Spins

Struqture can be used to represent spin operators, hamiltonians and open systems, such as:

\[ \hat{H} = \sum_{i, j=0}^N \alpha_{i, j} (\sigma^x_i \sigma^x_j + \sigma^y_i \sigma^y_j) + \sum_{i=0}^N \lambda_i \sigma^z_i \]

and \[ \dot{\rho} = \mathcal{L}(\rho) =-i [\hat{H}, \rho] + \sum_{j,k} \Gamma_{j,k} \left( L_{j}\rho L_{k}^{\dagger} - \frac{1}{2} \{ L_k^{\dagger} L_j, \rho \} \right) . \]

All spin objects in struqture are expressed based on products of either Pauli matrices {X, Y, Z} or operators which are better suited to express decoherence {X, iY, Z}.

The Pauli matrices (coherent dynamics):

• I: identity matrix \[ \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \]

• X: \( \sigma^x \) matrix \[ \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \]

• Y: \( \sigma^y \) matrix \[ \begin{pmatrix} 0 & -i\\ i & 0 \end{pmatrix} \]

• Z: \( \sigma^z \) matrix \[ \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \]

The modified Pauli matrices (decoherent dynamics):

• I: identity matrix \[ \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix} \]
• X: \( \sigma^x \) matrix \[ \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \]
• iY: \( \mathrm{i} \sigma^y \) \[ \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \]
• Z: \( \sigma^z \) matrix \[ \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \]

The simplest way that the user can interact with these matrices is by using symbolic representation: "0X1X" represents a \( \sigma^x\ \sigma^x_1 \) term. This is a very scalable approach, as indices not mentioned in this string representation are assumed to be acted on by the identity operator: "7Y25Z" represents a \( \sigma^y_7 \sigma^z_{25} \) term, where all other spins (0 to 6 and 8 to 24) are acted on by \(I\).

However, for more fine-grain control over the operators, we invite the user to look into the PauliProducts and DecoherenceProducts classes, in the Building blocks section. If not, please proceed to the coherent or decoherent dynamics section.

NOTE: There exists an alternative representation, the {+, -, Z} basis, detailed in the alternative basis section.