corrected codewords (|{circumflex over (ψ)} i includes obtaining the intentional error from said computed syndromes and computing the correction on the basis of said intentional error, in particular as inverse of the intentional error ({circumflex over (P)} i † ).