maximum_likelihood_ae

This module contains the MLAE class. Given a quantum oracle operator, this class estimates the probability of a given target state using the Maximum Likelihood Amplitude Estimation based on the paper:

Suzuki, Y., Uno, S., Raymond, R., Tanaka, T., Onodera, T., & Yamamoto, N. Amplitude estimation without phase estimation Quantum Information Processing, 19(2), 2020 arXiv: quant-ph/1904.10246v2

Author: Gonzalo Ferro Costas & Alberto Manzano Herrero

class QQuantLib.AE.maximum_likelihood_ae.MLAE(oracle: qat.lang.AQASM.QRoutine, target: list, index: list, **kwargs)

Class for using Maximum Likelihood Quantum Amplitude Estimation (MLAE) algorithm

Parameters:

oracle (QLM gate) – QLM gate with the Oracle for implementing the Grover operator: init_q_prog and q_gate will be interpreted as None
target (list of ints) – python list with the target for the amplitude estimation
index (list of ints) – qubits which mark the register to do the amplitude estimation
kwargs (dictionary) – dictionary that allows the configuration of the MLAE algorithm
qpu (kwargs, QLM solver) – solver for simulating the resulting circuits
schedule (kwargs, list of two lists) – the schedule for the algorithm
optimizer (kwargs,) – an optimizer with just one possible entry
delta (kwargs, float) – tolerance to avoid division by zero warnings
ns (kwargs, int) – number of grid points for brute scipy optimizer
mcz_qlm (kwargs, bool) – for using or not QLM implementation of the multi controlled Z gate

static cost_function(angle: float, m_k: list, n_k: list, h_k: list) → float

This method calculates the -Likelihood of angle theta for a given schedule m_k,n_k

Notes

\[L(\theta,\mathbf{h}) = -\sum_{k = 0}^M\log{l_k(\theta|h_k)}\]

Parameters:

angle (float) – Angle (radians) for calculating the probability of measure a positive event.
m_k (list of ints) – number of times the grover operator was applied.
n_k (list of ints) – number of total events measured for the specific m_k
h_k (list of ints) – number of positive events measured for each m_k

Returns:

cost – the aggregation of the individual likelihoods

Return type:

float

property index: creating index property

static likelihood(theta: float, m_k: int, n_k: int, h_k: int) → float

Calculates Likelihood from Suzuki paper. For h_k positive events of n_k total events, this function calculates the probability of this taking into account that the probability of a positive event is given by theta and by m_k The idea is use this function to minimize it for this reason it gives minus Likelihood

Notes

\[l_k(\theta|h_k) = \sin^2\left((2m_k+1)\theta\right)^{h_k} \ \cos^2 \left((2m_k+1)\theta\right)^{n_k-h_k}\]

Parameters:

theta (float) – Angle (radians) for calculating the probability of measure a positive event.
m_k (int) – number of times the grover operator was applied.
n_k (int) – number of total events measured for the specific m_k
h_k (int) – number of positive events measured for each m_k

Returns:

Gives the Likelihood p(h_k with m_k amplifications|theta)

Return type:

float

static log_likelihood(theta: float, m_k: int, n_k: int, h_k: int) → float

Calculates log of the likelihood from Suzuki paper.

Notes

\[\log{l_k(\theta|h_k)} = 2h_k\log\big[\sin\left((2m_k+1) \ \theta\right)\big] +2(n_k-h_k)\log\big[\cos\left((2m_k+1) \ \theta\right)\big]\]

Parameters:

theta (float) – Angle (radians) for calculating the probability of measure a positive event.
m_k (int) – number of times the grover operator was applied.
n_k (int) – number of total events measured for the specific m_k
h_k (int) – number of positive events measured for each m_k

Returns:

Gives the log Likelihood p(h_k with m_k amplifications|theta)

Return type:

float

mlae(schedule, optimizer)

This method executes a complete Maximum Likelihood Algorithm, including executing schedule, defining the correspondent cost function and optimizing it.

Parameters:

schedule (list of two lists) – the schedule for the algorithm
optimizer (optimization routine.) – the optimizer should receive a function of one variable the angle to be optimized. Using lambda functions is the recommended way.

Returns:

result (optimizer results) – the type of the result is the type of the result of the optimizer
h_k (list) – list with number of positive outcomes from quantum circuit for each pair element of the input schedule
cost_function_partial (function) – partial cost function with the m_k, n_k and h_k fixed to the obtained values of the different experiments.

property oracle: creating oracle property

run() → float

run method for the class.

Returns:: list with the estimation of a
Return type:: result

Notes

\[a^* = \sin^2(\theta^*) \; where \; \theta^* = \arg \ \min_{\theta} L(\theta,\mathbf{h})\]

run_schedule(schedule)

This method execute the run_step method for each pair of values of a given schedule.

Parameters:: schedule (list of two lists) – the schedule for the algorithm
Returns:: h_k – list with the h_k result of each pair of the input schedule
Return type:: list

run_step(m_k: int, n_k: int) → int

This method executes on step of the MLAE algorithm

Parameters:

m_k (int) – number of times to apply the self.q_gate to the quantum circuit
n_k (int) – number of shots

Returns:

h_k (int) – number of positive events
routine (QLM Routine object)

property schedule: creating schedule property

set_exponential_schedule(n_t: int, n_s: int)

Creates a scheduler of exponential increasing of m_ks.

Parameters:

n_t (int) – number of maximum applications of the grover operator
n_s (int) – number of shots for each m_k grover applications

set_linear_schedule(n_t: int, n_s: int)

Creates a scheduler of linear increasing of m_ks.

Parameters:

n_t (int) – number of maximum applications of the grover operator
n_s (int) – number of shots for each m_k grover applications

property target: creating target property