{ "stage": 3, "task": "Campaign Outcome Simulation (P(at least 3 of 6 selected customers subscribe))", "model_conditional_disclaimer": "All Stage 3 conversion probabilities are conditional on the frozen Stage 1 predicted probabilities for the 6 selected customers and assume independent customer responses; they are not observed ground-truth test outcomes.", "selected_customer_ids": [ "C0031", "C0170", "C0079", "C0265", "C0318", "C0256" ], "qubit_to_customer_mapping": [ { "qubit_index": 0, "bitstring_char_position_from_right": 0, "bitstring_char_index_in_6char_str": 5, "customer_id": "C0031", "predicted_probability": 0.2979882203218803, "ry_angle_rad": 1.1548851896798047, "expected_deposit_value": 13.0, "contact_cost": 2.0, "capacity_units": 1.0, "expected_net_value": 1.8738468641844443 }, { "qubit_index": 1, "bitstring_char_position_from_right": 1, "bitstring_char_index_in_6char_str": 4, "customer_id": "C0170", "predicted_probability": 0.2476515995507443, "ry_angle_rad": 1.0417656083294446, "expected_deposit_value": 22.0, "contact_cost": 1.0, "capacity_units": 1.0, "expected_net_value": 4.448335190116375 }, { "qubit_index": 2, "bitstring_char_position_from_right": 2, "bitstring_char_index_in_6char_str": 3, "customer_id": "C0079", "predicted_probability": 0.2401919300152541, "ry_angle_rad": 1.0243947119795815, "expected_deposit_value": 24.0, "contact_cost": 3.0, "capacity_units": 3.0, "expected_net_value": 2.7646063203660987 }, { "qubit_index": 3, "bitstring_char_position_from_right": 3, "bitstring_char_index_in_6char_str": 2, "customer_id": "C0265", "predicted_probability": 0.2146196879868689, "ry_angle_rad": 0.9633644404890409, "expected_deposit_value": 23.0, "contact_cost": 4.0, "capacity_units": 4.0, "expected_net_value": 0.936252823697985 }, { "qubit_index": 4, "bitstring_char_position_from_right": 4, "bitstring_char_index_in_6char_str": 1, "customer_id": "C0318", "predicted_probability": 0.19792520406404, "ry_angle_rad": 0.9220980759313568, "expected_deposit_value": 21.0, "contact_cost": 3.0, "capacity_units": 3.0, "expected_net_value": 1.1564292853448395 }, { "qubit_index": 5, "bitstring_char_position_from_right": 5, "bitstring_char_index_in_6char_str": 0, "customer_id": "C0256", "predicted_probability": 0.194275255852506, "ry_angle_rad": 0.9129052102278539, "expected_deposit_value": 22.0, "contact_cost": 1.0, "capacity_units": 4.0, "expected_net_value": 3.274055628755132 } ], "bit_ordering_convention": "Qiskit little-endian: in a 6-bit measurement string 'b5 b4 b3 b2 b1 b0', the rightmost character b0 is qubit q0 (selected customer index 0) and the leftmost character b5 is qubit q5 (selected customer index 5).", "summary_estimates_prob_at_least_3": { "exact_poisson_binomial_dp": 0.1412884529348319, "exact_64_outcome_enumeration": 0.14128845293483186, "ideal_quantum_statevector": 0.14128845293483192, "noiseless_quantum_shots_100k": 0.14047, "classical_monte_carlo_100k": 0.14102, "noisy_gate_depolarizing_only_100k": 0.14863, "noisy_readout_error_only_100k": 0.15369, "noisy_combined_depol_and_readout_100k": 0.16447 }, "exact_classical_dp": { "pmf_by_hamming_weight": { "0": 0.2036802335564143, "1": 0.3729231181536828, "2": 0.28210819535507103, "3": 0.1129204871895556, "4": 0.025235181988729417, "5": 0.0029865031353968775, "6": 0.00014628062114998796 }, "prob_at_least_3": 0.1412884529348319, "normalization_sum": 0.9999999999999999 }, "ideal_quantum_statevector": { "prob_at_least_3": 0.14128845293483192, "abs_error_vs_exact_dp": 