BP+OSD: A decoder for quantum LDPC codes

A Python library implementing belief propagation with ordered statistics post-processing for decoding sparse quantum LDPC codes as described in arXiv:2005.07016. Note, this library has recently been completly rewritten using Python and Cython. The bulk of the code now resides in the LDPC repository. The original C++ version can be found in the cpp_version branch of this repository.

Installation from PyPi (recommended method)

Installation from PyPi requires Python>=3.6. To install via pip, run:

pip install -U bposd

Documentation

This package buids upon the LDPC python package. The documentation for LDPC can be found here.

Attribution

If you use this software in your research, please cite the following research paper:

@article{roffe_decoding_2020,
   title={Decoding across the quantum low-density parity-check code landscape},
   volume={2},
   ISSN={2643-1564},
   url={http://dx.doi.org/10.1103/PhysRevResearch.2.043423},
   DOI={10.1103/physrevresearch.2.043423},
   number={4},
   journal={Physical Review Research},
   publisher={American Physical Society (APS)},
   author={Roffe, Joschka and White, David R. and Burton, Simon and Campbell, Earl},
   year={2020},
   month={Dec}
}

Please also cite the LDPC software package:

@software{Roffe_LDPC_Python_tools_2022,
author = {Roffe, Joschka},
title = {{LDPC: Python tools for low density parity check codes}},
url = {https://pypi.org/project/ldpc/},
year = {2022}
}

Basic usage

Constructing CSS codes

The bposd.css.css_code class can be used to create a CSS code from two classical codes. As an example, we can create a [[7,4,3]] Steane code from the classical Hamming code

from ldpc.codes import hamming_code
from bposd.css import css_code
h=hamming_code(3) #Hamming code parity check matrix
steane_code=css_code(hx=h,hz=h) #create Steane code where both hx and hz are Hamming codes
print("Hx")
print(steane_code.hx)
print("Hz")
print(steane_code.hz)

Hx
[[0 0 0 1 1 1 1]
 [0 1 1 0 0 1 1]
 [1 0 1 0 1 0 1]]
Hz
[[0 0 0 1 1 1 1]
 [0 1 1 0 0 1 1]
 [1 0 1 0 1 0 1]]

The bposd.css.css_code class automatically computes the logical operators of the code.

print("Lx Logical")
print(steane_code.lx)
print("Lz Logical")
print(steane_code.lz)

Lx Logical
[[1 1 1 0 0 0 0]]
Lz Logical
[[1 1 1 0 0 0 0]]

Not all combinations of the hx and hz matrices will produce a valid CSS code. Use the bposd.css.css_code.test function to check whether the code is valid. For example, we can easily check that the Steane code passes all the CSS code tests:

steane_code.test()

<Unnamed CSS code>, (3,4)-[[7,1,nan]]
 -Block dimensions: Pass
 -PCMs commute hz@hx.T==0: Pass
 -PCMs commute hx@hz.T==0: Pass
 -lx \in ker{hz} AND lz \in ker{hx}: Pass
 -lx and lz anticommute: Pass
 -<Unnamed CSS code> is a valid CSS code w/ params (3,4)-[[7,1,nan]]





True

As an example of a code that isn't valid, consider the case when hx and hz are repetition codes:

from ldpc.codes import rep_code

hx=hz=rep_code(7)
qcode=css_code(hx,hz)
qcode.test()

<Unnamed CSS code>, (2,2)-[[7,-5,nan]]
 -Block dimensions incorrect
 -PCMs commute hz@hx.T==0: Fail
 -PCMs commute hx@hz.T==0: Fail
 -lx \in ker{hz} AND lz \in ker{hx}: Pass
 -lx and lz anitcommute: Fail





False

Hypergraph product codes

The hypergraph product can be used to construct a valid CSS code from any pair of classical seed codes. To use the the hypergraph product, call the bposd.hgp.hgp function. Below is an example of how the distance-3 surface code can be constructed by taking the hypergraph product of two distance-3 repetition codes.

from ldpc.codes import rep_code
from bposd.hgp import hgp
h=rep_code(3)
surface_code=hgp(h1=h,h2=h,compute_distance=True) #nb. set compute_distance=False for larger codes
surface_code.test()

<Unnamed CSS code>, (2,4)-[[13,1,3]]
 -Block dimensions: Pass
 -PCMs commute hz@hx.T==0: Pass
 -PCMs commute hx@hz.T==0: Pass
 -lx \in ker{hz} AND lz \in ker{hx}: Pass
 -lx and lz anticommute: Pass
 -<Unnamed CSS code> is a valid CSS code w/ params (2,4)-[[13,1,3]]





True

BP+OSD Decoding

BP+OSD decoding is useful for codes that do not perform well under standard-BP. To use the BP+OSD decoder, we first call the bposd.bposd_decoder class:

import numpy as np
from ldpc import bposd_decoder

bpd=bposd_decoder(
    surface_code.hz,#the parity check matrix
    error_rate=0.05,
    channel_probs=[None], #assign error_rate to each qubit. This will override "error_rate" input variable
    max_iter=surface_code.N, #the maximum number of iterations for BP)
    bp_method="ms",
    ms_scaling_factor=0, #min sum scaling factor. If set to zero the variable scaling factor method is used
    osd_method="osd_cs", #the OSD method. Choose from:  1) "osd_e", "osd_cs", "osd0"
    osd_order=7 #the osd search depth
    )

We can then decode by passing a syndrome to the bposd.bposd_decoder.decode method:

error=np.zeros(surface_code.N).astype(int)
error[[5,12]]=1
syndrome=surface_code.hz@error %2
bpd.decode(syndrome)

print("Error")
print(error)
print("BP+OSD Decoding")
print(bpd.osdw_decoding)
#Decoding is successful if the residual error commutes with the logical operators
residual_error=(bpd.osdw_decoding+error) %2
a=(surface_code.lz@residual_error%2).any()
if a: a="Yes"
else: a="No"
print(f"Logical Error: {a}\n")

Error
[0 0 0 0 0 1 0 0 0 0 0 0 1]
BP+OSD Decoding
[0 0 0 0 0 0 0 0 1 0 0 0 0]
Logical Error: No