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Изучаем Q#. Обучаем перцептрон
Нерон, любующийся горящим Римом. Худ. Карл Теодор фон Пилоти, 1861 г.
Базовым элементом построения нейросетей, как мы знаем, является модель нейрона, а, соответственно, простейшей моделью нейрона, является перцептрон.
С математической точки зрения, перцептрон решает задачу разделения пространства признаков гиперплоскостью, на две части. То есть является простейшим линейным классификатором.
Обобщенная схема нейрона
Обобщенная схема нейрона представляет собой функцию f(SUM Wi*xi - W0)
Здесь
x1,...,xn – компоненты вектора признаков x=(x1,x2,...,xn);
SUM – сумматор;
W1,W2,...,Wn – синоптические веса;
f – функция активации; f(v)= { 0 при v < 0 и 1 при v>0 }
W0 – порог.
Таким образом, нейрон представляет собой линейный классификатор с дискриминантной функцией g(X)=f(SUM Wi*Xi - W0).
И задача построения линейного классификатора для заданного множества прецедентов (Xk,Yk) сводится к задаче обучения нейрона, т.е. подбора соответствующих весов W1,W2,...,Wn и порога W0.
Классический подход обучения перцептрона хорошо известен
Инициализируем W0,W1,W2,...Wn (обычно случайными значениями)
Для обучающей выборки (Xk,Yk) пока для всех значений не будет выполняться f(SUM WiXki - W0)==Yi повторяем последовательно для всех элементов:
W = W + r(Yk - f(SUM WiXki - W0))Xk, где 0 < r < 1 - коэффициент обучения
Для доказательства сходимости алгоритма применяется теорема Новикова, которая говорит, что если существует разделяющая гиперплоскость, то она может быть найдена указанным алгоритмом.
Что же нам может предложить модель квантовых вычислений для решения задачи обучения перцептрона - то есть для нахождения синоптических весов по заданной обучающей выборке?
Ответ - мы можем сразу опробовать все возможные значения весов и выбрать из них тот - который удовлетворяет нашим требованиям - то есть правильно разделяет обучающую выборку.
Для понимания данного туториала вам потребуются базовые знания по
нейросетям
квантовым вычислениям (кубиты и трансформации)
программированию на Q-sharp
Замутим небольшие рассуждения
Пусть мы имеем обучающую выборку (Xk,Yk), где Xk - двоичный вектор из {0,1}^n, и Yk - тоже является двоичным значением из {0,1} (это условие делается для упрощения задачи, и ни в коей мере не накладывает ограничение на обобщение).
Модель перцептрона, предполагает, что W0,W1,W2,...,Wn являются действительными числами
но математически легко доказать, что если существует решение в рациональных числах, то существует решение и в целых числах.
а для конечной обучающей выборки, если существует решение в действительных числах, то будет существовать решение и рациональных числах.
таким образом мы можем искать нужные нам коэффициенты W0,W1,W2,...,Wn в целых числах
Предположим, также, что значения W0,W1,W2,...,Wn могут быть представлены двоичными числами с не более чем m разрядов (соответственно, старший разряд будет показывать знак целого числа)
Таким образом, у нас получается, что
можно взять m(n+1) кубитов и рассматривать их как двоичное представление n+1 целого числа W0,W1,W2,...,Wn,
для обучающей выборки (Xk,Yk) вычислить число несовпадений Error(W) = SUM (sign(SUM Wi Xki - W0) xor Yk), где sign - взятие старшего (знакового) разряда
найти такой W, что Error(W)=0
И тут мы видим, что последний пункт - это есть условие для применения алгоритма Гровера!!! А для него уже давно и много чего написано.
Алгоритм Гровера представляет собой обобщённый, независящей от конкретной задачи поиск, функция которого представляет "чёрный ящик" f: {0,1}^n to {0,1}^n, для которой известно, что EXISTS!w:f(w)=a, где a - заданное значение (считаем, что для f и заданного a можно построить оракул Uf: { |w> to |1>, |x> to |0> if |x> != |w> })
Алгоритм Гровера достаточно прост
Задаём в регистре (массиве кубитов) начальное значение H|0>
Повторяем несколько раз (исходя из оценки) пару трансформаций над регистром
Отражение от решения Uw: { |w> to -|w>, |x> to |x> if |x> !=|w> } или Uw = I-2|w><w|
Отражение от s=H|0> Us = 2|s><s|-I
Забираем нужное решение из регистра (с большой долей вероятности, что оно правильное)
Таким образом, мы свели задачу обучения перцептрона по выборке, к известному алгоритму квантовых вычислений - алгоритму Гровера.
А зачем всё это, здесь же не удалось ускорить расчёты, и их даже стало больше?
Если, вы знакомы с алгоритмом Гровера, то понимаете, что по сравнению с полным перебором, вместо N/M итераций, он требует PI/4*SQRT(N/M) итераций, где N - число всех возможных вариантов перебора, а M - число возможных решений задачи.
