qutip_qip.decompose
Unitary decomposition. (experimental)
Functions
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An input 1-qubit gate is expressed as a product of rotation matrices \(\textrm{R}_i\) and \(\textrm{R}_j\) or as a product of rotation matrices \(\textrm{R}_i\) and \(\textrm{R}_j\) and a Pauli \(\sigma_k\). |
- qutip_qip.decompose.decompose_one_qubit_gate(input_gate, method)[source]
An input 1-qubit gate is expressed as a product of rotation matrices \(\textrm{R}_i\) and \(\textrm{R}_j\) or as a product of rotation matrices \(\textrm{R}_i\) and \(\textrm{R}_j\) and a Pauli \(\sigma_k\).
Here, \(i \neq j\) and \(i, j, k \in {x, y, z}\).
Based on Lemma 4.1 and Lemma 4.3 of https://arxiv.org/abs/quant-ph/9503016v1 respectively.
\[\begin{split}U = \begin{bmatrix} a & b \\ -b^* & a^* \\ \end{bmatrix} = \textrm{R}_i(\alpha) \textrm{R}_j(\theta) \textrm{R}_i(\beta) = \textrm{A} \sigma_k \textrm{B} \sigma_k \textrm{C}\end{split}\]Here,
\(\textrm{A} = \textrm{R}_i(\alpha) \textrm{R}_j\left(\frac{\theta}{2} \right)\)
\(\textrm{B} = \textrm{R}_j \left(\frac{-\theta}{2} \right)\textrm{R}_i \left(\frac{- \left(\alpha + \beta \right)}{2} \right)\)
\(\textrm{C} = \textrm{R}_i \left(\frac{\left(-\alpha + \beta\right)}{2} \right)\)
- Parameters
- input_gate
qutip.Qobj The matrix to be decomposed.
- methodstring
Name of the preferred decomposition method
Method Key
Method
ZYZ
\(\textrm{R}_z(\alpha) \textrm{R}_y(\theta) \textrm{R}_z(\beta)\)
ZXZ
\(\textrm{R}_z(\alpha) \textrm{R}_x(\theta) \textrm{R}_z(\beta)\)
ZYZ_PauliX
\(\textrm{A} \sigma_k \textrm{B} \sigma_k \textrm{C}\) \(\forall k =x, i =z, j=y\)
Note
This function is under construction. As more combinations are added, above table will be updated with their respective keys.
- input_gate
- Returns
- tuple
The gates in the decomposition are returned as a tuple of
Gateobjects.When the input gate is decomposed to product of rotation matrices - tuple will contain 4 elements per each \(1 \times 1\) qubit gate - \(\textrm{R}_i(\alpha)\), \(\textrm{R}_j(\theta)\) , \(\textrm{R}_i(\beta)\), and some global phase gate.
When the input gate is decomposed to product of rotation matrices and Pauli - tuple will contain 6 elements per each \(1 \times 1\) qubit gate - 2 gates forming \(\textrm{A}\), 2 gates forming \(\textrm{B}\), 1 gates forming \(\textrm{C}\), and some global phase gate.