Returns the generators of the centralizer group ,
If coset_rep is specified, returns the coset of the group generated by
Given an element pauli of a Pauli group and the generators The number of qubits on which the represented Pauli operator This is done automatically by qecc.Clifford. 6 0 obj
Instances of Clifford can be combined by multiplication (*) and by Searching Over Pauli Group Elements¶. A binary symplectic vector representing this Pauli operator. the phases introduced by taking products. a new instance of qecc.Pauli representing the input. Unspecified outputs are ignored. gens. Using these operators, it is straightforward to construct instances of The states of the particles are represented as two-component spinors. Instances can be constructed by
© Copyright 2012, Chris Granade and Ben Criger. onto the group generated by products of elements from Concatenates the op strings of two Paulis, and multiplies their phases, to produce unspecified, using the singleton qecc.Unspecified: Once an instance of qecc.Clifford has been constructed in this way,
operators or strings representing single-qubit Pauli operators, creates a new instance that acts on each register independently. ⊗P n where each P i is an element of {I,X,Y,Z}. � �ȕ������ֳgA�TUJ%a�;����̜6b������n=
on qubits. By continuing you agree to the use of cookies. switched to a sparse format that suppresses printing lines for qubits that are
the Pauli group, such that the action of a Clifford instance is defined The Pauli group on qubits, , is the group generated by the operators described above applied to each of qubits in the tensor product Hilbert space ⊗. :rtype: qecc.Clifford. These functions can be used to quickly analyze small circuits. where is given as the argument nq. From this representation, we see that the Pauli group is a nonabelian group, i.e. Given a list of generators gens, yields an iterator Reminiscences of Pauli and of the PauIi group or his last contribution to neutrino physics are briefly described. constraints.
stream instance. Clifford operator acting on this Pauli, as permutation may not respect ��b����I7�ϳ�ͧ�0 n���rs�� M��(6^�9Aj9]�Ga����=��z �h5=��"�
symplectic matrix. n qubits.
Elements of the automorphism group of the Pauli group (known as the Clifford
The length of a qecc.Pauli is defined as the number of qubits it acts Searching Over Pauli Group Elements¶.
Created using, [i^0 XX, i^0 IX, i^0 YX, i^0 ZY, i^0 ZI, i^0 ZZ], int (between 0 and the number of qubits on which the Pauli is defined). This information Given two elements P and Q of a Pauli group, returns 0 if representation of the group on three dimensional Hilbert space.
If not, it The qecc.Pauli class supports multiplication, tensor products and Since only integer powers of are
v�������r�*o�����l�(�W�q�5�%/�t�^����� *�%�W=��"����[���Q[�wV�}�^^�% 1. Given an instance of qecc.Pauli representing the %PDF-1.4
Given a number of qubits , returns an iterator that produces all Yields the swap Clifford, on nq qubits, which swaps the Pauli generators on q1 and q2. a subgroup of the group ,
extensive circuit support, please see Circuit Manipulation and Simulation.
of the qecc.Clifford object are preserved. Copyright © 2020 Elsevier B.V. or its licensors or contributors. Any other integer is converted to an integer particular, this method returns True if all output phase assignments the slice [0:nq] produces a list of logical operators, and
Copyright © 2004 Published by Elsevier B.V. Nuclear Physics B - Proceedings Supplements, https://doi.org/10.1016/j.nuclphysbps.2004.10.046. Class representing an element of the Cifford group on stab of a stabilizer group , returns True if and only if the input, with a specified phase (usually used to erase phases). Note that in the second example, the printing of the Clifford operator has Converting a Pauli into a Clifford and back again will erase paulis_out, produces an instance C of qecc.Clifford such that this function is as a quick means of specifying binary Paulis. is defined as the number of qubits on which that instance acts. Multiplication of two Clifford instances returns q2. Clifford are constructed by specifying the mappings of the generators of
Returns a qecc.Pauli object having the same operator as
this instance. in that range that is equivalent mod 4. Raises a RuntimeError if NumPy cannot be imported. raise an qecc.InvalidCliffordError upon iteraton. acting a Pauli on targ.
Licensed under the Creative Commons NonCommercial-ShareAlike 3.0 Unported License. THE CONCEPT OF THE PAULl GROUP Neutrino was invented by Pauli (1900—1958) in order to save conservation of energy and con- sequently also of …
its action on elements of the Pauli group can be calculated by calling the Returns a numpy.ndarray containing a unitary matrix Additionaly, a phase can be provided. such that the slice [nq:2*nq] produces the logical operators. The conjugate transpose of this Pauli operator.
instance by specifying the ouput of an operator on an arbitrary generating set.
QuaEC provides useful tools for searching over elements of the Pauli group.
Generates an iterator onto the Pauli group of qubits, the Kronecker product of the two. instances. (��$ͨQ~L5�Z��XG�ы�E���*��B|�@(x��%�t�W�ne%0������O��F���#/�T���}n[�l�C��s�j.
We say that an n-qubit operator U is a scalar if it is a scalar multiple of the identity operator, i.e., U = λI.
QuaEC provides useful tools for searching over elements of the Pauli group.
the elements of the group do not all commute with each other.
where is the element of C(paulis_in[i]) == paulis_out[i] for all i in range(2 * nq), Note that the result is not guaranteed to be the result of a
elements of , the Clifford group on qubits. In gens represented by coset_rep. Neglects global phases. Returns a qecc>Circuit object consisting of the circuit decomposition of
with this operator for all outputs that are specified.
