FA-91071 / Quantum circuit simulation / Open access

Bloch vector conjugates the |1> amplitude · case 01

The |+i> state is drawn at y = -1.

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

ROOT CAUSE

The coherence term is a * conj(b) instead of conj(a) * b, flipping the sign of the y component.

VERIFIED REPAIR

Use conj(a) * b for the coherence.

Unsuccessful approach: The attempted repair conjugates the product a * b, which is wrong whenever a is not real.

Case contract

Input [[ar, ai], [br, bi]] for a|0> + b|1> (possibly unnormalized). Return the Bloch vector [2Re(conj(a) b), 2Im(conj(a) b), |a|^2 - |b|^2] divided by |a|^2 + |b|^2, rounded to 6 decimals, or "zero-state" when the squared norm is below 1e-12.

Why this case matters

Bloch coordinates feed visualizers and single-qubit tomography checks; sign or scale slips put states on the wrong axis.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    (ar, ai), (br, bi) = x
    a, b = complex(ar, ai), complex(br, bi)
    nrm = abs(a) ** 2 + abs(b) ** 2
    if nrm < 1e-12:
        return 'zero-state'
    cross = a * b.conjugate()
    bx = 2 * cross.real / nrm
    by = 2 * cross.imag / nrm
    bz = (abs(a) ** 2 - abs(b) ** 2) / nrm
    return [round(v, 6) + 0.0 for v in (bx, by, bz)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: plus i', [[0.707107, 0], [0, 0.707107]], [0.0, 1.0, 0.0]], ['regression: minus i', [[0.707107, 0], [0, -0.707107]], [0.0, -1.0, 0.0]], ['regression: unnormalized', [[3, 0], [0, 4]], [0.0, 0.96, -0.28]], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state']], [['regression: random qubit 0', [[-1.176, -0.41], [1.26, -1.053]], [-0.494424, 0.826337, -0.26965]], ['regression: random qubit 1', [[1.052, -0.843], [-0.499, 1.97]], [-0.735014, 0.555478, -0.388843]], ['regression: unnormalized', [[3, 0], [0, 4]], [0.0, 0.96, -0.28]], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state'], ['control: noise on one amplitude', [[3e-08, 0], [0, 0]], 'zero-state'], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]]], [['regression: random qubit 3', [[1.769, -1.743], [0.441, -0.443]], [0.47339, -0.004576, 0.880841]], ['regression: random qubit 4', [[-1.063, 1.137], [0.11, -1.819]], [-0.760893, 0.629754, -0.156369]], ['regression: random qubit 1', [[1.052, -0.843], [-0.499, 1.97]], [-0.735014, 0.555478, -0.388843]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state'], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]]], [['regression: random qubit 6', [[1.624, -0.932], [-1.233, 0.708]], [-0.963265, 0.00023, 0.268554]], ['regression: random qubit 7', [[-1.454, -0.834], [1.157, -0.946]], [-0.354262, 0.928143, 0.114234]], ['regression: random qubit 3', [[1.769, -1.743], [0.441, -0.443]], [0.47339, -0.004576, 0.880841]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state'], ['control: noise on one amplitude', [[3e-08, 0], [0, 0]], 'zero-state'], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]]], [['regression: random qubit 9', [[0.467, -1.484], [-1.959, -0.631]], [0.006475, -0.962062, -0.272753]], ['regression: random qubit 10', [[-1.198, -0.992], [1.144, -1.926]], [0.145232, 0.925635, -0.349439]], ['regression: random qubit 5', [[1.781, 0.313], [-0.534, -1.473]], [-0.493327, -0.858114, 0.142371]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state'], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state']]]
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: plus i	`[0.0, -1.0, 0.0]`	`[0.0, 1.0, 0.0]`	Failed
regression: minus i	`[0.0, 1.0, 0.0]`	`[0.0, -1.0, 0.0]`	Failed
regression: unnormalized	`[0.0, -0.96, -0.28]`	`[0.0, 0.96, -0.28]`	Failed
control: ket zero	`[0.0, 0.0, 1.0]`	`[0.0, 0.0, 1.0]`	Passed
control: ket one	`[0.0, 0.0, -1.0]`	`[0.0, 0.0, -1.0]`	Passed
control: plus	`[1.0, 0.0, 0.0]`	`[1.0, 0.0, 0.0]`	Passed
control: zero vector	`zero-state`	`zero-state`	Passed