2.7755575615628914e-17, "normalization_sum": 1.0, "pmf_by_hamming_weight": { "0": 0.20368023355641426, "1": 0.3729231181536828, "2": 0.2821081953550711, "3": 0.11292048718955562, "4": 0.02523518198872942, "5": 0.0029865031353968775, "6": 0.00014628062114998796 } }, "noiseless_quantum_shot_simulation": { "shots": 100000, "success_shots_weight_ge_3": 14047, "estimated_prob_at_least_3": 0.14047, "abs_error_vs_exact": 0.0008184529348318759, "signed_error_vs_exact": -0.0008184529348318759, "standard_error": 0.0010988092605179482, "ci_95_clopper_pearson": [ 0.13832182244302998, 0.14263905310246577 ], "counts_by_hamming_weight": { "0": 20625, "1": 37144, "2": 28184, "3": 11278, "4": 2472, "5": 284, "6": 13 }, "pmf_by_hamming_weight": { "0": 0.20625, "1": 0.37144, "2": 0.28184, "3": 0.11278, "4": 0.02472, "5": 0.00284, "6": 0.00013 }, "raw_bitstring_counts": { "000000": 20625, "000001": 8496, "000010": 6618, "000011": 2876, "000100": 6475, "000101": 2722, "000110": 2143, "000111": 876, "001000": 5552, "001001": 2354, "001010": 1756, "001011": 808, "001100": 1764, "001101": 783, "001110": 605, "001111": 238, "010000": 5057, "010001": 2122, "010010": 1672, "010011": 680, "010100": 1590, "010101": 675, "010110": 528, "010111": 213, "011000": 1391, "011001": 619, "011010": 488, "011011": 182, "011100": 444, "011101": 183, "011110": 133, "011111": 65, "100000": 4946, "100001": 2114, "100010": 1592, "100011": 633, "100100": 1549, "100101": 647, "100110": 538, "100111": 221, "101000": 1319, "101001": 552, "101010": 442, "101011": 193, "101100": 401, "101101": 152, "101110": 122, "101111": 54, "110000": 1220, "110001": 486, "110010": 387, "110011": 166, "110100": 342, "110101": 162, "110110": 144, "110111": 49, "111000": 344, "111001": 134, "111010": 117, "111011": 45, "111100": 112, "111101": 46, "111110": 25, "111111": 13 } }, "classical_monte_carlo_simulation": { "trials": 100000, "seed": 42, "success_trials_weight_ge_3": 14102, "estimated_prob_at_least_3": 0.14102, "abs_error_vs_exact": 0.00026845293483188093, "signed_error_vs_exact": -0.00026845293483188093, "standard_error": 0.0011006060130673465, "ci_95_clopper_pearson": [ 0.13886828496589598, 0.14319255854044916 ], "pmf_by_hamming_weight": { "0": 0.20361, "1": 0.3732, "2": 0.28217, "3": 0.11234, "4": 0.0256, "5": 0.00296, "6": 0.00012 } }, "synthetic_noise_experiments": { "gate_depolarizing_only": { "shots": 100000, "success_shots_weight_ge_3": 14863, "estimated_prob_at_least_3": 0.14863, "abs_error_vs_exact": 0.007341547065168125, "signed_error_vs_exact": 0.007341547065168125, "standard_error": 0.0011248960978685987, "ci_95_clopper_pearson": [ 0.1464304571795505, 0.1508499431186139 ], "counts_by_hamming_weight": { "0": 19521, "1": 36752, "2": 28864, "3": 11817, "4": 2726, "5": 300, "6": 20 }, "pmf_by_hamming_weight": { "0": 0.19521, "1": 0.36752, "2": 0.28864, "3": 0.11817, "4": 0.02726, "5": 0.003, "6": 0.0002 }, "raw_bitstring_counts": { "000000": 19521, "000001": 8515, "000010": 6691, "000011": 2887, "000100": 6421, "000101": 2813, "000110": 2144, "000111": 961, "001000": 5531, "001001": 2376, "001010": 1913, "001011": 808, "001100": 1831, "001101": 749, "001110": 578, "001111": 248, "010000": 4798, "010001": 2134, "010010": 1690, "010011": 710, "010100": 1619, "010101": 720, "010110": 537, "010111": 255, "011000": 1413, "011001": 629, "011010": 494, "011011": 199, "011100": 467, "011101": 188, "011110": 159, "011111": 62, "100000": 4796, "100001": 