FA-91201 / Quantum circuit simulation / Open access

Grover planner omits the initial rotation from the success angle · case 01

Success probabilities are reported for angle 2k theta, underestimating the achievable probability.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

After k iterations the state angle is (2k+1) theta, but the planner uses 2k theta.

VERIFIED REPAIR

Use sin^2((2k+1) theta).

Unsuccessful approach: The attempted repair uses the right angle but returns sin instead of sin squared.

Case contract

Input [N, M]. "invalid" if N < 1, M < 0 or M > N; "no-solution" if M == 0; if 2M >= N return {"iterations": 0, "success": M/N}; else theta = asin(sqrt(M/N)), k = floor(pi/(4 theta)) and success = sin^2((2k+1) theta). Probabilities rounded to 6 decimals.

Why this case matters

Choosing the Grover iteration count wrongly overshoots the amplitude peak and collapses the success probability.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    N, M = x
    if N < 1 or M < 0 or M > N:
        return 'invalid'
    if M == 0:
        return 'no-solution'
    if 2 * M >= N:
        return {'iterations': 0, 'success': round(M / N, 6)}
    theta = math.asin(math.sqrt(M / N))
    k = math.floor(math.pi / (4 * theta))
    return {'iterations': k, 'success': round(math.sin((2 * k) * theta) ** 2, 6)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: odd N just under half', [5, 2], {'iterations': 1, 'success': 0.784}], ['regression: single marked N=4', [4, 1], {'iterations': 1, 'success': 1.0}], ['regression: N=7 M=3', [7, 3], {'iterations': 1, 'success': 0.708455}], ['control: half marked N=4 M=2', [4, 2], {'iterations': 0, 'success': 0.5}], ['control: no marked items', [8, 0], 'no-solution'], ['control: M greater than N', [4, 5], 'invalid'], ['control: negative M', [4, -1], 'invalid']], [['regression: N=9 M=4', [9, 4], {'iterations': 1, 'success': 0.663923}], ['regression: N=8 M=1', [8, 1], {'iterations': 2, 'success': 0.945312}], ['regression: N=8 M=3', [8, 3], {'iterations': 1, 'success': 0.84375}], ['control: empty space', [0, 0], 'invalid'], ['control: all marked', [8, 8], {'iterations': 0, 'success': 1.0}], ['control: one over the space', [4, 5], 'invalid'], ['control: all marked N=1', [1, 1], {'iterations': 0, 'success': 1.0}]], [['regression: N=8 M=3', [8, 3], {'iterations': 1, 'success': 0.84375}], ['regression: N=16 M=1', [16, 1], {'iterations': 3, 'success': 0.961319}], ['regression: N=16 M=2', [16, 2], {'iterations': 2, 'success': 0.945312}], ['control: all marked N=3', [3, 3], {'iterations': 0, 'success': 1.0}], ['control: N=8 M=5', [8, 5], {'iterations': 0, 'success': 0.625}], ['control: N=8 M=7', [8, 7], {'iterations': 0, 'success': 0.875}], ['control: N=16 M=13', [16, 13], {'iterations': 0, 'success': 0.8125}]], [['regression: N=16 M=3', [16, 3], {'iterations': 1, 'success': 0.949219}], ['regression: N=16 M=5', [16, 5], {'iterations': 1, 'success': 0.957031}], ['regression: N=16 M=2', [16, 2], {'iterations': 2, 'success': 0.945312}], ['control: half marked N=4 M=2', [4, 2], {'iterations': 0, 'success': 0.5}], ['control: no marked items', [8, 0], 'no-solution'], ['control: M greater than N', [4, 5], 'invalid'], ['control: negative M', [4, -1], 'invalid']], [['regression: N=32 M=1', [32, 1], {'iterations': 4, 'success': 0.999182}], ['regression: N=32 M=2', [32, 2], {'iterations': 3, 'success': 0.961319}], ['regression: N=16 M=5', [16, 5], {'iterations': 1, 'success': 0.957031}], ['control: empty space', [0, 0], 'invalid'], ['control: all marked', [8, 8], {'iterations': 0, 'success': 1.0}], ['control: one over the space', [4, 5], 'invalid'], ['control: all marked N=1', [1, 1], {'iterations': 0, 'success': 1.0}]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression: odd N just under half	`{'iterations': 1, 'success': 0.96}`	`{'iterations': 1, 'success': 0.784}`	Failed
regression: single marked N=4	`{'iterations': 1, 'success': 0.75}`	`{'iterations': 1, 'success': 1.0}`	Failed
regression: N=7 M=3	`{'iterations': 1, 'success': 0.979592}`	`{'iterations': 1, 'success': 0.708455}`	Failed
control: half marked N=4 M=2	`{'iterations': 0, 'success': 0.5}`	`{'iterations': 0, 'success': 0.5}`	Passed
control: no marked items	`no-solution`	`no-solution`	Passed
control: M greater than N	`invalid`	`invalid`	Passed
control: negative M	`invalid`	`invalid`	Passed

