NoiseTransformer

We convert the operators to a matrix by applying the channel to the four basis elements of the 2x2 matrix space representing density operators; this is standard linear algebra

Parameters

operators (list) – The list of operators to transform into a Matrix

Returns

The matrx representation of the operators

Return type

sympy.Matrix

This method creates the matrix P in the f(x) = 1/2(x*P*x)+q*x representation of the objective function :param As: list of symbolic matrices repersenting the channel matrices :type As: list

Returns

The matrix P for the description of the quadaric program

Return type

matrix

Given a quantum state’s density function rho, the effect of the channel on this state is: rho -> sum_{i=1}^n E_i * rho * E_i^dagger

Parameters

• rho (number) – Density function
• operators (list) – List of operators

Returns

The result of applying the list of operators

Return type

number

This method creates the vector q for the f(x) = 1/2(x*P*x)+q*x representation of the objective function :param As: list of symbolic matrices repersenting the quadratic program :type As: list :param C: matrix representing the the constant channel matrix :type C: matrix

Returns

The vector q for the description of the quadaric program

Return type

list

Calculates channel fidelity

Parameters

m (Matrix) – The matrix to flatten

Returns

A row vector repesenting the flattened matrix

Return type

list

Generate symbolic channel matrices.

Generates a list of 4x4 symbolic matrices describing the channel defined from the given operators. The identity matrix is assumed to be the first element in the list:

[(I, ), (A1, B1, ...), (A2, B2, ...), ..., (An, Bn, ...)]

E.g. for a Pauli channel, the matrices are:

[(I,), (X,), (Y,), (Z,)]

For relaxation they are:

[(I, ), (|0><0|, |0><1|), |1><0|, |1><1|)]

We consider this input to symbolically represent a channel in the following manner: define indeterminates $x_0, x_1, ..., x_n$ which are meant to represent probabilities such that $x_i \ge 0$ and $x0 = 1-(x_1 + ... + x_n)$.

Now consider the quantum channel defined via the Kraus operators ${\sqrt(x_0)I, \sqrt(x_1) A_1, \sqrt(x1) B_1, ..., \sqrt(x_m)A_n, \sqrt(x_n) B_n, ...}$ This is the channel C symbolically represented by the operators.

Parameters

transform_channel_operators_list (list) – A list of tuples of matrices which represent Kraus operators.

Returns

A list of 4x4 complex matrices ([D1, D2, ..., Dn], E) such that the matrix $x_1 D_1 + ... + x_n D_n + E$ represents the operation of the channel C on the density operator. we find it easier to work with this representation of C when performing the combinatorial optimization.

Return type

list

Generate matrices for quadratic program.

Parameters

• channel (Matrix) – a 4x4 symbolic matrix
• symbols (list) – the symbols x1, …, xn which may occur in the matrix

Returns

A list of 4x4 complex matrices ([D1, D2, …, Dn], E) such that: channel == x1*D1 + … + xn*Dn + E

Return type

list

Extract the numeric constant matrix.

Parameters

• channel (matrix) – a 4x4 symbolic matrix.
• symbols (list) – The full list [x1, …, xn] of symbols used in the matrix.

Returns

a 4x4 numeric matrix.

Return type

matrix

Additional Information:

Each entry of the 4x4 symbolic input channel matrix is assumed to be a polynomial of the form a1x1 + … + anxn + c. The corresponding entry in the output numeric matrix is c.

Extract the numeric parameter matrix.

Parameters

• channel (matrix) – a 4x4 symbolic matrix.
• symbol (list) – a symbol xi

Returns

a 4x4 numeric matrix.

Return type

matrix

Additional Information:

Each entry of the 4x4 symbolic input channel matrix is assumed to be a polynomial of the form a1x1 + … + anxn + c. The corresponding entry in the output numeric matrix is ai.

Converts an operator representation to noise circuit.

Parameters

operator (operator) – operator representation. Can be a noise circuit or a matrix or a list of matrices.

Returns

The operator, converted to noise circuit representation.

Return type

List

Converts an operator representation to Kraus matrix representation

Parameters

operator (operator) – operator representation. Can be a noise circuit or a matrix or a list of matrices.

Returns

the operator, converted to Kraus representation.

Return type

Kraus

Prepares a list of channel operators.

Parameters

ops_list (List) – The list of operators to prepare

Returns

The channel operator list

Return type

List

Prepares the honesty constraint.

Parameters

• transform_channel_operators_list (list) – A list of tuples of matrices which represent
• operators. (Kraus) –

Solve the quadratic program optimization problem.

This function solved the quadratic program to minimize the objective function f(x) = 1/2(x*P*x)+q*x subject to the additional constraints Gx <= h

Where P, q are given and G,h are computed to ensure that x represents a probability vector and subject to honesty constraints if required :param P: A matrix representing the P component of the objective function :type P: matrix :param q: A vector representing the q component of the objective function :type q: list

Returns

The solution of the quadratic program (represents probabilities)

Return type

list

Additional information:

This method is the only place in the code where we rely on the cvxpy library should we consider another library, only this method needs to change.

Transform by by quantum channels.

This method creates objective function representing the Hilbert-Schmidt norm of the matrix (A-B) obtained as the difference of the input noise channel and the output channel we wish to determine.

This function is represented by a matrix P and a vector q, such that f(x) = 1/2(x*P*x)+q*x where x is the vector we wish to minimize, where x represents probabilities for the noise operators that construct the output channel

Parameters

• channel_matrices (list) – A list of 4x4 symbolic matrices
• const_channel_matrix (matrix) – a 4x4 constant matrix

Returns

a list of the optimal probabilities for the channel matrices, determined by the quadratic program solver

Return type

list

Transform input Kraus operators.

Allows approximating a set of input Kraus operators as in terms of a different set of Kraus matrices.

For example, setting $[X, Y, Z]$ allows approximating by a Pauli channel, and $[(|0 \langle\rangle 0|, |0\langle\rangle 1|), |1\langle\rangle 0|, |1 \langle\rangle 1|)]$ represents the relaxation channel

In the case the input is a list $[A_1, A_2, ..., A_n]$ of transform matrices and $[E_0, E_1, ..., E_m]$ of noise Kraus operators, the output is a list $[p_1, p_2, ..., p_n]$ of probabilities such that:

1. $p_i \ge 0$
2. $p_1 + ... + p_n \le 1$
3. $[\sqrt(p_1) A_1, \sqrt(p_2) A_2, ..., \sqrt(p_n) A_n, \sqrt(1-(p_1 + ... + p_n))I]$ is a list of Kraus operators that define the output channel (which is “close” to the input channel given by $[E_0, ..., E_m]$.)

This channel can be thought of as choosing the operator $A_i$ in probability $p_i$ and applying this operator to the quantum state.

More generally, if the input is a list of tuples (not necessarily of the same size): $[(A_1, B_1, ...), (A_2, B_2, ...), ..., (A_n, B_n, ...)]$ then the output is still a list $[p_1, p_2, ..., p_n]$ and now the output channel is defined by the operators: $[\sqrt(p_1)A1, \sqrt(p_1)B_1, ..., \sqrt(p_n)A_n, \sqrt(p_n)B_n, ..., \sqrt(1-(p_1 + ... + p_n))I]$

Parameters

• noise_kraus_operators (List) – a list of matrices (Kraus operators) for the input channel.
• transform_channel_operators (List) – a list of matrices or tuples of matrices representing Kraus operators that can construct the output channel.

Returns

A list of amplitudes that define the output channel.

Return type

List