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Autocallables with Integration Amplitude Loading

This notebook covers the implementation of the Integration Amplitude Loading Method for the autocallables based on [1] and [2] using the Classiq platform’s Qmod language.

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Data Definitions

import numpy as np
import scipy

S = 18  # notional value (can be asset initial price)
dt = 1  # in year
NUM_MARKET_DAYS_IN_YEAR = 250

DAILY_SIGMA = 0.0150694  # daily std log return
DAILY_MU = 0.00050963  # daily mean log return

SIGMA = DAILY_SIGMA * np.sqrt(NUM_MARKET_DAYS_IN_YEAR)  # annual
MU = DAILY_MU * NUM_MARKET_DAYS_IN_YEAR  # annual

b = 0.7  # binary barrier (to check with returns)
K_comp = 1  # K put (to check with returns)
K = K_comp * S  # K put (to to use for the payoff)
r = 0.04  # annual risk free rate

K_bin_1 = 1.1  # K binaries (to check with returns)
K_bin_2 = 1.1  # K binaries (to check with returns)
K_bin_norm = [
    np.log(K_bin_1),
    np.log(K_bin_2),
]  # K binaries (to check with log returns)
bin_1_payoff = 2 * np.exp(-r * 1)  # payoff binaries already discounted
bin_2_payoff = 3 * np.exp(-r * 2)  # payoff binaries already discounted


NUM_QUBITS = 2
TIME_STEPS = 3  # 3 years


PRECISION = 2

​
Gaussian State Preparation

def gaussian_discretization(num_qubits, mu=0, sigma=1, stds_around_mean_to_include=3):
    lower = mu - stds_around_mean_to_include * sigma
    upper = mu + stds_around_mean_to_include * sigma
    num_of_bins = 2**num_qubits
    sample_points = np.linspace(lower, upper, num_of_bins + 1)

    def single_gaussian(x: np.ndarray, _mu: float, _sigma: float) -> np.ndarray:
        cdf = scipy.stats.norm.cdf(x, loc=_mu, scale=_sigma)
        return cdf[1:] - cdf[0:-1]

    non_normalized_pmf = (single_gaussian(sample_points, mu, sigma),)
    real_probs = non_normalized_pmf / np.sum(non_normalized_pmf)
    return sample_points[:-1], real_probs[0].tolist()


grid_points, probabilities = gaussian_discretization(
    NUM_QUBITS, stds_around_mean_to_include=3
)

STEP_X = grid_points[1] - grid_points[0]
MIN_X = grid_points[0]

​
Rescalings

Compute RTmaxR_T^{max} resulting from discretization: 
R_T_MAX_PROP = 0

if MU > 0 and SIGMA > 0:
    R_T_MAX_PROP = TIME_STEPS * (MU * dt + SIGMA * np.sqrt(dt) * (grid_points[-1]))

R_T_MIN_PROP = 0

if MU > 0 and SIGMA > 0:
    R_T_MIN_PROP = TIME_STEPS * (MU * dt + SIGMA * np.sqrt(dt) * (grid_points[0]))

R_T_MAX_PROP = np.max([np.abs(R_T_MIN_PROP), np.abs(R_T_MAX_PROP)])

R_T_MAX = np.log(
    np.max(
        [
            (np.exp(TIME_STEPS * r) * bin_2_payoff) + K,
            (np.exp(TIME_STEPS * r) * bin_1_payoff) + K,
            K,
        ]
    )
    / S
)
R_T_MAX = max(R_T_MAX_PROP, R_T_MAX)
In two’s complement, given NN as the number of qubits, represent from −2N−1-2^{N-1} and 2N−1−12^{N-1}-1: 
a = 1 / (2**PRECISION)


if R_T_MAX < 1:
    int_places = 1
else:
    int_places = np.ceil(np.log2(np.ceil(R_T_MAX))) + 1


num_norm_factor = np.exp(a * (2 ** (int_places + PRECISION))) - 1
den_norm_factor = np.exp(a * (2 ** (int_places + PRECISION - 1) + 1))


norm_factor = S * (num_norm_factor / den_norm_factor)

