qrisp.QuantumVariable.app_phase_function

QuantumVariable.app_phase_function(phi)

Applies a previously specified phase function to each computational basis state of the QuantumVariable using Gray-Synthesis.

For a given phase function \(\phi(x)\) and a QuantumVariable in state \(\ket{\psi} = \sum_{x \in \text{Labels}} a_x \ket{x}\) this method acts as:

\[U_{\phi} \sum_{x \in \text{Labels}} a_x \ket{x} = \sum_{x \in \text{Labels}} \text{exp}(i\phi(x)) a_x \ket{x}\]

Parameters:

phiPython function

A Python function which turns the labels of the QuantumVariable into floats.

Examples

We create a QuantumFloat and encode the k-th basis state of the Fourier basis. Finally, we will apply an inverse Fourier transformation to measure k in the computational basis.

>>> import numpy as np
>>> from qrisp import QuantumFloat, h, QFT
>>> n = 5
>>> qf = QuantumFloat(n, signed = False)
>>> h(qf)

After this, qf is in the state

\[\ket{\text{qf}} = \frac{1}{\sqrt{2^n}} \sum_{x = 0}^{2^n} \ket{x}\]

We specify phi

>>> k = 4
>>> def phi(x):
>>>     return 2*np.pi*x*k/2**n

And apply phi as a phase function

>>> qf.app_phase_function(phi)

qf is now in the state

\[\ket{\text{qf}} = \frac{1}{\sqrt{2^n}} \sum_{x = 0}^{2^n} \text{exp}\left( \frac{2\pi ikx}{2^n}\right) \ket{x}\]

Finally we apply the inverse Fourier transformation and measure:

>>> QFT(qf, inv = True)
>>> print(qf)
{4: 1.0}