2211, "100010": 1631, "100011": 730, "100100": 1608, "100101": 682, "100110": 547, "100111": 264, "101000": 1356, "101001": 625, "101010": 438, "101011": 204, "101100": 476, "101101": 190, "101110": 148, "101111": 66, "110000": 1238, "110001": 523, "110010": 405, "110011": 169, "110100": 393, "110101": 161, "110110": 141, "110111": 47, "111000": 345, "111001": 142, "111010": 131, "111011": 49, "111100": 127, "111101": 40, "111110": 36, "111111": 20 }, "analytical_noisy_prob_at_least_3": 0.14928271907293786, "noise_specification": "1-qubit depolarizing_error(p=0.02) on all 'ry' gates; zero readout error." }, "readout_error_only": { "shots": 100000, "success_shots_weight_ge_3": 15369, "estimated_prob_at_least_3": 0.15369, "abs_error_vs_exact": 0.012401547065168106, "signed_error_vs_exact": 0.012401547065168106, "standard_error": 0.001140479653040772, "ci_95_clopper_pearson": [ 0.15145976762562344, 0.15594033807004665 ], "counts_by_hamming_weight": { "0": 19301, "1": 36565, "2": 28765, "3": 12102, "4": 2897, "5": 359, "6": 11 }, "pmf_by_hamming_weight": { "0": 0.19301, "1": 0.36565, "2": 0.28765, "3": 0.12102, "4": 0.02897, "5": 0.00359, "6": 0.00011 }, "raw_bitstring_counts": { "000000": 19301, "000001": 8252, "000010": 6600, "000011": 2899, "000100": 6145, "000101": 2790, "000110": 2107, "000111": 959, "001000": 5501, "001001": 2479, "001010": 1863, "001011": 835, "001100": 1725, "001101": 803, "001110": 609, "001111": 266, "010000": 5155, "010001": 2163, "010010": 1678, "010011": 713, "010100": 1631, "010101": 724, "010110": 536, "010111": 240, "011000": 1432, "011001": 609, "011010": 483, "011011": 216, "011100": 500, "011101": 206, "011110": 190, "011111": 76, "100000": 4912, "100001": 2097, "100010": 1618, "100011": 696, "100100": 1547, "100101": 707, "100110": 549, "100111": 253, "101000": 1447, "101001": 630, "101010": 508, "101011": 216, "101100": 523, "101101": 234, "101110": 170, "101111": 70, "110000": 1289, "110001": 526, "110010": 422, "110011": 163, "110100": 410, "110101": 189, "110110": 140, "110111": 55, "111000": 360, "111001": 168, "111010": 121, "111011": 58, "111100": 125, "111101": 57, "111110": 43, "111111": 11 }, "analytical_noisy_prob_at_least_3": 0.15335212258214423, "noise_specification": "Symmetric ReadoutError(p(1|0)=p(0|1)=0.015) on all measurements; zero gate error." }, "combined_depolarizing_and_readout": { "shots": 100000, "success_shots_weight_ge_3": 16447, "estimated_prob_at_least_3": 0.16447, "abs_error_vs_exact": 0.023181547065168118, "signed_error_vs_exact": 0.023181547065168118, "standard_error": 0.0011722611445407545, "ci_95_clopper_pearson": [ 0.16217716525453124, 0.16678231302797272 ], "counts_by_hamming_weight": { "0": 18383, "1": 35846, "2": 29324, "3": 12956, "4": 3023, "5": 447, "6": 21 }, "pmf_by_hamming_weight": { "0": 0.18383, "1": 0.35846, "2": 0.29324, "3": 0.12956, "4": 0.03023, "5": 0.00447, "6": 0.00021 }, "raw_bitstring_counts": { "000000": 18383, "000001": 8164, "000010": 6547, "000011": 2813, "000100": 6050, "000101": 2822, "000110": 2102, "000111": 1029, "001000": 5217, "001001": 2498, "001010": 1943, "001011": 887, "001100": 1892, "001101": 841, "001110": 629, "001111": 286, "010000": 5092, "010001": 2274, "010010": 1771, "010011": 806, "010100": 1618, "010101": 803, "010110": 663, "010111": 259, "011000": 1482, "011001": 668, "011010": 511, "011011": 230, "011100": 514, "011101": 184, "011110": 162, "011111": 69, "100000": 4776, "100001": 2139, "100010": 