dq.signm

signm(x: QArrayLike) -> QArray

Returns the operator sign function of a hermitian qarray.

The operator sign function \(\mathrm{sign}(A)\) of a hermitian matrix \(A\) with eigendecomposition \(A = U\, \text{diag}(\lambda_1,\dots,\lambda_n)\, U^\dagger\), with \((\lambda_1,\dots,\lambda_n)\in\R^n\) the eigenvalues of \(A\), is defined by $$ \mathrm{sign}(A) = U\,\mathrm{diag}(\mathrm{sign}(\lambda_1),\dots,\mathrm{sign}(\lambda_n))\,U^\dagger, $$ where \(\mathrm{sign}(x)\) is the sign of \(x\in\R\).

Parameters

• x (qarray-like of shape (..., n, n)) –

  Square hermitian matrix.

Returns

(qarray of shape (..., n, n)) Operator sign function of x.

Note

The operator sign is generally dense, and is different from the element-wise sign of the operator.

Examples

>>> dq.signm(dq.sigmax())
QArray: shape=(2, 2), dims=(2,), dtype=complex64, layout=dense
[[0.+0.j 1.+0.j]
 [1.+0.j 0.+0.j]]
>>> dq.position(4)
QArray: shape=(4, 4), dims=(4,), dtype=complex64, layout=dia, ndiags=2
[[    ⋅     0.5  +0.j     ⋅         ⋅    ]
 [0.5  +0.j     ⋅     0.707+0.j     ⋅    ]
 [    ⋅     0.707+0.j     ⋅     0.866+0.j]
 [    ⋅         ⋅     0.866+0.j     ⋅    ]]
>>> dq.signm(dq.position(4))
QArray: shape=(4, 4), dims=(4,), dtype=complex64, layout=dense
[[ 0.   +0.j  0.888+0.j  0.   +0.j -0.46 +0.j]
 [ 0.888+0.j -0.   +0.j  0.46 +0.j -0.   +0.j]
 [ 0.   +0.j  0.46 +0.j -0.   +0.j  0.888+0.j]
 [-0.46 +0.j -0.   +0.j  0.888+0.j -0.   +0.j]]