SHA-256 / a469bbe6d5be01c48dfe87d3cc4e83c306e2851c930a8ea4b718796334a03b9d

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    N, M = x
    if N < 1 or M < 0 or M > N:
        return 'invalid'
    if M == 0:
        return 'no-solution'
    if 2 * M >= N:
        return {'iterations': 0, 'success': round(M / N, 6)}
    theta = math.asin(math.sqrt(M / N))
    k = math.floor(math.pi / (4 * theta))
    return {'iterations': k, 'success': round(math.sin((2 * k + 1) * theta), 6)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: odd N just under half', [5, 2], {'iterations': 1, 'success': 0.784}], ['regression: single marked N=4', [4, 1], {'iterations': 1, 'success': 1.0}], ['regression: N=7 M=3', [7, 3], {'iterations': 1, 'success': 0.708455}], ['control: half marked N=4 M=2', [4, 2], {'iterations': 0, 'success': 0.5}], ['control: no marked items', [8, 0], 'no-solution'], ['control: M greater than N', [4, 5], 'invalid'], ['control: negative M', [4, -1], 'invalid']], [['regression: N=9 M=4', [9, 4], {'iterations': 1, 'success': 0.663923}], ['regression: N=8 M=1', [8, 1], {'iterations': 2, 'success': 0.945312}], ['regression: N=8 M=3', [8, 3], {'iterations': 1, 'success': 0.84375}], ['control: empty space', [0, 0], 'invalid'], ['control: all marked', [8, 8], {'iterations': 0, 'success': 1.0}], ['control: one over the space', [4, 5], 'invalid'], ['control: all marked N=1', [1, 1], {'iterations': 0, 'success': 1.0}]], [['regression: N=8 M=3', [8, 3], {'iterations': 1, 'success': 0.84375}], ['regression: N=16 M=1', [16, 1], {'iterations': 3, 'success': 0.961319}], ['regression: N=16 M=2', [16, 2], {'iterations': 2, 'success': 0.945312}], ['control: all marked N=3', [3, 3], {'iterations': 0, 'success': 1.0}], ['control: N=8 M=5', [8, 5], {'iterations': 0, 'success': 0.625}], ['control: N=8 M=7', [8, 7], {'iterations': 0, 'success': 0.875}], ['control: N=16 M=13', [16, 13], {'iterations': 0, 'success': 0.8125}]], [['regression: N=16 M=3', [16, 3], {'iterations': 1, 'success': 0.949219}], ['regression: N=16 M=5', [16, 5], {'iterations': 1, 'success': 0.957031}], ['regression: N=16 M=2', [16, 2], {'iterations': 2, 'success': 0.945312}], ['control: half marked N=4 M=2', [4, 2], {'iterations': 0, 'success': 0.5}], ['control: no marked items', [8, 0], 'no-solution'], ['control: M greater than N', [4, 5], 'invalid'], ['control: negative M', [4, -1], 'invalid']], [['regression: N=32 M=1', [32, 1], {'iterations': 4, 'success': 0.999182}], ['regression: N=32 M=2', [32, 2], {'iterations': 3, 'success': 0.961319}], ['regression: N=16 M=5', [16, 5], {'iterations': 1, 'success': 0.957031}], ['control: empty space', [0, 0], 'invalid'], ['control: all marked', [8, 8], {'iterations': 0, 'success': 1.0}], ['control: one over the space', [4, 5], 'invalid'], ['control: all marked N=1', [1, 1], {'iterations': 0, 'success': 1.0}]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression: odd N just under half	`{'iterations': 1, 'success': 0.885438}`	`{'iterations': 1, 'success': 0.784}`	Failed
regression: single marked N=4	`{'iterations': 1, 'success': 1.0}`	`{'iterations': 1, 'success': 1.0}`	Passed
regression: N=7 M=3	`{'iterations': 1, 'success': 0.841698}`	`{'iterations': 1, 'success': 0.708455}`	Failed
control: half marked N=4 M=2	`{'iterations': 0, 'success': 0.5}`	`{'iterations': 0, 'success': 0.5}`	Passed
control: no marked items	`no-solution`	`no-solution`	Passed
control: M greater than N	`invalid`	`invalid`	Passed
control: negative M	`invalid`	`invalid`	Passed