В данном случае, N=2^(m(n+1)) и решение задачи потребует оценку O(2^(m(n+1)/2)) итераций в худшем случае, что при больших m и n делает сомнительным выгоду от использования квантовых вычислений.
Отвечу - да, вы правы в своих сомнениях на текущий момент времени. Лично я уверен, что (уже сейчас) существуют более быстрые алгоритмы по сравнению с алгоритмом Гровера - просто общественности может быть о них неизвестно о них на текущий момент времени (вспомним как это было с алгоритмом, известным впоследствии как RSA). А пока, примите данное решение, как один из вариантов обучения перцептрона с помощью квантовых вычислений.
Рон Ривест, Ади Шамир и Леонард Адлеман из Массачусетского технологического института в течение года предприняли несколько попыток создать одностороннюю функцию, которую было бы трудно инвертировать. Ривест и Шамир, будучи компьютерными учеными, предложили множество потенциальных функций, а Адлеман, будучи математиком, отвечал за поиск их слабых мест. Они опробовали множество подходов, включая "ранцевый" и "перестановочные полиномы". Какое-то время они думали, что-то, чего они хотели достичь, невозможно из-за противоречивых требований. В апреле 1977 года они провели Песах в доме одного из студентов и выпили много манишевицкого вина, а затем вернулись к себе домой около полуночи. Ривест, не в силах заснуть, лег на диван с учебником математики и начал думать о своей односторонней функции. Остаток ночи он провел, формализуя свою идею, и к рассвету большая часть статьи была готова. Алгоритм теперь известен как RSA - инициалы их фамилий в том же порядке, что и в их статье.
Клиффорд Кокс, английский математик, работавший в британской разведывательной службе Government Communications Headquarters (GCHQ), описал эквивалентную систему во внутреннем документе в 1973 г. Однако, учитывая относительно дорогие компьютеры, необходимые для ее реализации в то время, она считалась в основном курьезом и, насколько известно, так и не была применена. Однако его открытие было раскрыто только в 1997 году из-за его сверхсильного засекречивания.
Так, что кодим ...
Нам потребуется реализация следующих методов
Метод арифметики над регистром из кубитов
увеличение значения на единицу, то есть трансформация Inc:|k> to |k+1>
Сложение, то есть трансформация Add:|a>|b> to |a+b>|b>
Добавление константы, то есть трансформация: AddiInt:|k>,value to |k+value>
/// # Описание
/// увеличение на единицу числового значения в массиве кубитов (рассматриваемых как регистр)
/// то есть трансформация вида |k> -> |k+1>
operation Inc(target: Qubit[]) : Unit is Ctl {
let n = Length(target);
for idx in 1..n {
Controlled X(target[0..n-idx-1], target[n-idx]);
}
}
/// # Описание
/// увеличение на указанную величину числового значения в массиве кубитов (рассматриваемых как регистр)
/// то есть трансформация вида |k>|value> -> |k+value>|value>
operation Add(target: Qubit[], value: Qubit[]) : Unit {
let n = Length(target);
use (qubits) = (Qubit[2]) {
for idx in 0..n-1 {
let carry = qubits[idx%2];
let next = qubits[1-(idx%2)];
// вычисляем следующее значение флага переноса разряда
// next = carry+target[idx]+value[idx]>=2
Controlled X([carry, target[idx]], next);
Controlled X([carry, value[idx]], next);
Controlled X([target[idx], value[idx]], next);
Controlled X([carry, target[idx], value[idx]], next);
// target[idx] = carry^target[idx]^value[idx]
Controlled X([value[idx]], target[idx]);
Controlled X([carry], target[idx]);
Reset(carry);
}
ResetAll(qubits);
}
}
/// # Описание
/// увеличение на указанную величину числового значения в массиве кубитов (рассматриваемых как регистр)
/// то есть трансформация вида |k> -> |k+value>
operation AddInt(target: Qubit[], value: Int) : Unit {
let n = Length(target);
mutable bools = [false, size=n];
if(value>0) {
let tmp = IntAsBoolArray(value, n);
for i in 0..n-1 {
set bools w/= i <- tmp[i];
}
}
if(value<0) {
// value = ~(-value)+1
let tmp = IntAsBoolArray(-value, n);
mutable carry = true;
