qecsim demos

Simulating error correction with a planar stabilizer code

This demo shows verbosely how to simulate one error correction run.

For normal use, the code in this demo is encapsulated in the function:
qecsim.app.run_once(code, error_model, decoder, error_probability),
and the simulation of many error correction runs is encapsulated in the function:
qecsim.app.run(code, error_model, decoder, error_probability, max_runs, max_failures).

Notes:

  • Operators can be visualised in binary symplectic form (bsf) or Pauli form, e.g. [1 1 0|0 1 0] = XYI.

  • The binary symplectic product is denoted by \(\odot\) and defined as \(A \odot B \equiv A \Lambda B \bmod 2\) where \(\Lambda = \left[\begin{matrix} 0 & I \\ I & 0 \end{matrix}\right]\).

  • Binary addition is denoted by \(\oplus\) and defined as addition modulo 2, or equivalently exclusive-or.

Initialise the models

%run qsu.ipynb  # color-printing functions
import numpy as np
from qecsim import paulitools as pt
from qecsim.models.generic import DepolarizingErrorModel
from qecsim.models.planar import PlanarCode, PlanarMWPMDecoder

# initialise models
my_code = PlanarCode(5, 5)
my_error_model = DepolarizingErrorModel()
my_decoder = PlanarMWPMDecoder()
# print models
print(my_code)
print(my_error_model)
print(my_decoder)

PlanarCode(5, 5)
DepolarizingErrorModel()
PlanarMWPMDecoder()

Generate a random error

# set physical error probability to 10%
error_probability = 0.1
# seed random number generator for repeatability
rng = np.random.default_rng(59)

# error: random error based on error probability
error = my_error_model.generate(my_code, error_probability, rng)
qsu.print_pauli('error:\n{}'.format(my_code.new_pauli(error)))

error:
·─┬─·─┬─·─┬─·─┬─·
  ·   ·   ·   Y
·─┼─·─┼─Z─┼─·─┼─Y
  Y   ·   Z   ·
·─┼─·─┼─·─┼─·─┼─·
  ·   Z   ·   ·
·─┼─·─┼─·─┼─·─┼─·
  X   ·   ·   ·
·─┴─·─┴─·─┴─·─┴─·

Evaluate the syndrome

The syndrome is a binary array indicating the stabilizers with which the error does not commute. It is calculated as \(syndrome = error \odot stabilisers^T\).

# syndrome: stabilizers that do not commute with the error
syndrome = pt.bsp(error, my_code.stabilizers.T)
qsu.print_pauli('syndrome:\n{}'.format(my_code.ascii_art(syndrome)))

syndrome:
──┬───┬───┬───X──
  │   │   │ Z │
──X───X───┼───┼──
Z │ Z │   │   │ Z
──X───X───X───┼──
  │   │   │   │
──┼───X───┼───┼──
Z │ Z │   │   │
──┴───┴───┴───┴──

Find a recovery operation

In this case, the recovery operation is found by a minimum weight perfect matching decoder that finds the recovery operation as follows:

  • The syndrome is resolved to plaquettes using: PlanarCode.syndrome_to_plaquette_indices.

  • For each plaquette, the nearest corresponding off-boundary plaquette is found using: PlanarCode.virtual_plaquette_index.

  • A graph between plaquettes is built with weights given by: PlanarMWPMDecoder.distance.

  • A MWPM algorithm is used to match plaquettes into pairs.

  • A recovery operator is constructed by applying the shortest path between matching plaquette pairs using: PlanarPauli.path.

# recovery: best match recovery operation based on decoder
recovery = my_decoder.decode(my_code, syndrome)
qsu.print_pauli('recovery:\n{}'.format(my_code.new_pauli(recovery)))

recovery:
·─┬─·─┬─·─┬─·─┬─Z
  ·   ·   ·   X
·─┼─·─┼─·─┼─·─┼─X
  Y   Z   ·   ·
·─┼─·─┼─Z─┼─·─┼─·
  ·   Z   ·   ·
·─┼─·─┼─·─┼─·─┼─·
  X   ·   ·   ·
·─┴─·─┴─·─┴─·─┴─·

As a sanity check, we expect \(recovery \oplus error\) to commute with all stabilizers, i.e. \((recovery \oplus error) \odot stabilisers^T = 0\).

# check recovery ^ error commutes with stabilizers (by construction)
print(pt.bsp(recovery ^ error, my_code.stabilizers.T))

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 0 0 0]

Visualise \(recovery \oplus error\)

Just out of curiosity, we can see what \(recovery \oplus error\) looks like. If successful, it should be a product of stabilizer plaquette / vertex operators.

# print recovery ^ error (out of curiosity)
qsu.print_pauli('recovery ^ error:\n{}'.format(my_code.new_pauli(recovery ^ error)))

recovery ^ error:
·─┬─·─┬─·─┬─·─┬─Z
  ·   ·   ·   Z
·─┼─·─┼─Z─┼─·─┼─Z
  ·   Z   Z   ·
·─┼─·─┼─Z─┼─·─┼─·
  ·   ·   ·   ·
·─┼─·─┼─·─┼─·─┼─·
  ·   ·   ·   ·
·─┴─·─┴─·─┴─·─┴─·

Test if the recovery operation is successful

The recovery operation is successful iff \(recovery \oplus error\) commutes with all logical operators, i.e. \((recovery \oplus error) \odot logicals^T = 0.\)

# success iff recovery ^ error commutes with logicals
print(pt.bsp(recovery ^ error, my_code.logicals.T))

[0 0]

Note: The decoder is not guaranteed to find a successful recovery operation. The planar 5 x 5 code has distance \(d = 5\) so we can only guarantee to correct errors up to weight \((d - 1)/2=2\).

Equivalent code in single call

The above demo is equivalent to the following code.

# repeat demo in single call
from qecsim import app
print(app.run_once(my_code, my_error_model, my_decoder, error_probability))

{'error_weight': 3, 'success': True}