QEC Codes

We provide a number of quantum error correction codes.

using TensorQEC

Toric Code

The Toric code is a 2D topological code^[Kitaev]. We can define a Toric code instance by ToricCode.

tc = ToricCode(2, 3)

ToricCode(2, 3)

Here is a schematic diagram of 2*3 Toric code:

The Toric code has two types of stabilizers: X stabilizers and Z stabilizers. X stabilizers are plaquettes of the lattice, and Z stabilizers are vertices of the lattice. We can get the stabilizers of the toric code by

st = stabilizers(tc)

10-element Vector{PauliString{12}}:
 XIXIIIXXIIII
 IXIXIIXXIIII
 IIXIXIIIXXII
 IIIXIXIIXXII
 XIIIXIIIIIXX
 ZZIIIIZIIIZI
 ZZIIIIIZIIIZ
 IIZZIIZIZIII
 IIZZIIIZIZII
 IIIIZZIIZIZI

Note the order of pauli strings is following the big-endian convention. For example, the Pauli string XYZ means $X_1Y_2Z_3$.

Surface Code

The surface code is a 2D topological code^[Kitaev]. Similarly to Toric code, we can define a surface code instance by SurfaceCode and get the stabilizers of the surface code by stabilizers.

sc = SurfaceCode(3, 3)
st = stabilizers(sc)

8-element Vector{PauliString{9}}:
 XXIXXIIII
 IIIIXXIXX
 IIXIIXIII
 IIIXIIXII
 IZZIZZIII
 IIIZZIZZI
 ZZIIIIIII
 IIIIIIIZZ

Here is a schematic diagram of 3*3 surface code:

Shor Code

The [[9,1,3]] Shor code^[Shor] functions by concatenating each qubit of a phase-flip with a bit-flip repetition code, allowing it to correct both types of errors at the same time. We can define a Shor code instance by ShorCode.

shor = ShorCode()
st = stabilizers(shor)

8-element Vector{PauliString{9}}:
 XXXXXXIII
 XXXIIIXXX
 ZZIIIIIII
 ZIZIIIIII
 IIIZZIIII
 IIIZIZIII
 IIIIIIZZI
 IIIIIIZIZ

Steane Code

The [[7,1,3]] Steane is constructed using the classical binary [7,4,3] Hamming code^[Steane]. We can define a Steane code instance by SteaneCode.

steane = SteaneCode()
st = stabilizers(steane)

6-element Vector{PauliString{7}}:
 XIXIXIX
 IXXIIXX
 IIIXXXX
 ZIZIZIZ
 IZZIIZZ
 IIIZZZZ

[[8,3,2]] Code

The [[8,3,2]] CSS code is the smallest non-trivial 3D color code^{[Menendez][Code832]}. We can define a CSS [[8,3,2]] code instance by Code832.

code832 = Code832()
st = stabilizers(code832)

5-element Vector{PauliString{8}}:
 XXXXXXXX
 ZZZZZZZZ
 ZIZIZIZI
 ZZZZIIII
 ZZIIZZII

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