SHA-256 / 0ca175b9abb8d42b6cb2a42e0e6322cc2fd8e5c8b8f055c15af462fa05d4a842

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    (ar, ai), (br, bi) = x
    a, b = complex(ar, ai), complex(br, bi)
    nrm = abs(a) ** 2 + abs(b) ** 2
    if nrm < 1e-12:
        return 'zero-state'
    cross = (a * b).conjugate()
    bx = 2 * cross.real / nrm
    by = 2 * cross.imag / nrm
    bz = (abs(a) ** 2 - abs(b) ** 2) / nrm
    return [round(v, 6) + 0.0 for v in (bx, by, bz)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: plus i', [[0.707107, 0], [0, 0.707107]], [0.0, 1.0, 0.0]], ['regression: minus i', [[0.707107, 0], [0, -0.707107]], [0.0, -1.0, 0.0]], ['regression: unnormalized', [[3, 0], [0, 4]], [0.0, 0.96, -0.28]], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state']], [['regression: random qubit 0', [[-1.176, -0.41], [1.26, -1.053]], [-0.494424, 0.826337, -0.26965]], ['regression: random qubit 1', [[1.052, -0.843], [-0.499, 1.97]], [-0.735014, 0.555478, -0.388843]], ['regression: unnormalized', [[3, 0], [0, 4]], [0.0, 0.96, -0.28]], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state'], ['control: noise on one amplitude', [[3e-08, 0], [0, 0]], 'zero-state'], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]]], [['regression: random qubit 3', [[1.769, -1.743], [0.441, -0.443]], [0.47339, -0.004576, 0.880841]], ['regression: random qubit 4', [[-1.063, 1.137], [0.11, -1.819]], [-0.760893, 0.629754, -0.156369]], ['regression: random qubit 1', [[1.052, -0.843], [-0.499, 1.97]], [-0.735014, 0.555478, -0.388843]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state'], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]]], [['regression: random qubit 6', [[1.624, -0.932], [-1.233, 0.708]], [-0.963265, 0.00023, 0.268554]], ['regression: random qubit 7', [[-1.454, -0.834], [1.157, -0.946]], [-0.354262, 0.928143, 0.114234]], ['regression: random qubit 3', [[1.769, -1.743], [0.441, -0.443]], [0.47339, -0.004576, 0.880841]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state'], ['control: noise on one amplitude', [[3e-08, 0], [0, 0]], 'zero-state'], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]]], [['regression: random qubit 9', [[0.467, -1.484], [-1.959, -0.631]], [0.006475, -0.962062, -0.272753]], ['regression: random qubit 10', [[-1.198, -0.992], [1.144, -1.926]], [0.145232, 0.925635, -0.349439]], ['regression: random qubit 5', [[1.781, 0.313], [-0.534, -1.473]], [-0.493327, -0.858114, 0.142371]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state'], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state']]]
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: plus i	`[0.0, -1.0, 0.0]`	`[0.0, 1.0, 0.0]`	Failed
regression: minus i	`[0.0, 1.0, 0.0]`	`[0.0, -1.0, 0.0]`	Failed
regression: unnormalized	`[0.0, -0.96, -0.28]`	`[0.0, 0.96, -0.28]`	Failed
control: ket zero	`[0.0, 0.0, 1.0]`	`[0.0, 0.0, 1.0]`	Passed
control: ket one	`[0.0, 0.0, -1.0]`	`[0.0, 0.0, -1.0]`	Passed
control: plus	`[1.0, 0.0, 0.0]`	`[1.0, 0.0, 0.0]`	Passed
control: zero vector	`zero-state`	`zero-state`	Passed