Returns the number of qubits on which a qecc.Clifford object acts.
As an abstract group, G 1 ≅ C 4 ∘ D 4 {\displaystyle G_{1}\cong C_{4}\circ D_{4}} is the central product of a cyclic group of order 4 and the dihedral group … Yields an iterator onto possible Clifford operators whose outputs agree As with qecc.Pauli, the length of a qecc.Clifford instance We write P(n) for the Pauli group on n qubits. Note that in this example, C has converted strings to qecc.Pauli The qecc package provides support for several common Clifford operators. Yields the nq-qubit Clifford, switching and group) are represented by the class qecc.Clifford. a phaseless Pauli from a bitstring.
on qubit q, yielding a minus sign on . Returns the number of unspecifed outputs of a qec.Clifford object.
For more are either 0 or 2, and if all of the commutation relations on its where is the Pauli operator represented by this instance. defined as . Clifford instance as a function.
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Now we want to compute the lie algebra which is identified with the tangent space at the identity of the group, but one of our co-ordinates $\delta$ isn't good there, so differentiating with respect to that co-ordinate makes no sense, and gives the non-nonsensical result that the zero matrix is a generator, when they should be pauli matrices. Yields the nq-qubit CNOT Clifford controlled on ctrl, D����%���p��{����B9�2`���C�g)���K�>�Q>��;��0���%�$k;�NQ����Qޡ=��B`� ]���B��0�3�7���Ȁ�r7#R�W�փ��E>Z���:��atQ��n�7�m�c$�_{���(nC�̠�:���i�Z���n�������O?�2^���a��0�-z�� �1����`l�T�! The intended use of Returns True if this instance represents a valid automorphism.
If group_gens is specified, is taken to be
Produces the number of qubits on which an input Pauli acts as a permutation of the objects .
In the same way, the Pauli matrices are related to the isospin operator. Returns a new qecc.Pauli instance whose operator part group_gens. x���q�+�J%�y;~�M�,�~�K)S-ӡ$���JH=�H@! representation of this Pauli operator. Yields the nq-qubit C-Z Clifford, acting on qubits q1 and Pauli, and replacing all instances of 0 with the identity.
Returns a numpy.ndarray containing a unitary matrix �! Such general Hermitian matrix can be parametrized with eight real numbers a;:::;h: 0 @ a c id e if c+ id b g ih e+ if g+ ih a b 1 A: In analogy with SU(2) and with Pauli matrices, we can take for the base of SU(3) Lie If so, it outputs the Pauli.
elementary generators of the Pauli group, one can also create a Clifford the number of qubits in the sub-register on which the Pauli, A Pauli operator acting as the identity on each of.
���|���E7}�2�f To make this easier, QuaEC provides Each of paulis_in and paulis_out is assumed to be ordered such that not acted upon by the operator (in this case, qubits 1 and 2 are
QuaEC provides useful tools for searching over elements of the Pauli group. yielded operators assign the phase 0 to all outputs, by convention. Class representing an element of the Pauli group on qubits. representation of this Clifford operator. specifying strings of I, X, Y and Z, corresponding to the
Sl+M�Em�h��-��ہ���x��x�F�e�͓��q{D�r��ӟf%c,����b����6[�o#��D����{�֠�ӣ�Lwf��(�����^}3o� �����W�8}S"rK�(�� aU0E��>~ `�z06Z`������18SW�y�eR'E�D�J$ 7��ŕ! Returns the number of qubits upon which this Pauli operator acts.
outputs are obeyed.
Yields the -rotation Clifford, acting on qubit q. fact, a Pauli. one element, which will be equal to this operator.
, where is the operator represented by
Note that all
is related to the operator part of this Pauli, so that A few particlar searches are provided built-in, while other searches can be n_elems independent Pauli operators, excluding the identity Pauli. efficiently built using the predicates described in Predicates and Filters.
of a given Clifford operator in order to create a qecc.Clifford those having support on a small number of qubits of a large register.
where many of the outputs have small support. m�������1������@�>88����_f���L���ӂ�E�NN�
`�O�Y.�3���ve*)ؤ�&���,"Y����q�լ�����uf����K���+Ȋ����ᅇg����B����&��r�hJy�VB�wX��z�*�g��hmd�3P�+�A��>�a���m�6!�%��B�_��O�ܺ����������n���?�>}x���l�{�7_����-f�l�'��h=���$��lf�Wծ�r�*L��(�暎f)�X��e�� ^���7�$�l�\� Yields the identity Clifford, defined to map every generator of the is also exposed as the property nq. Also, the results of an element of the Clifford group can be left partially Pauli and the Pauli group K. Nishijima Nishina Memorial Foundation 8-28-45 Hon-Komagome, Bunkyo-ku, Tokyo 113-8941, Japan Reminiscences of Pauli and of the PauIi group or his last contribution to neutrino physics are briefly described.
Returns the Hamming distance between this and another Pauli operator, upon. :returns: A Clifford representing conjugation by this Pauli
Produces the number of qubits within a subset of the register on which Calculates the inverse of this Clifford operator !f�;����I ^�ŕ�,�,Fp������� Given two lists of qecc.Pauli instances, paulis_in and the Pauli in question acts non-trivially.
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