for i in 0..n-1 {
let next = carry and not tmp[i];
set bools w/= i <- (not tmp[i] and not carry) or (tmp[i] and carry);
set carry = next;
}
}
use (qubits) = (Qubit[2]) {
for idx in 0..n-1 {
let carry = qubits[idx%2];
let next = qubits[1-(idx%2)];
// вычисляем следующее значение флага переноса разряда
if(bools[idx]) {
// next = carry*target[idx]^carry^target[idx] = carry|target[idx]
Controlled X([carry, target[idx]], next);
Controlled X([carry], next);
Controlled X([target[idx]], next);
}
else {
// next = carry*target[idx] = carry&target[idx]
Controlled X([carry, target[idx]], next);
}
// добавляем текушее значение флага переноса и добавляемого бита
Controlled X([carry], target[idx]);
if(bools[idx]) {
X(target[idx]);
}
Reset(carry);
}
ResetAll(qubits);
}
}
вспомогательные методы
/// # Описание
/// вспомогательный метод для копирования значений массива кубитов
operation Copy(source: Qubit[], target: Qubit[]) : Unit {
let n = Length(source);
for i in 0..(n-1) {
Controlled X([source[i]], target[i]);
}
}
/// # Описание
/// измерение значений (коллапсирование) кубитов в массиве (который рассматриваем как один регистр)
operation Measure(qubits: Qubit[]) : Bool[] {
let results = ForEach(M, qubits);
return ResultArrayAsBoolArray(results);
}
/// # Описание
/// измерение значений (коллапсирование) кубитов в массиве (который рассматриваем как один регистр)
operation MeasureWeights(n: Int, m: Int, qubits: Qubit[]) : Int[] {
let results = ForEach(M, qubits);
mutable w = [0, size=n+1];
for i in 0..n {
let j = ResultArrayAsInt(results[i*m..(i+1)*m-1]);
if((j&&&(2^(m-1))) == 0) {
set w w/= i <- j;
}
else {
set w w/= i <- -((j^^^(2^m-1))+1);
}
}
return w;
}
Вычисление значения SUM Wi xi - W0 (где Wi - регистр из m кубитов, xi - значение из {0,1}) и оракул для алгоритма Гровера U(W)=SUM (sign(SUM Wi Xki - W0) xor Yk)
/// # Описание
/// подсчёт суммы весов для заданного значения аргумента
operation SumOfWeights(n: Int, m: Int, arg: Bool[], W: Qubit[], target: Qubit[]) : Unit {
// target = -W0 = ~W0 + 1
Copy(W[0..m-1], target);
ApplyToEach(X, target);
Inc(target);
for i in 1..n {
if(arg[i-1]) {
Add(target, W[i*m..(i+1)*m-1]);
}
}
}
/// # Описание
/// подсчёт числа ошибок для заданной выборки
operation CountErrors(n: Int, m: Int, l: Int, args: Bool[][], values: Bool[], W: Qubit[], target: Qubit[]) : Unit {
for i in 0..l-1 {
use (qubits) = (Qubit[m]) {
SumOfWeights(n, m, args[i], W, qubits);
if(not values[i]) {
X(qubits[m-1]);
}
Controlled Inc([qubits[m-1]],target);
ResetAll(qubits);
}
}
}
/// # Описание
/// реализация оракла, необходимого для алгоритма гровера
/// соответственно, мы считаем, что правильное решение - это то, которое не имеет ошибок
operation NoErrorOracle(n: Int, m: Int, l: Int, args: Bool[][], values: Bool[], W: Qubit[], target: Qubit) : Unit {
use (error) = (Qubit[m]) {
CountErrors(n, m, l, args, values, W, error);
ApplyToEach(X, error);
Controlled X(error, target);
ResetAll(error);
}
}
Методы алгоритма Гровера (взято с https://learn.microsoft.com/ru-ru/azure/quantum/tutorial-qdk-grovers-search?tabs=tabid-visualstudio)
Отражение от решения
Отражение от H|0>
И, собственно, основной цикл алгоритма Гровера
/// # Описание
/// шаг для алгоритма гровера
/// отражение от решения
operation ReflectAboutSolution(oracle : (Qubit[], Qubit) => Unit, register : Qubit[]) : Unit {
use (target)=(Qubit()){
within {
X(target);
H(target);
}
apply {
oracle(register, target);
}
}
}
/// # Описание
/// шаг для алгоритма гровера
/// отражение от H|0>
operation ReflectAboutUniform(inputQubits : Qubit[]) : Unit {
within {
ApplyToEachA(H, inputQubits);
ApplyToEachA(X, inputQubits);
}
apply {
Controlled Z(Most(inputQubits), Tail(inputQubits));
}
}
/// # Описание
/// алгоритм гровера
operation RunGroversSearch(register : Qubit[], oracle : (Qubit[], Qubit) => Unit, iterations : Int) : Unit {
ApplyToEach(H, register);
for _ in 1 .. iterations {
ReflectAboutSolution(oracle, register);
ReflectAboutUniform(register);
