Iterative Quantum Amplitude Estimation#

IQAE(qargs, state_function, eps, alpha, mes_kwargs={})[source]#

Accelerated Quantum Amplitude Estimation (IQAE). This function performs QAE with a fraction of the quantum resources of the well-known QAE algorithm. See Accelerated Quantum Amplitude Estimation without QFT.

The problem of iterative quantum amplitude estimation is described as follows:

• Given a unitary operator $\mathcal{A}$, let $|\mathrm{\Psi }⟩=\mathcal{A}|0⟩|\text{False}⟩$.

• Write $|\mathrm{\Psi }⟩=\sqrt{a}|{\mathrm{\Psi }}_1⟩|\text{True}⟩+\sqrt{1-a}|{\mathrm{\Psi }}_0⟩|\text{False}⟩$ as a superposition of the orthogonal good and bad components of $|\mathrm{\Psi }⟩$.

• Find an estimate for $a$, the probability that a measurement of $|\mathrm{\Psi }⟩$ yields a good state.

Parameters:

qargslist[QuantumVariable]

The list of QuantumVariables which represent the state, the quantum amplitude estimation is performed on. The last variable in the list must be of type QuantumBool.

state_functionfunction

A Python function preparing the state $|\mathrm{\Psi }⟩$. This function will receive the variables in the list qargs as arguments in the course of this algorithm.

epsfloat

Accuracy $ϵ>0$ of the algorithm.

alphafloat

Confidence level $\alpha \in (0,1)$ of the algorithm.

mes_kwargsdict, optional

The keyword arguments for the measurement function. Default is an empty dictionary.

Returns:

afloat

An estimate $\hat{a}$ of $a$ such that

$$\mathbb{P}\{|\hat{a}-a|<ϵ\}\ge 1-\alpha$$

Examples

We show the same Numerical integration example which can also be found in the QAE documentation.

We wish to evaluate

$$A={\int }_0^{1}f(x)\mathrm{d}x.$$

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

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

n = 6 
inp = QuantumFloat(n,-n)
tar = QuantumBool()
input_list = [inp, 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 IQAE and obtain an estimate $a$ for the value of the integral $A=0.27268$.

a = IQAE(input_list, state_function, eps=0.01, alpha=0.01)

>>> a 
0.26782038552705856