SHA-256 / d0ef7748952228d301b95952cb61765ed9ff2638d9a3e0432bdcdd64537feb88

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    N, M = x
    if N < 1 or M < 0 or M > N:
        return 'invalid'
    if M == 0:
        return 'no-solution'
    if 2 * M >= N:
        return {'iterations': 0, 'success': round(M / N, 6)}
    theta = math.asin(math.sqrt(M / N))
    k = math.floor(math.pi / (4 * theta))
    return {'iterations': k, 'success': round(math.sin((2 * k + 1) * theta) ** 2, 6)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: odd N just under half', [5, 2], {'iterations': 1, 'success': 0.784}], ['regression: single marked N=4', [4, 1], {'iterations': 1, 'success': 1.0}], ['regression: N=7 M=3', [7, 3], {'iterations': 1, 'success': 0.708455}], ['control: half marked N=4 M=2', [4, 2], {'iterations': 0, 'success': 0.5}], ['control: no marked items', [8, 0], 'no-solution'], ['control: M greater than N', [4, 5], 'invalid'], ['control: negative M', [4, -1], 'invalid']], [['regression: N=9 M=4', [9, 4], {'iterations': 1, 'success': 0.663923}], ['regression: N=8 M=1', [8, 1], {'iterations': 2, 'success': 0.945312}], ['regression: N=8 M=3', [8, 3], {'iterations': 1, 'success': 0.84375}], ['control: empty space', [0, 0], 'invalid'], ['control: all marked', [8, 8], {'iterations': 0, 'success': 1.0}], ['control: one over the space', [4, 5], 'invalid'], ['control: all marked N=1', [1, 1], {'iterations': 0, 'success': 1.0}]], [['regression: N=8 M=3', [8, 3], {'iterations': 1, 'success': 0.84375}], ['regression: N=16 M=1', [16, 1], {'iterations': 3, 'success': 0.961319}], ['regression: N=16 M=2', [16, 2], {'iterations': 2, 'success': 0.945312}], ['control: all marked N=3', [3, 3], {'iterations': 0, 'success': 1.0}], ['control: N=8 M=5', [8, 5], {'iterations': 0, 'success': 0.625}], ['control: N=8 M=7', [8, 7], {'iterations': 0, 'success': 0.875}], ['control: N=16 M=13', [16, 13], {'iterations': 0, 'success': 0.8125}]], [['regression: N=16 M=3', [16, 3], {'iterations': 1, 'success': 0.949219}], ['regression: N=16 M=5', [16, 5], {'iterations': 1, 'success': 0.957031}], ['regression: N=16 M=2', [16, 2], {'iterations': 2, 'success': 0.945312}], ['control: half marked N=4 M=2', [4, 2], {'iterations': 0, 'success': 0.5}], ['control: no marked items', [8, 0], 'no-solution'], ['control: M greater than N', [4, 5], 'invalid'], ['control: negative M', [4, -1], 'invalid']], [['regression: N=32 M=1', [32, 1], {'iterations': 4, 'success': 0.999182}], ['regression: N=32 M=2', [32, 2], {'iterations': 3, 'success': 0.961319}], ['regression: N=16 M=5', [16, 5], {'iterations': 1, 'success': 0.957031}], ['control: empty space', [0, 0], 'invalid'], ['control: all marked', [8, 8], {'iterations': 0, 'success': 1.0}], ['control: one over the space', [4, 5], 'invalid'], ['control: all marked N=1', [1, 1], {'iterations': 0, 'success': 1.0}]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression: odd N just under half	`{'iterations': 1, 'success': 0.784}`	`{'iterations': 1, 'success': 0.784}`	Passed
regression: single marked N=4	`{'iterations': 1, 'success': 1.0}`	`{'iterations': 1, 'success': 1.0}`	Passed
regression: N=7 M=3	`{'iterations': 1, 'success': 0.708455}`	`{'iterations': 1, 'success': 0.708455}`	Passed
control: half marked N=4 M=2	`{'iterations': 0, 'success': 0.5}`	`{'iterations': 0, 'success': 0.5}`	Passed
control: no marked items	`no-solution`	`no-solution`	Passed
control: M greater than N	`invalid`	`invalid`	Passed
control: negative M	`invalid`	`invalid`	Passed

SHA-256 / fa4e071c071fe739d089653857396ae84e2cb51f6340bcd5d3a141f4afa6c381

Verification & scope

A deterministic bounded teaching model with a stipulated toy contract; amplitudes are rounded to fixed decimals for strict JSON output. It is not a production quantum SDK and claims no standards conformance. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:51:33.713948+00:00.

Case digest / 4990702aa9c94c9f74b677dc053a8e118b673e71a1efe0a8be53e5545a000d32