}
}
Подготовим тест
Сгенерируем параметры гиперплоскости и сгенерируем обучающую выборку
/// # Описание
/// генерация параметров случайный гиперплоскости
operation RandomWeights(n: Int, m: Int) : Int[] {
use (qubits)=(Qubit[m*(n+1)]){
ApplyToEach(H, qubits);
let w = MeasureWeights(n, m, qubits);
ResetAll(qubits);
return w;
}
}
/// # Описание
/// генерация случайных данных, как обучающей последовательности
operation RandomTrain(n: Int, m: Int, w: Int[], l: Int) : (Bool[][],Bool[]) {
mutable args = [[false, size=n], size = l];
mutable values = [false, size = l];
use qubits = Qubit[n] {
for idx in 0..l-1 {
ApplyToEach(H, qubits);
let arg = Measure(qubits);
mutable s = -w[0];
for j in 1..n {
if(arg[j-1]) {
set s+=w[j];
}
}
let value = s>=0;
set args w/= idx <- arg;
set values w/= idx <- value;
ResetAll(qubits);
}
}
return (args,values);
}
// подготавливаем случайные данные
mutable rand_w = [0, size=n+1];
mutable (args,values)=([[false, size=n],size=l],[false,size=l]);
mutable isAllZero = false;
mutable isAllOne = false;
repeat {
let temp_w = RandomWeights(n, 2);
let (temp_args,temp_values) = RandomTrain(n, m, temp_w, l);
set isAllOne = true;
set isAllZero = true;
for i in 0..n {
set rand_w w/= i <- temp_w[i];
}
for i in 0..l-1 {
set args w/= i <- temp_args[i];
set values w/= i <- temp_values[i];
set isAllOne = isAllOne and values[i];
set isAllZero = isAllZero and not values[i];
}
}
until (not isAllOne and not isAllZero);
Message($"n={n} m={m} l={l} m*(n+1)={m*(n+1)} rand_w={rand_w}");
for i in 0..l-1 {
Message($"TrainData: {args[i]} -> {values[i]}");
}
let noErrorOracle = NoErrorOracle(n, m, l, args, values, _, _);
Проверим правильность работы построенного оракла
let noErrorOracle = NoErrorOracle(n, m, l, args, values, _, _);
// проверка правильности работы оракла
use (qubits, oracle)=(Qubit[m*(n+1)],Qubit()) {
for i in 0..n {
AddInt(qubits[i*m..(i+1)*m-1], rand_w[i]);
}
noErrorOracle(qubits, oracle);
let w = MeasureWeights(n, m, qubits);
Message($"SelfCheck: {w} ... oracle = {M(oracle)}");
if(M(oracle)==One){
Message($"SelfCheck: Success!!! {w}");
}
ResetAll(qubits);
Reset(oracle);
}
Запустим алгоритм Гровера
// применяем алгоритм гровера
// точное число шагов у алгоритма мы не знаем (знаем только оценку)
// поэтому запускаем с разными значениями итераций
// Повторение итераций после groverIterations сопровождается снижением этой вероятности
// вплоть до практически нулевой вероятности успеха на итерации 2*groverIterations.
// После этого вероятность снова возрастает до итерации 3*groverIterations и т. д.
// В практических приложениях обычно неизвестно, сколько решений имеет ваша задача,
// прежде чем вы решите ее. Эффективной стратегией для решения этой проблемы является
// "предположение" количества решений путем постепенного увеличения степени двойки (т. е. 1,2,4,8,...).
// Одно из этих предположений будет достаточно близким для того, чтобы алгоритм нашел решение
// со средним числом итераций около SQRT(2^n/S)
mutable currenIterations = 1;
set isSuccess = false;
repeat{
let repeatTests = 3;
mutable currentTest = 0;
repeat {
set currentTest += 1;
use (qubits, oracle) = (Qubit[m*(n+1)], Qubit()){
RunGroversSearch(qubits, noErrorOracle, currenIterations);
noErrorOracle(qubits, oracle);
let w = MeasureWeights(n, m, qubits);
Message($"GroversSearch: iterations = {currenIterations} ... {w} ... oracle = {M(oracle)}");
if(M(oracle)==One){
set isSuccess = true;
Message($"GroversSearch: Success!!! {w}");
}
ResetAll(qubits);
Reset(oracle);
}
}
until(currentTest>=repeatTests or isSuccess);
set currenIterations *= 2;
}
until (isSuccess);
Полный текст кода
namespace qperceptron {
open Microsoft.Quantum.Canon;
open Microsoft.Quantum.Intrinsic;
open Microsoft.Quantum.Arrays;
open Microsoft.Quantum.Convert;
open Microsoft.Quantum.Math;
open Microsoft.Quantum.Logical;
open Microsoft.Quantum.Diagnostics;
/// # Описание
/// увеличение на единицу числового значения в массиве кубитов (рассматриваемых как регистр)