​
Compute Constant Rotations

def compute_constant_rotation(const_payoff):
    b1 = const_payoff * np.exp(r * TIME_STEPS)
    num1 = (b1 + K) * den_norm_factor
    den1 = S * num_norm_factor
    den2 = num_norm_factor
    num2 = 1
    angle = (num1 / den1) - (num2 / den2)
    return 2 * np.arcsin(np.sqrt(angle))


bin_1_payoff_rotation = compute_constant_rotation(bin_1_payoff)
bin_2_payoff_rotation = compute_constant_rotation(bin_2_payoff)
zero_rotation = compute_constant_rotation(0)


bit_T_normalize = (S * np.exp(-r * TIME_STEPS)) / den_norm_factor


def postprocessing(x):
    return (
        (x * norm_factor * np.exp(-r * TIME_STEPS))
        + bit_T_normalize
        - (K * np.exp(-r * TIME_STEPS))
    )

​
Verifications

bin_1_payoff
Output:
np.float64(1.9215788783046464)
  


postprocessing((np.sin((compute_constant_rotation(bin_1_payoff) / 2))) ** 2)
Output:
np.float64(1.9215788783046523)
  


bin_2_payoff
Output:
np.float64(2.7693490391599074)
  


postprocessing((np.sin((compute_constant_rotation(bin_2_payoff) / 2))) ** 2)
Output:
np.float64(2.7693490391599074)
  


postprocessing((np.sin((compute_constant_rotation(0) / 2))) ** 2)
Output:
np.float64(1.7763568394002505e-15)
Compute constants for comparisons 
b_norm = np.log(b)
K_norm = np.log(K_comp)

​
Classical Payoff

from itertools import product

import pandas as pd

simulation = pd.DataFrame.from_dict(
    {
        "quantum_samples": list(range(len(grid_points))),
        "classical_samples": grid_points,
        "probabilities": probabilities,
    }
)

binary_combinations = list(product(range(2**NUM_QUBITS), repeat=TIME_STEPS))
sim = pd.DataFrame(binary_combinations)
new_col = ["time_" + str(i) for i in range(TIME_STEPS)]
sim.columns = new_col


def round_down(a):
    precision_factor = 2 ** (PRECISION)
    return np.floor(a * precision_factor) / precision_factor


def round_factor(a):
    # precision_factor = 2 ** (PRECISION)
    # return np.round(a * precision_factor) / precision_factor
    return floor_factor(a)


def floor_factor(a):
    precision_factor = 2 ** (PRECISION)

    if a >= 0:
        return np.floor(a * precision_factor) / precision_factor
    else:
        return np.ceil(a * precision_factor) / precision_factor


for i in range(TIME_STEPS):
    sim = sim.merge(simulation, left_on="time_" + str(i), right_on="quantum_samples")
    sim = sim.drop("quantum_samples", axis=1)
    new_col = new_col + ["c_" + str(i), "q_prob_" + str(i)]
    sim.columns = new_col
    sim["log_ret_val_" + str(i)] = (
        round_factor(MU * dt + np.sqrt(dt) * SIGMA * MIN_X)
        + round_factor(SIGMA * np.sqrt(dt) * STEP_X) * sim["time_" + str(i)]
    )
    new_col = new_col + ["log_ret_val_" + str(i)]
    sim.columns = new_col
    if i != 0:
        sim["log_ret_val_" + str(i)] = (
            sim["log_ret_val_" + str(i)] + sim["log_ret_val_" + str(i - 1)]
        )
    sim["ret_val_" + str(i)] = np.exp(sim["log_ret_val_" + str(i)])
    new_col = new_col + ["ret_val_" + str(i)]


sim["prob"] = 1
for i in range(TIME_STEPS):
    sim["prob"] = sim["prob"] * sim["q_prob_" + str(i)]


for i in range(TIME_STEPS):
    sim["b_crossed_" + str(i)] = sim["log_ret_val_" + str(i)] < round_factor(b_norm)

sim["bin_1_activate"] = sim["log_ret_val_0"] > round_factor(K_bin_norm[0])
sim["bin_2_activate"] = sim["log_ret_val_1"] > round_factor(K_bin_norm[1])

sim["K_put"] = sim["log_ret_val_" + str(TIME_STEPS - 1)] < round_factor(K_norm)

sim["payoff"] = 0.0

sim.loc[sim["bin_1_activate"], "payoff"] = bin_1_payoff
sim.loc[(~sim["bin_1_activate"]) & (sim["bin_2_activate"]), "payoff"] = bin_2_payoff

barrier_crossed_once = sim["b_crossed_0"]
for i in range(1, TIME_STEPS):
    barrier_crossed_once = barrier_crossed_once | sim["b_crossed_" + str(i)]

put_condition = (
    (~((sim["bin_1_activate"]) | (sim["bin_2_activate"])))
    & (barrier_crossed_once)
    & (sim["K_put"])
)

sim.loc[put_condition, "payoff"] = (
    S * (sim[put_condition]["ret_val_2"] 

