Quantum Amplitude Estimation#

QAE(args, state_function, oracle_function, kwargs_oracle={}, precision=None, target=None)[source]#

This method implements the canonical quantum amplitude estimation (QAE) algorithm by Brassard et al..

The problem of quantum amplitude estimation 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 \(\ket{\Psi}\).
Find an estimate for \(a=\langle\Psi_1|\Psi_1\rangle\), the probability that a measurement of \(\ket{\Psi}\) yields a good state.

Parameters:

argsQuantumVariable or list[QuantumVariable]: The (list of) QuantumVariables which represent the state, the quantum amplitude estimation 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 {}.
precisionint, optional: The precision of the estimation. The default is None.
targetQuantumFloat, optional: A target QuantumFloat to perform the estimation into. The default is None. If given neither a precision nor a target, an Exception will be raised.

Returns:

resQuantumFloat

A QuantumFloat encoding the angle \(\theta\) as a fraction of \(\pi\), such that \(\tilde{a}=\sin^2(\theta)\) is an estimate for \(a\).

More precisely, we have \(\theta=\pi\frac{y}{M}\) for \(y\in\{0,\dotsc,M-1\}\) and \(M=2^{\text{precision}}\). After measurement, the estimate \(\tilde{a}=\sin^2(\theta)\) satisfies

\[|a-\tilde{a}|\leq\frac{2\pi}{M}+\frac{\pi^2}{M^2}\]

with probability of at least \(8/\pi^2\).

Examples

We define a function that prepares the state \(\ket{\Psi}=\cos(\frac{\pi}{8})\ket{0}+\sin(\frac{\pi}{8})\ket{1}\) and an oracle that tags the good state \(\ket{1}\). In this case, we have \(a=\sin^2(\frac{\pi}{8})\).

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

def state_function(qb):
    ry(np.pi/4,qb)

def oracle_function(qb):   
    z(qb)

qb = QuantumBool()

res = QAE([qb], state_function, oracle_function, precision=3)

>>> res.get_measurement()
{0.125: 0.5, 0.875: 0.5}

That is, after measurement we find \(\theta=\frac{\pi}{8}\) or \(\theta=\frac{7\pi}{8}\) with probability \(\frac12\), respectively. Therefore, we obtain the estimate \(\tilde{a}=\sin^2(\frac{\pi}{8})\) or \(\tilde{a}=\sin^2(\frac{7\pi}{8})\). In this case, both results coincide with the exact value \(a\).

Numerical integration

Here, we demonstarate how to use QAE for numerical integration.

Consider a continuous function \(f\colon[0,1]\rightarrow[0,1]\). We wish to evaluate

\[A=\int_0^1f(x)\mathrm dx.\]

For this, we set up the corresponding state_function acting on the input_list:

from qrisp import QuantumFloat, QuantumBool, control, z, h, ry, QAE
import numpy as np

n = 6 
inp = QuantumFloat(n,-n)
tar = QuantumBool()
input_list = [inp, tar]

Here, \(N=2^n\) is the number of sampling points the function \(f\) is evaluated on. The state_function acts as

\[\ket{0}\ket{0}\rightarrow\frac{1}{\sqrt{N}}\sum\limits_{x=0}^{N-1}\ket{x}\left(\sqrt{1-f(x/N)}\ket{0}+\sqrt{f(x/N)}\ket{1}\right).\]

Then the probability of measuring \(1\) in the target state tar is

\[p(1)=\frac{1}{N}\sum\limits_{x=0}^{N-1}f(x/N),\]

which acts as an approximation for the value of the integral \(A\).

The oracle_function, therefore, tags the \(\ket{1}\) state of the target state:

def oracle_function(inp, tar):
    z(tar)

For example, if \(f(x)=\sin^2(x)\) the state_function can be implemented as follows:

def state_function(inp, tar):
    h(inp)

    N = 2**inp.size
    for k in range(inp.size):
        with control(inp[k]):
            ry(2**(k+1)/N,tar)

Finally, we apply QAE and obtain an estimate \(a\) for the value of the integral \(A=0.27268\).

prec = 6
res = QAE(input_list, state_function, oracle_function, precision=prec)
meas_res = res.get_measurement()

theta = np.pi*max(meas_res, key=meas_res.get)
a = np.sin(theta)**2

>>> a
0.26430