/// то есть трансформация вида |k> -> |k+1>
operation Inc(target: Qubit[]) : Unit is Ctl {
let n = Length(target);
for idx in 1..n {
Controlled X(target[0..n-idx-1], target[n-idx]);
}
}
/// # Описание
/// увеличение на указанную величину числового значения в массиве кубитов (рассматриваемых как регистр)
/// то есть трансформация вида |k>|value> -> |k+value>|value>
operation Add(target: Qubit[], value: Qubit[]) : Unit {
let n = Length(target);
use (qubits) = (Qubit[2]) {
for idx in 0..n-1 {
let carry = qubits[idx%2];
let next = qubits[1-(idx%2)];
// вычисляем следующее значение флага переноса разряда
// next = carry+target[idx]+value[idx]>=2
Controlled X([carry, target[idx]], next);
Controlled X([carry, value[idx]], next);
Controlled X([target[idx], value[idx]], next);
Controlled X([carry, target[idx], value[idx]], next);
// target[idx] = carry^target[idx]^value[idx]
Controlled X([value[idx]], target[idx]);
Controlled X([carry], target[idx]);
Reset(carry);
}
ResetAll(qubits);
}
}
/// # Описание
/// увеличение на указанную величину числового значения в массиве кубитов (рассматриваемых как регистр)
/// то есть трансформация вида |k> -> |k+value>
operation AddInt(target: Qubit[], value: Int) : Unit {
let n = Length(target);
mutable bools = [false, size=n];
if(value>0) {
let tmp = IntAsBoolArray(value, n);
for i in 0..n-1 {
set bools w/= i <- tmp[i];
}
}
if(value<0) {
// value = ~(-value)+1
let tmp = IntAsBoolArray(-value, n);
mutable carry = true;
for i in 0..n-1 {
let next = carry and not tmp[i];
set bools w/= i <- (not tmp[i] and not carry) or (tmp[i] and carry);
set carry = next;
}
}
use (qubits) = (Qubit[2]) {
for idx in 0..n-1 {
let carry = qubits[idx%2];
let next = qubits[1-(idx%2)];
// вычисляем следующее значение флага переноса разряда
if(bools[idx]) {
// next = carry*target[idx]^carry^target[idx] = carry|target[idx]
Controlled X([carry, target[idx]], next);
Controlled X([carry], next);
Controlled X([target[idx]], next);
}
else {
// next = carry*target[idx] = carry&target[idx]
Controlled X([carry, target[idx]], next);
}
// добавляем текушее значение флага переноса и добавляемого бита
Controlled X([carry], target[idx]);
if(bools[idx]) {
X(target[idx]);
}
Reset(carry);
}
ResetAll(qubits);
}
}
/// # Описание
/// вспомогательный метод для копирования значений массива кубитов
operation Copy(source: Qubit[], target: Qubit[]) : Unit {
let n = Length(source);
for i in 0..(n-1) {
Controlled X([source[i]], target[i]);
}
}
/// # Описание
/// измерение значений (коллапсирование) кубитов в массиве (который рассматриваем как один регистр)
operation Measure(qubits: Qubit[]) : Bool[] {
let results = ForEach(M, qubits);
return ResultArrayAsBoolArray(results);
}
/// # Описание
/// измерение значений (коллапсирование) кубитов в массиве (который рассматриваем как один регистр)
operation MeasureWeights(n: Int, m: Int, qubits: Qubit[]) : Int[] {
let results = ForEach(M, qubits);
mutable w = [0, size=n+1];
for i in 0..n {
let j = ResultArrayAsInt(results[i*m..(i+1)*m-1]);
if((j&&&(2^(m-1))) == 0) {
set w w/= i <- j;
}
else {
set w w/= i <- -((j^^^(2^m-1))+1);
}
}
return w;
}
/// # Описание
/// подсчёт суммы весов для заданного значения аргумента
operation SumOfWeights(n: Int, m: Int, arg: Bool[], W: Qubit[], target: Qubit[]) : Unit {
// target = -W0 = ~W0 + 1
Copy(W[0..m-1], target);
ApplyToEach(X, target);
Inc(target);
for i in 1..n {
if(arg[i-1]) {
Add(target, W[i*m..(i+1)*m-1]);
}
}
}
/// # Описание
/// подсчёт числа ошибок для заданной выборки
operation CountErrors(n: Int, m: Int, l: Int, args: Bool[][], values: Bool[], W: Qubit[], target: Qubit[]) : Unit {
for i in 0..l-1 {
use (qubits) = (Qubit[m]) {
SumOfWeights(n, m, args[i], W, qubits);
if(not values[i]) {
X(qubits[m-1]);
}
Controlled Inc([qubits[m-1]],target);
ResetAll(qubits);
}
}
}
/// # Описание
/// реализация оракла, необходимого для алгоритма гровера
/// соответственно, мы считаем, что правильное решение - это то, которое не имеет ошибок
