Quantum Amplitude Amplification#

amplitude_amplification(args, state_function, oracle_function, kwargs_oracle={}, iter=1)[source]#

This method performs quantum amplitude amplification.

The problem of quantum amplitude amplification is described as follows:

  • Given a unitary operator \(\mathcal{A}\), let \(\ket{\Psi}=\mathcal{A}\ket{0}\).

  • Write \(\ket{\Psi}=\ket{\Psi_1}+\ket{\Psi_0}\) as a superposition of the orthogonal good and bad components of \(\ket{\Psi}\).

  • Enhance the probability \(a=\langle\Psi_1|\Psi_1\rangle\) that a measurement of \(\ket{\Psi}\) yields a good state.

Let \(\theta_a\in [0,\pi/2]\) such that \(\sin^2(\theta_a)=a\). Then the amplitude amplification operator \(\mathcal Q\) acts as

\[\mathcal Q^j\ket{\Psi}=\frac{1}{\sqrt{a}}\sin((2j+1)\theta_a)\ket{\Psi_1}+\frac{1}{\sqrt{1-a}}\cos((2j+1)\theta_a)\ket{\Psi_0}.\]

Therefore, after \(m\) iterations the probability of measuring a good state is \(\sin^2((2m+1)\theta_a)\).

Parameters
argsQuantumVariable or list[QuantumVariable]

The (list of) QuantumVariables which represent the state, the amplitude amplification is performed on.

state_functionfunction

A Python function preparing the state \(\ket{\Psi}\). This function will receive the variables in the list args as arguments in the course of this algorithm.

oracle_functionfunction

A Python function tagging the good state \(\ket{\Psi_1}\). This function will receive the variables in the list args as arguments in the course of this algorithm.

kwargs_oracledict, optional

A dictionary containing keyword arguments for the oracle. The default is {}.

iterint, optional

The amount of amplitude amplification iterations to perform. The default is 1.

Examples

We define a function that prepares the state \(\ket{\Psi}=\cos(\frac{\pi}{16})\ket{0}+\sin(\frac{\pi}{16})\ket{1}\) and an oracle that tags the good state \(\ket{1}\). In this case, we have \(a=\sin^2(\frac{\pi}{16})\approx 0.19509\).

from qrisp import z, ry, QuantumBool, amplitude_amplification
import numpy as np

def state_function(qb):
    ry(np.pi/8,qb)

def oracle_function(qb):   
    z(qb)

qb = QuantumBool()

state_function(qb)

>>> qb.qs.statevector(decimals=5)
0.98079∣False⟩+0.19509∣True⟩

We can enhance the probability of measuring the good state with amplitude amplification:

>>> amplitude_amplification([qb], state_function, oracle_function)
>>> qb.qs.statevector(decimals=5)
0.83147*|False> + 0.55557*|True> 

>>> amplitude_amplification([qb], state_function, oracle_function)
>>> qb.qs.statevector(decimals=5)
0.55557*|False> + 0.83147*|True> 

>>> amplitude_amplification([qb], state_function, oracle_function)
>>> qb.qs.statevector(decimals=5)
0.19509*|False> + 0.98079*|True>