qecsim demos

Simulating error correction with a basic 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, NaiveDecoder
from qecsim.models.basic import FiveQubitCode

# initialise models
my_code = FiveQubitCode()
my_error_model = DepolarizingErrorModel()
my_decoder = NaiveDecoder()
# print models
print(my_code)
print(my_error_model)
print(my_decoder)

FiveQubitCode()
DepolarizingErrorModel()
NaiveDecoder(10)

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(8)

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

error: [0 0 0 0 0 0 1 0 0 0] IZIII

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)
print('syndrome: {}'.format(syndrome))

syndrome: [0 1 0 1]

Find a recovery operation

In this case, the recovery operation is found by using a naive decoder that iterates through all possible Pauli operations, in ascending weight, until it finds a recovery operation that gives the same syndrome as the random error, i.e. \(recovery \odot stabilisers^T = syndrome\).

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

recovery: [0 0 0 0 0 0 1 0 0 0] IZIII

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]

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 five qubit code has distance \(d = 3\) so we can only guarantee to correct errors up to weight \((d - 1)/2=1\).

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': 1, 'success': True}