operation NoErrorOracle(n: Int, m: Int, l: Int, args: Bool[][], values: Bool[], W: Qubit[], target: Qubit) : Unit {
use (error) = (Qubit[m]) {
CountErrors(n, m, l, args, values, W, error);
ApplyToEach(X, error);
Controlled X(error, target);
ResetAll(error);
}
}
/// # Описание
/// шаг для алгоритма гровера
/// отражение от решения
operation ReflectAboutSolution(oracle : (Qubit[], Qubit) => Unit, register : Qubit[]) : Unit {
use (target)=(Qubit()){
within {
X(target);
H(target);
}
apply {
oracle(register, target);
}
}
}
/// # Описание
/// шаг для алгоритма гровера
/// отражение от H|0>
operation ReflectAboutUniform(inputQubits : Qubit[]) : Unit {
within {
ApplyToEachA(H, inputQubits);
ApplyToEachA(X, inputQubits);
}
apply {
Controlled Z(Most(inputQubits), Tail(inputQubits));
}
}
/// # Описание
/// алгоритм гровера
operation RunGroversSearch(register : Qubit[], oracle : (Qubit[], Qubit) => Unit, iterations : Int) : Unit {
ApplyToEach(H, register);
for _ in 1 .. iterations {
ReflectAboutSolution(oracle, register);
ReflectAboutUniform(register);
}
}
/// # Описание
/// генерация параметров случайный гиперплоскости
operation RandomWeights(n: Int, m: Int) : Int[] {
use (qubits)=(Qubit[m*(n+1)]){
ApplyToEach(H, qubits);
let w = MeasureWeights(n, m, qubits);
ResetAll(qubits);
return w;
}
}
/// # Описание
/// генерация случайных данных, как обучающей последовательности
operation RandomTrain(n: Int, m: Int, w: Int[], l: Int) : (Bool[][],Bool[]) {
mutable args = [[false, size=n], size = l];
mutable values = [false, size = l];
use qubits = Qubit[n] {
for idx in 0..l-1 {
ApplyToEach(H, qubits);
let arg = Measure(qubits);
mutable s = -w[0];
for j in 1..n {
if(arg[j-1]) {
set s+=w[j];
}
}
let value = s>=0;
set args w/= idx <- arg;
set values w/= idx <- value;
ResetAll(qubits);
}
}
return (args,values);
}
@EntryPoint()
operation Main(n: Int, m: Int, l: Int) : Unit {
Message("Hello quantum world!");
// подготавливаем случайные данные
mutable rand_w = [0, size=n+1];
mutable (args,values)=([[false, size=n],size=l],[false,size=l]);
mutable isAllZero = false;
mutable isAllOne = false;
repeat {
let temp_w = RandomWeights(n, 2);
let (temp_args,temp_values) = RandomTrain(n, m, temp_w, l);
set isAllOne = true;
set isAllZero = true;
for i in 0..n {
set rand_w w/= i <- temp_w[i];
}
for i in 0..l-1 {
set args w/= i <- temp_args[i];
set values w/= i <- temp_values[i];
set isAllOne = isAllOne and values[i];
set isAllZero = isAllZero and not values[i];
}
}
until (not isAllOne and not isAllZero);
Message($"n={n} m={m} l={l} m*(n+1)={m*(n+1)} rand_w={rand_w}");
for i in 0..l-1 {
Message($"TrainData: {args[i]} -> {values[i]}");
}
let noErrorOracle = NoErrorOracle(n, m, l, args, values, _, _);
// проверка правильности работы оракла
use (qubits, oracle)=(Qubit[m*(n+1)],Qubit()) {
for i in 0..n {
AddInt(qubits[i*m..(i+1)*m-1], rand_w[i]);
}
noErrorOracle(qubits, oracle);
let w = MeasureWeights(n, m, qubits);
Message($"SelfCheck: {w} ... oracle = {M(oracle)}");
if(M(oracle)==One){
Message($"SelfCheck: Success!!! {w}");
}
ResetAll(qubits);
Reset(oracle);
}
let groverIterations = Round(PI()/4.0*Sqrt(IntAsDouble(2^(m*(n+1)))));
Message($"GroversSearch: groverIterations = {groverIterations}?");
mutable isSuccess = false;
// применяем алгоритм гровера
// точное число шагов у алгоритма мы не знаем (знаем только оценку)
// поэтому запускаем с разными значениями итераций
// Повторение итераций после groverIterations сопровождается снижением этой вероятности
// вплоть до практически нулевой вероятности успеха на итерации 2*groverIterations.
// После этого вероятность снова возрастает до итерации 3*groverIterations и т. д.
// В практических приложениях обычно неизвестно, сколько решений имеет ваша задача,
// прежде чем вы решите ее. Эффективной стратегией для решения этой проблемы является
// "предположение" количества решений путем постепенного увеличения степени двойки (т. е. 1,2,4,8,...).