- K_comp) * np.exp(-r * TIME_STEPS)
)


expected_payoff = sum(sim["prob"] * sim["payoff"])


print("expected payoff classical: " + str(expected_payoff))
Output:
expected payoff classical: -3.5032073946209352

​
Integration Method Circuit Synthesis

from classiq import *
from classiq.qmod.symbolic import sqrt


@qfunc
def integrator(exp_rate: CReal, x: Const[QNum], ref: QNum, res: QBit) -> None:
    prepare_exponential_state(-exp_rate, ref)
    res ^= x >= ref


def affine_py(x: QNum):
    return MU * dt + SIGMA * sqrt(dt) * (x * STEP_X + MIN_X)


@qperm
def add_minimum(x: QArray):
    X(x[x.size - 1])


round_K_bin_norm = []
round_b_norm = round_factor(b_norm)
round_K_bin_norm.append(round_factor(K_bin_norm[0]))
round_K_bin_norm.append(round_factor(K_bin_norm[1]))
round_k_norm = round_factor(K_norm)


@qfunc
def integration_load_amplitudes(y: Const[QNum], aux_reg: QNum, ind_reg: QBit):
    exp_rate = 1 / (2**PRECISION)
    integrator(exp_rate, y, aux_reg, ind_reg)


@qfunc
def integration_payoff(log_return: Const[QNum], aux_reg: QNum, ind_reg: QBit):
    log_return_unsigned = QNum(size=log_return.size)

    within_apply(
        lambda: [
            bind(log_return, log_return_unsigned),
            add_minimum(log_return_unsigned),
        ],
        lambda: integration_load_amplitudes(log_return_unsigned, aux_reg, ind_reg),
    )


@qperm
def check_barrier_crossed(log_return: Const[QNum], barrier_crossed: QBit):
    barrier_crossed ^= log_return < round_b_norm


@qperm
def check_binary(log_return: Const[QNum], round_K_bin_norm: CReal, binary_valid: QBit):
    binary_valid ^= log_return > round_K_bin_norm


@qperm
def check_K_put(
    log_return: Const[QNum[PRECISION + int_places, SIGNED, PRECISION]],
    k_put_valid: QBit,
):
    k_put_valid ^= log_return < round_k_norm


@qfunc
def binary_payoff(binary_payoff: CReal, target: QBit):
    RY(binary_payoff, target)


@qfunc
def zero_payoff(target: QBit):
    RY(zero_rotation, target)


@qperm
def check_put_activate(
    barriers_crossed: Const[QArray],
    k_put_valid: Const[QBit],
    put_alone_valid: QBit,
):
    put_alone_valid ^= (
        (
            (barriers_crossed[0] == 1)
            | (barriers_crossed[1] == 1)
            | (barriers_crossed[2] == 1)
        )
        & (k_put_valid == 1)
    ) == 1


@qperm
def populate_put_alone_valid(
    sum_log_return: Const[QNum],
    barriers_crossed: QArray,
    put_alone_valid: Output[QBit],
):
    k_put_valid = QBit()

    allocate(put_alone_valid)

    within_apply(
        lambda: [
            allocate(k_put_valid),
            check_K_put(sum_log_return, k_put_valid),
            check_barrier_crossed(
                sum_log_return, barriers_crossed[2]
            ),  # magari qui si può condizionare
        ],
        lambda: check_put_activate(barriers_crossed, k_put_valid, put_alone_valid),
    )


@qfunc
def final_payoff(
    check_all_zeros: Const[QNum],
    sum_log_return: Const[QNum],
    aux_reg: QNum,
    ind_reg: QBit,
):
    control(check_all_zeros == 0, lambda: zero_payoff(ind_reg))
    control(
        check_all_zeros == 1,
        lambda: integration_payoff(sum_log_return, aux_reg, ind_reg),
    )


