paulitools: pauli and binary symplectic tools¶
qecsim.paulitools¶
This module contains functions for Pauli strings and binary symplectic vectors / matrices.
In qecsim, a Pauli operator on N qubits is represented, without phase, by either of the following:
a string of I, X, Y, Z of length N, such as
'XIZIY', where the nth element operates on the nth qubit.a binary 1d numpy.array of length 2N, such as
np.array([1,0,0,0,1,0,0,1,0,1]), where the nth element operates as X on the nth qubit, the N+nth element operates as Z on the nth qubit, and XZ operates as Y (since phase is ignored).
A group of Pauli operators is specified as a list of string, or a 2d numpy.array, respectively.
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qecsim.paulitools.bsf_to_pauli(bsf)¶ Convert the given binary symplectic form to Pauli operator(s).
(1 0 0 0 1 | 0 0 1 0 1) -> XIZIY
Assumptions:
bsf is a numpy.array (1d or 2d) in binary symplectic form.
- Parameters
bsf (numpy.array (1d or 2d)) – Binary symplectic vector or matrix.
- Returns
Pauli operators.
- Return type
str or list of str
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qecsim.paulitools.bsf_wt(bsf)¶ Return weight of given binary symplectic form.
- Parameters
bsf (numpy.array (1d or 2d)) – Binary symplectic vector or matrix.
- Returns
Weight
- Return type
int
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qecsim.paulitools.bsp(a, b)¶ Return the binary symplectic product of A with B.
The binary symplectic product \(\odot\) is defined as \(A \odot B \equiv A \Lambda B \bmod 2\) where \(\Lambda = \left[\begin{matrix} 0 & I \\ I & 0 \end{matrix}\right]\).
Assumptions:
A and B should be 1d (vector) or 2d (matrix) numpy arrays with elements 0 or 1.
A should have an even number of columns (or elements if 1d).
B should have an even number of rows (or elements if 1d).
A and B should have compatible dimensions for a dot product, as per numpy requirements.
- Parameters
a (numpy.array (1d or 2d)) – LHS binary symplectic vector or matrix.
b (numpy.array (1d or 2d)) – RHS binary symplectic vector or matrix.
- Returns
Binary symplectic product of A with B.
- Return type
int if A and B vectors; numpy.array (1d if A or B vector, 2d if A and B matrices)
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qecsim.paulitools.ibsf(n_qubits, min_weight=0, max_weight=None)¶ Return an iterator of binary symplectic representations of Paulis in ascending weight.
- Parameters
n_qubits (int) – Number of physical qubits.
min_weight (int) – Minimum weight. (default=0)
max_weight (int) – Maximum weight. (default=n_qubits)
- Returns
Iterator of binary symplectic representation of Pauli.
- Return type
iterator of numpy.array (1d)
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qecsim.paulitools.ipauli(n_qubits, min_weight=0, max_weight=None)¶ Return an iterator of Paulis in ascending weight.
Notes:
Each Pauli is a string of I, X, Y, Z such as ‘XIZIY’
- Parameters
n_qubits (int) – Number of qubits.
min_weight (int) – Minimum weight. (default=0)
max_weight (int) – Maximum weight. (default=n_qubits)
- Returns
Iterator of Paulis.
- Return type
iterator of str
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qecsim.paulitools.pack(binary_array)¶ Return packed representation of the given binary array (e.g. binary symplectic form of Pauli).
- Parameters
binary_array (numpy.array (1d)) – Binary array.
- Returns
Packed binary array as (integer value of binary array, length of binary array)
- Return type
(str, int)
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qecsim.paulitools.pauli_to_bsf(pauli)¶ Convert the given Pauli operator(s) to binary symplectic form.
XIZIY -> (1 0 0 0 1 | 0 0 1 0 1)
Assumptions:
pauli is a string of I, X, Y, Z such as ‘XIZIY’ or a list of such strings of the same length.
- Parameters
pauli (str or list of str) – A single or list of Pauli operators.
- Returns
Binary symplectic representation of Pauli.
- Return type
numpy.array (1d or 2d)
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qecsim.paulitools.pauli_wt(pauli)¶ Return weight of given Pauli operator(s).
- Parameters
pauli (str or list of str) – A single or list of Pauli operators.
- Returns
Weight
- Return type
int
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qecsim.paulitools.unpack(packed_binary_array)¶ Return a binary array corresponding to the packed representation.
- Parameters
packed_binary_array ((str, int)) – Packed binary array as (hex string of binary array, length of binary array)
- Returns
Binary array
- Return type
numpy.array (1d)