Multivariate Multicycle Codes For Complete Single Shot Decoding

Authors:

Feroz Ahmed Mian, Owen Gwilliam, Stefan Krastanov

---

This repository provides a database of X-type and Z-type parity check matrices (HX, HZ) and metacheck matrices for the Multivariate Multicycle (MM) family of new quantum error-correcting codes. All matrices are stored in the standard Matrix Market (.mtx) format, ensuring direct usage with modern tools such as Stim.

The dataset can be directly used for running code distance algorithms and calculating confinement profiles using the dist‑m4ri program.

All code instances in this database were generated explicitly using QuantumClifford.jl. For a comprehensive overview of the Multivariate Multicycle code construction and its subfamilies, please refer to the MM code documentation.

Confinement Profile Computation using dist‑m4ri

A convenient method to compute confinement is using Google Colab. Here is a simple guide:

Install Dependencies and Clone dist‑m4ri Program

!apt-get update
!apt-get install -y libm4ri-dev build-essential git
!git clone https://github.com/QEC-pages/dist-m4ri.git
%cd dist-m4ri/src
!make dist_m4ri

Verify Installation

!ls -l dist_m4ri
!./dist_m4ri --help

Find Installation Path of dist‑m4ri Program

%%bash
find /content -name dist_m4ri 2>/dev/null

Running Confinement Computation

Once installed, navigate to your QEC code directory and run the computation:

%%bash
cd "/content/drive/MyDrive/QEC_CODE"
DIST="/content/dist-m4ri/src/dist_m4ri" # Example path
HZ="MM_96_12_4_HZ.mtx"
echo "Using dist-4mri:"
ls -l "$DIST"
$DIST method=2 finH="$HZ" wmax=6

Output

Using dist-4mri:
-rwxr-xr-x 1 root root 229688 Jan 29 05:56 /content/dist-m4ri/src/dist-m4ri/src/dist_m4ri
# read H <- file 'MM_96_12_4_HZ.mtx'
# recursively searching for w=1 codewords wmax=6 beg=0 end=95
# recursively searching for w=2 codewords wmax=6 beg=0 end=94
# recursively searching for w=3 codewords wmax=6 beg=0 end=93
# recursively searching for w=4 codewords wmax=6 beg=0 end=92
# w=1 min non-zero syndrome weight 8
# w=2 min non-zero syndrome weight 8
# w=3 min non-zero syndrome weight 8
# w=4 min non-zero syndrome weight 8
### Cluster (actual min-weight codeword found): d=4

And

%%bash
cd "/content/drive/MyDrive/mm_codes"
DIST="/content/dist-m4ri/src/dist-m4ri/src/dist_m4ri"
HZ="MM_96_12_4_HZ.mtx"
echo "Using binary:"
ls -l "$DIST"
$DIST method=2 finH="$HZ" wmax=6 debug=0

Output

Using binary:
-rwxr-xr-x 1 root root 229688 Jan 29 05:56 /content/dist-m4ri/src/dist-m4ri/src/dist_m4ri
# confinement: 8,8,8,8,
4 # code distance 

Example I: Confinement Profile of [[96, 12, 4]] multivariate multicycle code from Table I of reference

%%bash
cd "PATH_TO_QEC_CODE_DIR"
DIST="PATH_TO_DIST_M4RI_BINARY"

hx="MM_96_12_4_HX.mtx"
hz="MM_96_12_4_HZ.mtx"

if [ -f "$hx" ] && [ -f "$hz" ]; then
    for wmax in 6; do
        $DIST method=2 finH="$hx" finG="$hz" wmax=$wmax 2>&1 debug=0
    done

    $DIST method=2 finH="$hx" finG="$hz" wmax=6  2>&1 debug=0

else
    echo "Files not found!"
fi

Output

# confinement: 8,8,8,8,
4 # code distance
# confinement: 8,8,8,8,
4 # code distance

Example II: Confinement Profile of [[96, 6, 4]] 4D toric code from Table I of reference

%%bash
cd "PATH_TO_QEC_CODE_DIR"
DIST="PATH_TO_DIST_M4RI_BINARY"

hx="T_96_6_4_HX.mtx"
hz="T_96_6_4_HZ.mtx"

if [ -f "$hx" ] && [ -f "$hz" ]; then
    for wmax in 6; do
        $DIST method=2 finH="$hx" finG="$hz" wmax=$wmax 2>&1 debug=0
    done

    $DIST method=2 finH="$hx" finG="$hz" wmax=6  2>&1 debug=0

else
    echo "Files not found!"
fi

Output

# confinement: 4,4,4,
4 # code distance
# confinement: 4,4,4,
4 # code distance

Example III: Confinement Profile of [[84, 6, 7]] abelian multicycle code from Table I of reference

%%bash
cd "PATH_TO_QEC_CODE_DIR"
DIST="PATH_TO_DIST_M4RI_BINARY"

hx="AM_84_6_7_HX.mtx"
hz="AM_84_6_7_HZ.mtx"

if [ -f "$hx" ] && [ -f "$hz" ]; then
    for wmax in 7; do
        $DIST method=2 finH="$hx" finG="$hz" wmax=$wmax 2>&1 debug=0
    done

    $DIST method=2 finH="$hx" finG="$hz" wmax=7  2>&1 debug=0

else
    echo "Files not found!"
fi

Output

# confinement: 4,6,6,6,4,4,4
7 # code distance
# confinement: 4,6,6,6,4,4,4
7 # code distance

Citation

In any publication or work that utilizes this database, please cite the following.

@misc{mian2026multivariatemulticyclecodescomplete,
      title={Multivariate Multicycle Codes for Complete Single-Shot Decoding}, 
      author={Feroz Ahmed Mian and Owen Gwilliam and Stefan Krastanov},
      year={2026},
      eprint={2601.18879},
      archivePrefix={arXiv},
      primaryClass={quant-ph},
      url={https://arxiv.org/abs/2601.18879}, 
}