Method for decoding colour codes over a quantum erasure channel

A method for correcting a colour code over a quantum erasure channel, said colour code being defined by a bidimensional three-colourable tiling, any two adjacent faces of the tiling having different colours, the edges of the faces of the tiling defining a lattice, each vertex of the lattice being three-valent, each vertex representing a physical qubit, each face being associated with an X-type stabiliser and an Z-type stabiliser where X and Z are respectively the bit-flip Pauli operator and the phase-flip Pauli operator, the X-type stabiliser and an Z-type stabiliser being implemented as circuits comprising quantum gates, wherein:

• (a) a forest F comprised of trees, spanning the set Ω of the erased qubits is determined;
• (b) a syndrome is calculated for every face of the tiling by measuring the X-type stabilizer over the qubits incident of said face to obtain a first syndrome bit and by measuring the Z-type stabiliser over the qubits incident of said face to obtain a second syndrome bit;
• (c) the forest F is split into the said trees, each tree spanning a connected component of F;
• (d) the faces of the tiling that are touched by at least one vertex of a tree of forest F are identified;
• (e) if no face of the tiling is touched by more than one tree of forest F, the decoding method proceeds with an iterative loop, parsing each leaf of each branch of each tree of forest F and when a current leaf is selected, a pendant face f for said current leaf is determined and,
• (f1) if there is another vertex of forest F belonging to face f, the qubit associated with the current leaf is left unchanged;
• (f2) if there is no other vertex of forest F belonging to face f, the qubit associated with the current leaf is corrected by applying a correction operator X(1-zf)/2Z(1-xf)/2 where zf is the first syndrome bit and x is the second syndrome bit calculated for face f.