@qfunc
def autocallable_integration(
    x: QArray[QNum, TIME_STEPS],
    aux_reg: QNum,
    ind_reg: QBit,
    sum_log_return: QNum,
    first_bin_valid: QBit,
    second_bin_valid: QBit,
    barriers_crossed: QArray[QBit],
) -> None:
    repeat(x.len, lambda i: inplace_prepare_state(probabilities, 0, x[i]))

    sum_log_return ^= affine_py(x[0])
    check_binary(sum_log_return, round_K_bin_norm[0], first_bin_valid)
    control(first_bin_valid, lambda: binary_payoff(bin_1_payoff_rotation, ind_reg))
    # check_barrier_crossed(sum_log_return, barriers_crossed[0])
    check_barrier_crossed(sum_log_return, barriers_crossed[0])

    sum_log_return += affine_py(x[1])
    check_binary(sum_log_return, round_K_bin_norm[1], second_bin_valid)
    control(
        ((first_bin_valid == 0) & (second_bin_valid == 1)) == 1,
        lambda: binary_payoff(bin_2_payoff_rotation, ind_reg),
    )  # check order
    # check_barrier_crossed(sum_log_return, barriers_crossed[1]) #magari qui si può condizionare
    check_barrier_crossed(sum_log_return, barriers_crossed[1])

    sum_log_return += affine_py(x[2])

    put_alone_valid = QBit()
    within_apply(
        lambda: populate_put_alone_valid(
            sum_log_return,
            barriers_crossed,
            put_alone_valid,
        ),
        lambda: final_payoff(
            [put_alone_valid, first_bin_valid, second_bin_valid],
            sum_log_return,
            aux_reg,
            ind_reg,
        ),
    )

​
IQAE Functions and QStruct

from classiq.applications.iqae.iqae import IQAE


class OracleVars(QStruct):
    x: QArray[QNum[NUM_QUBITS, False, 0], TIME_STEPS]
    aux_reg: QNum[int_places + PRECISION]
    sum_log_return: QNum[int_places + PRECISION, SIGNED, PRECISION]
    first_bin_valid: QBit
    second_bin_valid: QBit
    barriers_crossed: QArray[QBit, TIME_STEPS]


@qfunc
def iqae_state_preparation(state: OracleVars, ind: QBit):
    autocallable_integration(
        state.x,
        state.aux_reg,
        ind,
        state.sum_log_return,
        state.first_bin_valid,
        state.second_bin_valid,
        state.barriers_crossed,
    )

​
Base Simulator Synthesis

iqae = IQAE(
    state_prep_op=iqae_state_preparation,
    problem_vars_size=NUM_QUBITS * TIME_STEPS
    + 2 * (int_places + PRECISION)
    + 2
    + TIME_STEPS,
    constraints=Constraints(optimization_parameter="width"),
    preferences=Preferences(
        optimization_level=1, machine_precision=PRECISION, timeout_seconds=2000
    ),
)

qmod = iqae.get_model()

print("Starting synthesis")
qprog = iqae.get_qprog()
show(qprog)
Output:

Starting synthesis
  Quantum program link: https://platform.classiq.io/circuit/3564N6oLHen6uvAwSGk0stVkwht
  


print("Circuit width: ", qprog.data.width)
Output:

Circuit width:  24
Execution takes a lot of time. Examine the results: 
# takes a lot of time

# EPSILON = 0.001
# ALPHA = 0.002
# res_iqae= iqae.run(EPSILON, ALPHA, execution_preferences=ExecutionPreferences(shots=100000))


# print("the expected result from classical computation is: " + str(expected_payoff))
# print("the result from IQAE is: " + str(postprocessing(res_iqae.estimation)))
# print("the confidence interval of the quantum estimation is: " + str(postprocessing(res_iqae.confidence_interval[0]))+"," +str(postprocessing(res_iqae.confidence_interval[1])) )
# print(f"Synthesis and execution time: {time() - start_time} seconds")


print("the expected result from classical computation is: " + str(expected_payoff))
print("the result from IQAE is: " + str(-3.5079217726515903))
print(
    "the confidence interval of the quantum estimation is: "
    + str(postprocessing(0.119158741))
    + ","
    + str(postprocessing(0.1197666))
)
Output:
the expected result from classical computation is: -3.5032073946209352
  the result from IQAE is: -3.5079217726515903
  the confidence interval of the quantum estimation is: -3.535334467336666,-3.4805134318653383

​
References

[1] [Francesca Cibrario et al. (2024). Quantum Amplitude Loading for Rainbow Options Pricing. Preprint.](https://arxiv.org/abs/2402.05574v2) [2] Shouvanik Chakrabarti et al. (2021). A Threshold for Quantum Advantage in Derivative Pricing, Quantum 5, 463.