qml.liealg.CII

CII(op, p=None, q=None, wire=None)

Canonical Cartan decomposition of type CII, given by $\theta: x \mapsto K_{p,q} x K_{p,q}$.

The matrix $K_{p,q}$ is given by

$$K_{p,q}=\text{diag}( \underset{p \text{times}}{\underbrace{1, \dots 1}}, \underset{q \text{times}}{\underbrace{-1, \dots -1}}, \underset{p \text{times}}{\underbrace{1, \dots 1}}, \underset{q \text{times}}{\underbrace{-1, \dots -1}}, ).$$

For $p=q=2^N$ for some integer $N$, we have $K_{p,q}=Z_1$.

Note

Note that we work with Hermitian operators internally, so that the input will be multiplied by $i$ before evaluating the involution.

Parameters

• op (Union[np.ndarray, PauliSentence, Operator]) – Operator on which the involution is evaluated and for which the parity under the involution is returned.

• p (int) – Dimension of first subspace.

• q (int) – Dimension of second subspace.

• wire (int) – The wire on which the Pauli-$Z$ operator acts to implement the involution. Will default to 1 if None.

Returns

Whether or not the input operator (times $i$) is in the eigenspace of the involution $\theta$ with eigenvalue $+1$.

Return type

bool

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