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Getting started

This page takes a code from constructor inputs to the standard CSS interface. The example is deliberately small so that every matrix can be inspected directly.

Installation

Install the package from GitHub in Julia's package mode:

pkg> add https://github.com/QuantumSavory/QuantumExpanders.jl.git

Then load the package together with Oscar:

using QuantumExpanders, Oscar

Julia 1.12 or later is required.

Construct a lifted code

Let $G=C_2$ and write its nonidentity element as $g$. We use the two-element multisets $A=B=[1_G,g]$ and the length-two repetition code with parity-check matrix $H=(1\;1)$:

G = cyclic_group(2)
g = gens(G)[1]

A = [one(G), g]
B = [one(G), g]
H = [1 1]

code = QuantumTannerViaLeftRightActions(G, A, B, H, H)

The parity-only constructor computes the dual generator matrices internally. The blocklength is

\[n=|G|\,|A|\,|B|=2\cdot2\cdot2=8.\]

code_n(code), code_k(code)
# (8, 2)

Inspect and validate the CSS code

QuantumTannerViaLeftRightActions is an AbstractCSSCode, so it implements the usual QECCore interface:

hx = parity_matrix_x(code)
hz = parity_matrix_z(code)

size(hx), size(hz)
iszero(mod.(hx * hz', 2))

The final expression checks CSS orthogonality, $H_XH_Z^\mathsf{T}=0$. The convenience accessor returns both matrices:

hx, hz = parity_matrix_xz(code)

The code can now be passed directly to functions from QECCore and QuantumClifford.ECC.

Accepted matrix types

The lifted constructors accept ordinary Julia integer matrices and Oscar matrices over $\mathbb{F}_2$. For example, this is equivalent to the previous choice of H:

H = matrix(GF(2), [1 1])
code = QuantumTannerViaLeftRightActions(G, A, B, H, H)

The parity-only constructor also accepts the matrix returned by dual_code:

H = dual_code([1 1])
code = QuantumTannerViaLeftRightActions(G, A, B, H, H)

Inputs are reduced modulo two, and every supplied parity-check/generator pair is checked for duality.

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