"""
Copyright 2024 Entropica Labs Pte Ltd
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
"""
import numpy as np
from .stab_array import StabArray, reindex_stabarray, stabarray_standard_form
[docs]
def find_logical_operator_set(stabarr: StabArray) -> tuple[StabArray, StabArray]:
"""Given a stabilizer array of a code, find a set of X and Z logical operators.
The Z logical operators shall contain only Z operators while the X logical operators
may contain both X and Z operators. They may not be in their simplified form.
This algorithm follows the notation and algorithm described in:
Nielsen, M. A., & Chuang, I. L. (2011). (p.470-471) Quantum Computation and Quantum
Information: 10th Anniversary Edition. Cambridge University Press,
How it works (ignoring signs):
1. The stabilizer array is first put into standard form. The standard form is of
the form:
(I, A1, A2 | B, 0, C)
(0, 0, 0 | D, I, E)
Note that the stabilizer array has to be reindexed to the standard form.
2. The Z logical operators are then given by:
(0, 0, 0 | A2^T, 0, I)
and the X logical operators are given by:
(0, E^T, I | C^T , 0, 0)
3. Reindex the logical operators to the original qubit order.
Parameters
----------
stabarr : StabArray
Returns
-------
tuple[StabArray, StabArray]
A tuple containing the X and Z logical operator StabArray objects.
"""
stabarray_std_form, r, inv_qbit_map = stabarray_standard_form(stabarr)
n = stabarr.nqubits # Number of qubits
k = n - stabarray_std_form.nstabs # Number of logical qubits
# Find the auxiliary arrays A2, E and C
a2_array = stabarray_std_form.array[:r, n - k : n]
e_array = stabarray_std_form.array[r:, -k - 1 : -1] # -1 to exclude the sign column
c_array = stabarray_std_form.array[:r, -k - 1 : -1] # -1 to exclude the sign column
# Construct the logical operator arrays and cast the data to the correct type
z_logop_arr = np.hstack(
[
np.zeros((k, stabarr.nqubits), dtype=StabArray.DTYPE),
a2_array.transpose(),
np.zeros((k, n - k - r), dtype=StabArray.DTYPE),
np.eye(k, dtype=StabArray.DTYPE),
np.zeros((k, 1), dtype=StabArray.DTYPE), # the sign column
],
dtype=StabArray.DTYPE,
)
x_logop_arr = np.hstack(
[
np.zeros((k, r), dtype=StabArray.DTYPE),
e_array.transpose(),
np.eye(k, dtype=StabArray.DTYPE),
c_array.transpose(),
np.zeros((k, n - r), dtype=StabArray.DTYPE),
np.zeros((k, 1), dtype=StabArray.DTYPE), # the sign column
],
dtype=StabArray.DTYPE,
)
# Cast the logical operator arrays to StabArray objects and reindex them to the
# original
# NOTE: We bypass the validation of the StabArray objects since we know that the
# logical operator arrays are valid
z_log_ops_stabarr = reindex_stabarray(
StabArray(z_logop_arr, validated=True), inv_qbit_map
)
x_log_ops_stabarr = reindex_stabarray(
StabArray(x_logop_arr, validated=True), inv_qbit_map
)
return x_log_ops_stabarr, z_log_ops_stabarr