1651, "100011": 782, "100100": 1650, "100101": 697, "100110": 592, "100111": 267, "101000": 1348, "101001": 676, "101010": 514, "101011": 246, "101100": 495, "101101": 218, "101110": 164, "101111": 102, "110000": 1321, "110001": 583, "110010": 459, "110011": 213, "110100": 453, "110101": 185, "110110": 170, "110111": 81, "111000": 354, "111001": 158, "111010": 142, "111011": 72, "111100": 139, "111101": 76, "111110": 47, "111111": 21 }, "analytical_noisy_prob_at_least_3": 0.16138129451522248, "noise_specification": "Combined 1-qubit depolarizing_error(p=0.02) on 'ry' + symmetric ReadoutError(p=0.015) on measurements (unmitigated)." }, "physical_interpretation": "Both single-qubit depolarizing noise and symmetric readout assignment error contract each qubit's marginal Bernoulli parameter p_k toward 0.5. Whenever the customer probabilities p_k lie below 0.5, this contraction systematically inflates the unmitigated tail probability P(Hamming weight >= 3)." }, "extreme_input_verification": { "all_zeros": { "expected": 0.0, "dp_exact": 0.0, "statevector_exact": 0.0, "abs_diff": 0.0 }, "all_ones": { "expected": 1.0, "dp_exact": 1.0, "statevector_exact": 1.0, "abs_diff": 0.0 }, "three_ones_three_zeros": { "expected": 1.0, "dp_exact": 1.0, "statevector_exact": 1.0, "abs_diff": 0.0 }, "two_ones_four_zeros": { "expected": 0.0, "dp_exact": 0.0, "statevector_exact": 0.0, "abs_diff": 0.0 }, "uniform_half": { "expected": 0.65625, "dp_exact": 0.65625, "statevector_exact": 0.6562500000000002, "abs_diff": 2.220446049250313e-16 } }, "analytical_sensitivity_derivatives": { "identity": "dP(S >= 3)/dp_i = P(S_{-i} = 2)", "analytical_gradient_wrt_p": { "C0031": 0.2286429674228685, "C0170": 0.24114106231171617, "C0079": 0.2429211259760477, "C0265": 0.24880941592496986, "C0318": 0.2524528923938858, "C0256": 0.2532267789594629 }, "central_diff_gradient_wrt_p": { "C0031": 0.22864296738479575, "C0170": 0.24114106231241414, "C0079": 0.24292112596444723, "C0265": 0.24880941594504588, "C0318": 0.25245289239614066, "C0256": 0.2532267789759589 }, "min_derivative": 0.2286429674228685, "max_derivative": 0.2532267789594629, "sum_of_derivatives_uniform_shift_rate": 1.4671942429889508, "max_abs_diff_vs_central_difference": 3.8072739405592415e-11 }, "error_propagation_and_sensitivity": { "stage2_objective_gradient_wrt_p": { "C0031": 13.0, "C0170": 22.0, "C0079": 24.0, "C0265": 23.0, "C0318": 21.0, "C0256": 22.0 }, "stage3_analytical_derivative_identity": "dP(S >= 3)/dp_i = P(S_{-i} = 2)", "stage3_conversion_gradient_wrt_p": { "C0031": 0.2286429674228685, "C0170": 0.24114106231171617, "C0079": 0.2429211259760477, "C0265": 0.24880941592496986, "C0318": 0.2524528923938858, "C0256": 0.2532267789594629 }, "stage3_central_diff_gradient_wrt_p": { "C0031": 0.22864296738479575, "C0170": 0.24114106231241414, "C0079": 0.24292112596444723, "C0265": 0.24880941594504588, "C0318": 0.25245289239614066, "C0256": 0.2532267789759589 }, "stage3_min_derivative": 0.2286429674228685, "stage3_max_derivative": 0.2532267789594629, "stage3_sum_of_derivatives_uniform_shift_rate": 1.4671942429889508, "stage3_max_abs_diff_analytical_vs_central_diff": 3.8072739405592415e-11, "calibration_shift_scenarios": { "delta_p_-0.05": { "shifted_mean_prob": 0.18210864963188225, "stage2_expected_net_value": 8.203526112464873, "stage3_prob_at_least_3": 0.07713031091878561 }, "delta_p_-0.02": { "shifted_mean_prob": 0.21210864963188228, "stage2_expected_net_value": 11.953526112464875, "stage3_prob_at_least_3": 