// Одно из этих предположений будет достаточно близким для того, чтобы алгоритм нашел решение
// со средним числом итераций около SQRT(2^n/S)
mutable currenIterations = 1;
set isSuccess = false;
repeat{
let repeatTests = 3;
mutable currentTest = 0;
repeat {
set currentTest += 1;
use (qubits, oracle) = (Qubit[m*(n+1)], Qubit()){
RunGroversSearch(qubits, noErrorOracle, currenIterations);
noErrorOracle(qubits, oracle);
let w = MeasureWeights(n, m, qubits);
Message($"GroversSearch: iterations = {currenIterations} ... {w} ... oracle = {M(oracle)}");
if(M(oracle)==One){
set isSuccess = true;
Message($"GroversSearch: Success!!! {w}");
}
ResetAll(qubits);
Reset(oracle);
}
}
until(currentTest>=repeatTests or isSuccess);
set currenIterations *= 2;
}
until (isSuccess);
}
}
И потестим ...
PS C:\Projects\qperceptron> dotnet run -n 3 -m 3 -l 5
Hello quantum world!
n=3 m=3 l=5 m*(n+1)=12 rand_w=[0,1,-1,0]
TrainData: [False,True,False] -> False
TrainData: [False,True,True] -> False
TrainData: [False,False,True] -> True
TrainData: [False,False,True] -> True
TrainData: [True,False,True] -> True
SelfCheck: [0,1,-1,0] ... oracle = One
SelfCheck: Success!!! [0,1,-1,0]
GroversSearch: groverIterations = 50?
GroversSearch: iterations = 1 ... [2,-2,0,-3] ... oracle = Zero
GroversSearch: iterations = 1 ... [-2,-4,-2,-1] ... oracle = Zero
GroversSearch: iterations = 1 ... [2,-4,-3,-3] ... oracle = Zero
GroversSearch: iterations = 2 ... [3,1,1,-1] ... oracle = Zero
GroversSearch: iterations = 2 ... [-2,3,-1,-3] ... oracle = Zero
GroversSearch: iterations = 2 ... [3,-1,-3,3] ... oracle = Zero
GroversSearch: iterations = 4 ... [-1,-1,-1,1] ... oracle = Zero
GroversSearch: iterations = 4 ... [3,-3,2,-1] ... oracle = Zero
GroversSearch: iterations = 4 ... [2,2,1,-1] ... oracle = Zero
GroversSearch: iterations = 8 ... [-1,0,3,2] ... oracle = One
GroversSearch: Success!!! [-1,0,3,2]
— Ваше Сиятельство, может мы английскую машинку опробуем?
— А, ну что же, не зря же вы её покупали... Да и случай подходящий.
— Новенькая! Тут, Ваше Сиятельство, человек вроде бы и не нужен, а нужен только его пальчик!
— Да. В смысле техники нам, конечно, до них далеко.
— Так, Афонька... Ты у неё будешь первый. Вещь хорошая, останешься довольный!
Да. В смысле техники нам, конечно, до них далеко...Нам бы только иностранное ругать...
Ссылки
https://learn.microsoft.com/ru-ru/azure/quantum/tutorial-qdk-grovers-search?tabs=tabid-visualstudio
https://learn.microsoft.com/ru-ru/azure/quantum/user-guide/host-programs?tabs=tabid-copilot
Ранее
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09.10.25 11:05
marcushenderson624
Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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11.10.25 04:41
luciajessy3
Don’t be deceived by different testimonies online that is most likely wrong. I have made use of several recovery options that got me disappointed at the end of the day but I must confess that the tech genius I eventually found is the best out here. It’s better you devise your time to find the valid professional that can help you recover your stolen or lost crypto such as bitcoins rather than falling victim of other amateur hackers that cannot get the job done. ADAMWILSON . TRADING @ CONSULTANT COM / WHATSAPP ; +1 (603) 702 ( 4335 ) is the most reliable and authentic blockchain tech expert you can work with to recover what you lost to scammers. They helped me get back on my feet and I’m very grateful for that. Contact their email today to recover your lost coins ASAP…
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11.10.25 10:44
Tonerdomark
A thief took my Dogecoin and wrecked my life. Then Mr. Sylvester stepped in and changed everything. He got back €211,000 for me, every single cent of my gains. His calm confidence and strong tech skills rebuilt my trust. Thanks to him, I recovered my cash with no issues. After months of stress, I felt huge relief. I had full faith in him. If a scam stole your money, reach out to him today at { yt7cracker@gmail . com } His help sparked my full turnaround.
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12.10.25 01:12
harristhomas7376
"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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12.10.25 01:12
harristhomas7376
"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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12.10.25 19:53
Tonerdomark
A crook swiped my Dogecoin. It ruined my whole world. Then Mr. Sylvester showed up. He fixed it all. He pulled back €211,000 for me. Not one cent missing from my profits. His steady cool and sharp tech know-how won back my trust. I got my money smooth and sound. After endless worry, relief hit me hard. I trusted him completely. Lost cash to a scam? Hit him up now at { yt7cracker@gmail . com }. His aid turned my life around. WhatsApp at +1 512 577 7957.
