PauliChannel
From QETLAB
| PauliChannel | |
| Generates a Pauli channel | |
| Other toolboxes required | none |
|---|---|
| Related functions | Pauli |
| Function category | Superoperators |
PauliChannel is a function that generates the Choi matrix of a Pauli channel (either a random Pauli channel on a specified number of qubits, or a specific Pauli channel requested by the user). The output of this function is sparse.
Contents
Syntax
- PHI = PauliChannel(P)
Argument descriptions
- P: Either a positive integer, indicating that the user wants a randomly-generated P-qubit Pauli channel. Otherwise, P should be a probability vector of length $4^q$ for some integer $q \geq 1$, whose entries specify the weights of the Pauli operators in the channel's Kraus decomposition. More specifically, the weight of the $j$-th Pauli operator (in lexicographical order) in the Kraus operator representation of this channel will be $\sqrt{P(j)}$.
Examples
A random 1-qubit Pauli channel
The following code returns the Choi matrix a randomly-generated 1-qubit Pauli channel:
>> Phi = full(PauliChannel(1)) % generate the random Pauli channel Phi = 0.5737 0 0 0.1346 0 0.4263 0.3196 0 0 0.3196 0.4263 0 0.1346 0 0 0.5737 >> celldisp(KrausOperators(Phi)) % look at its Kraus operators ans{1} = 0 0.6107 0.6107 0 ans{2} = 0.5951 0 0 0.5951 ans{3} = 0.4686 0 0 -0.4686 ans{4} = 0 0.2310 -0.2310 0
A specific Pauli channel
The following code generates a Pauli channel with Kraus operators $\left\{\tfrac{1}{\sqrt{15}}I,\tfrac{2}{\sqrt{15}}X,\tfrac{3}{\sqrt{15}}Y,\tfrac{4}{\sqrt{15}}Z\right\}$, where $I,X,Y,Z$ are the usual Pauli operators:
>> Phi = full(PauliChannel([1,2,3,4]/15)) Phi = 0.3333 0 0 -0.2000 0 0.3333 -0.0667 0 0 -0.0667 0.3333 0 -0.2000 0 0 0.3333 >> celldisp(KrausOperators(Phi)) % look at its Kraus operators ans{1} = -0.5164 0.0000 0 0.5164 ans{2} = -0.0000 0.4472 -0.4472 -0.0000 ans{3} = -0.0000 0.3651 0.3651 -0.0000 ans{4} = -0.2582 -0.0000 0 -0.2582
Source code
Click on "expand" to the right to view the MATLAB source code for this function.
%% PAULICHANNEL Generates a Pauli channel% This function has one required argument:% P: either a probability vector or a scalar indicating a number of% qubits%% PHI = PauliChannel(P) is the Choi matrix of a Pauli channel. If P is a% scalar then P is a random P-qubit Pauli channel. If P is a probability% vector then it must have length 4^Q for some integer Q (which is the% number of qubits that PHI acts on), and the weight of the J-th Pauli% operator (in lexicographical order) in PHI's Kraus decomposition is% sqrt(P(J)). The output of this function is sparse.%% URL: http://www.qetlab.com/PauliChannel% requires: ChoiMatrix.m, RandomProbabilities.m, update_odometer.m% author: Nathaniel Johnston (nathaniel@njohnston.ca)% package: QETLAB% last updated: December 16, 2014function Phi = PauliChannel(p)
len_p = numel(p);
if(len_p == 1) % P is a scalar, so generate a random P-qubit Pauli channel
q = p;
len_p = 4^q;p = RandomProbabilities(len_p);
% Otherwise, do some error checking to make sure that P is a probability% vector.else% Let the user enter either a single vector P or a 4-by-4-by-...-by-4% array P.p = reshape(p,len_p,1);
tol = len_p*eps^(3/4);
if(isa(p,'cvx') == 0 && (min(p) < -tol || sum(p) > 1+tol))
error('PauliChannel:InvalidP','P must be a probability vector: its entries must be non-negative and sum to 1.');
end% Make sure that its length is a power of 2.[pow_of_2,double_q] = log2(len_p);
if(abs(pow_of_2 - 1/2) > tol || mod(double_q,2) == 0)
error('PauliChannel:InvalidP','P must contain 4^Q entries for some positive integer Q, since there are 4^Q Pauli operators on Q qubits.');
endq = (double_q-1)/2; % number of qubits
end% Finally, construct the channel.Phi = sparse(len_p,len_p);
ind = zeros(1,q);
b4ones = 4*ones(1,q);
for j = 1:len_p
Phi = Phi + p(j)*ChoiMatrix({Pauli(ind)});
ind = update_odometer(ind,b4ones); % this loops over all base-4 strings of length Q
end