0.11336803533410501 }, "delta_p_+0.00": { "shifted_mean_prob": 0.23210864963188227, "stage2_expected_net_value": 14.453526112464875, "stage3_prob_at_least_3": 0.1412884529348319 }, "delta_p_+0.02": { "shifted_mean_prob": 0.2521086496318823, "stage2_expected_net_value": 16.95352611246487, "stage3_prob_at_least_3": 0.17197687567155281 }, "delta_p_+0.05": { "shifted_mean_prob": 0.2821086496318823, "stage2_expected_net_value": 20.70352611246487, "stage3_prob_at_least_3": 0.22262805219974524 } }, "interpretation": "Stage 2 expected net value is linear in predicted probabilities (dZ/dp_i = expected_deposit_value_i), whereas by multilinear expansion of independent Bernoulli variables, Stage 3 target probability satisfies the exact identity dP(S >= 3)/dp_i = P(S_{-i} = 2)." }, "circuit_resources": { "circuit_name": "Stage3_6Q_Bernoulli", "logical_qubits": 6, "physical_qubits": "N/A (local AerSimulator; 6 logical qubits)", "original_depth": 2, "transpiled_depth": 2, "original_gate_counts": { "ry": 6, "measure": 6 }, "transpiled_gate_counts": { "ry": 6, "measure": 6 }, "one_qubit_gate_count": 6, "two_qubit_gate_count": 0, "swap_count": 0, "bit_ordering_convention": "Qiskit little-endian: in a 6-bit measurement string 'b5 b4 b3 b2 b1 b0', the rightmost character b0 is qubit q0 (selected customer index 0) and the leftmost character b5 is qubit q5 (selected customer index 5).", "diagram_text": " \u250c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\u250c\u2500\u2510 \nq_0: \u2500\u2524 Ry(1.1549) \u251c\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524\u2514\u2565\u2518\u250c\u2500\u2510 \nq_1: \u2500\u2524 Ry(1.0418) \u251c\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 \u2551 \u2514\u2565\u2518\u250c\u2500\u2510 \nq_2: \u2500\u2524 Ry(1.0244) \u251c\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n \u250c\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 \u2551 \u2551 \u2514\u2565\u2518\u250c\u2500\u2510 \nq_3: \u2524 Ry(0.96336) \u251c\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\u2500\u2500\u2500\n \u2514\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 \u2551 \u2551 \u2551 \u2514\u2565\u2518\u250c\u2500\u2510 \nq_4: \u2500\u2524 Ry(0.9221) \u251c\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\u2500\u2500\u2500\n \u250c\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2524 \u2551 \u2551 \u2551 \u2551 \u2514\u2565\u2518\u250c\u2500\u2510\nq_5: \u2524 Ry(0.91291) \u251c\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2500\u256b\u2500\u2524M\u251c\n \u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518 \u2551 \u2551 \u2551 \u2551 \u2551 \u2514\u2565\u2518\nc: 6/\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\u2550\u2569\u2550\n 0 1 2 3 4 5 " }, "runtime_sec": 0.24474062491208315, "environment": { "python_executable": "/Users/nafeeskhan/Desktop/atomic hack/.venv/bin/python", "python_version": "3.13.13", "cwd": "/Users/nafeeskhan/Documents/Codex/2026-11-11/i/outputs/AtomiQ_Submission", "platform": "macOS-26.3-arm64-arm-64bit-Mach-O", "pid": 25585, "package_versions": { "numpy": "2.5.4", "scipy": "1.18.1", "pandas": "3.1.1", "openpyxl": "3.1.5", "scikit-learn": "1.9.1", "qiskit": "2.6.2", "qiskit-aer": "0.18.2", "matplotlib": "3.11.2", "pytest": "9.2.0" }, "seeds": { "workbook_sampling_seed": 42, "workbook_train_test_split_seed": 42, "stage1_cv_and_model_seed": 42, "stage2_qaoa_optimizer_and_simulator_seed": 42, "stage3_simulator_and_mc_seed": 42 } } }