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12.10.25 21:36
blessing
Writing this review is a joy. Marie has provided excellent service ever since I started working with her in early 2018. I was worried I wouldn't be able to get my coins back after they were stolen by hackers. I had no idea where to begin, therefore it was a nightmare for me. However, things became easier for me after my friend sent me to [email protected] and +1 7127594675 on WhatsApp. I'm happy that she was able to retrieve my bitcoin so that I could resume trading.
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13.10.25 01:11
elizabethrush89
God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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13.10.25 01:11
elizabethrush89
God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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14.10.25 01:15
tyleradams
Hi. Please be wise, do not make the same mistake I had made in the past, I was a victim of bitcoin scam, I saw a glamorous review showering praises and marketing an investment firm, I reached out to them on what their contracts are, and I invested $28,000, which I was promised to get my first 15% profit in weeks, when it’s time to get my profits, I got to know the company was bogus, they kept asking me to invest more and I ran out of patience then requested to have my money back, they refused to answer nor refund my funds, not until a friend of mine introduced me to the NVIDIA TECH HACKERS, so I reached out and after tabling my complaints, they were swift to action and within 36 hours I got back my funds with the due profit. I couldn’t contain the joy in me. I urge you guys to reach out to NVIDIA TECH HACKERS on their email: [email protected]
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14.10.25 08:46
robertalfred175
CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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14.10.25 08:46
robertalfred175
CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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14.10.25 08:46
robertalfred175
CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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15.10.25 18:07
crypto
Cryptocurrency's digital realm presents many opportunities, but it also conceals complex frauds. It is quite painful to lose your cryptocurrency to scam. You can feel harassed and lost as a result. If you have been the victim of a cryptocurrency scam, this guide explains what to do ASAP. Following these procedures will help you avoid further issues or get your money back. Communication with Marie ([email protected] and WhatsApp: +1 7127594675) can make all the difference.
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15.10.25 21:52
harristhomas7376
"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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15.10.25 21:52
harristhomas7376
"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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17.10.25 20:17
tyleradams
As time passes, there are an increasing number of frauds involving Bitcoin and other cryptocurrencies. Although there are many individuals who advertise recovering money online, people should use caution in dealing, especially when money is involved. You can trust NVIDIA TECH HACKERS [[email protected]], I promise. They are the top internet recovery company, and as their names indicate, your money is reclaimed as soon as feasible. My bitcoin was successfully retrieved in large part thanks to NVIDIA TECH HACKERS. Ensure that you get top-notch service; NVIDIA TECH HACKERS provides evidence of its work; and payment is only made when the service has been completed to your satisfaction. Reach them via email: [email protected] on google mail
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17.10.25 20:20
lindseyvonn
Have you gotten yourself involved in a cryptocurrency scam or any scam at all? If yes, know that you are not alone, there are a lot of people in this same situation. I'm a Health Worker and was a victim of a cryptocurrency scam that cost me a lot of money. This happened a few weeks ago, there’s only one solution which is to talk to the right people, if you don’t do this you will end up being really depressed. I was really devastated until went on LinkedIn one evening after my work hours and i saw lots of reviews popped up on my feed about [email protected], I sent an email to the team who came highly recommended - [email protected] I started seeing some hope for myself from the moment I sent them an email. The good part is they made the entire process stress free for me, i literally sat and waited for them to finish and I received what I lost in my wallet
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17.10.25 20:22
richardcharles
I would recommend NVIDIA TECH HACKERS to anyone that needs this service. I decided to get into crypto investment and I ended up getting my crypto lost to an investor late last year. The guy who was supposed to be managing my account turned out to be a scammer all along. I invested 56,000 USD and at first, my reading and profit margins were looking good. I started getting worried when I couldn’t make withdrawals and realized that I’ve been scammed. I came across some of the testimonials that people said about NVIDIA TECH HACKERS and how helpful he has been in recovering their funds. I immediately contacted him in his mail at [email protected] so I can get his assistance. One week into the recovery process the funds were traced and recovered back from the scammer. I can't appreciate him enough for his professionalism.
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17.10.25 20:23
stevekalfman
If you need a hacker for scam crypto recovery or mobile spy access remotely kindly reach out to [email protected] for quick response, I hired this hacker and he did a nice job. before NVIDIA TECH HACKERS, I met with different hacker's online which turns out to be scam, this NVIDIA TECH HACKERS case was different and he is the trusted hacker I can vote and refer.
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17.10.25 21:42
marcushenderson624
Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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17.10.25 21:42
marcushenderson624
Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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17.10.25 21:42
marcushenderson624
Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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