SHA-256 / b1ada6083b3923e9ef9d61562cf2a234dd096f03f40e92fa7a2e984ee01b48db

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    (ar, ai), (br, bi) = x
    a, b = complex(ar, ai), complex(br, bi)
    nrm = abs(a) ** 2 + abs(b) ** 2
    if nrm < 1e-12:
        return 'zero-state'
    cross = a.conjugate() * b
    bx = 2 * cross.real / nrm
    by = 2 * cross.imag / nrm
    bz = (abs(a) ** 2 - abs(b) ** 2) / nrm
    return [round(v, 6) + 0.0 for v in (bx, by, bz)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: plus i', [[0.707107, 0], [0, 0.707107]], [0.0, 1.0, 0.0]], ['regression: minus i', [[0.707107, 0], [0, -0.707107]], [0.0, -1.0, 0.0]], ['regression: unnormalized', [[3, 0], [0, 4]], [0.0, 0.96, -0.28]], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state']], [['regression: random qubit 0', [[-1.176, -0.41], [1.26, -1.053]], [-0.494424, 0.826337, -0.26965]], ['regression: random qubit 1', [[1.052, -0.843], [-0.499, 1.97]], [-0.735014, 0.555478, -0.388843]], ['regression: unnormalized', [[3, 0], [0, 4]], [0.0, 0.96, -0.28]], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state'], ['control: noise on one amplitude', [[3e-08, 0], [0, 0]], 'zero-state'], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]]], [['regression: random qubit 3', [[1.769, -1.743], [0.441, -0.443]], [0.47339, -0.004576, 0.880841]], ['regression: random qubit 4', [[-1.063, 1.137], [0.11, -1.819]], [-0.760893, 0.629754, -0.156369]], ['regression: random qubit 1', [[1.052, -0.843], [-0.499, 1.97]], [-0.735014, 0.555478, -0.388843]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state'], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]]], [['regression: random qubit 6', [[1.624, -0.932], [-1.233, 0.708]], [-0.963265, 0.00023, 0.268554]], ['regression: random qubit 7', [[-1.454, -0.834], [1.157, -0.946]], [-0.354262, 0.928143, 0.114234]], ['regression: random qubit 3', [[1.769, -1.743], [0.441, -0.443]], [0.47339, -0.004576, 0.880841]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state'], ['control: noise on one amplitude', [[3e-08, 0], [0, 0]], 'zero-state'], ['control: ket zero', [[1, 0], [0, 0]], [0.0, 0.0, 1.0]], ['control: ket one', [[0, 0], [1, 0]], [0.0, 0.0, -1.0]]], [['regression: random qubit 9', [[0.467, -1.484], [-1.959, -0.631]], [0.006475, -0.962062, -0.272753]], ['regression: random qubit 10', [[-1.198, -0.992], [1.144, -1.926]], [0.145232, 0.925635, -0.349439]], ['regression: random qubit 5', [[1.781, 0.313], [-0.534, -1.473]], [-0.493327, -0.858114, 0.142371]], ['control: plus', [[0.707107, 0], [0.707107, 0]], [1.0, 0.0, 0.0]], ['control: zero vector', [[0, 0], [0, 0]], 'zero-state'], ['control: global phase i', [[0, 1], [0, 0]], [0.0, 0.0, 1.0]], ['control: numerical noise state', [[1e-07, 0], [0, 1e-07]], 'zero-state']]]
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: plus i	`[0.0, 1.0, 0.0]`	`[0.0, 1.0, 0.0]`	Passed
regression: minus i	`[0.0, -1.0, 0.0]`	`[0.0, -1.0, 0.0]`	Passed
regression: unnormalized	`[0.0, 0.96, -0.28]`	`[0.0, 0.96, -0.28]`	Passed
control: ket zero	`[0.0, 0.0, 1.0]`	`[0.0, 0.0, 1.0]`	Passed
control: ket one	`[0.0, 0.0, -1.0]`	`[0.0, 0.0, -1.0]`	Passed
control: plus	`[1.0, 0.0, 0.0]`	`[1.0, 0.0, 0.0]`	Passed
control: zero vector	`zero-state`	`zero-state`	Passed

SHA-256 / 893ebbded1a611d1a61b679e57073d4b00d3f354c5a3d9a82f3dfe9ec9e1a1c0

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 2026-09-29T14:51:32.457716+00:00.

Case digest / 612a990b04cc6c219766aec5de53dde0380652dfa8dff8a0ffd06b365f8d16cf