{"abstract":"The |+i> state is drawn at y = -1.","category":"Quantum circuit simulation","checks":7,"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.","evaluation_group":"w2-quantum_circuit_simulation-bloch-coordinates","failed_approach":"The attempted repair conjugates the product a * b, which is wrong whenever a is not real.","family":"w2-quantum_circuit_simulation-bloch-coordinates-coherence-conjugation-side","id":"FA-91071","implementations":{"attempt":{"sha256":"b1ada6083b3923e9ef9d61562cf2a234dd096f03f40e92fa7a2e984ee01b48db","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x):\n    (ar, ai), (br, bi) = x\n    a, b = complex(ar, ai), complex(br, bi)\n    nrm = abs(a) ** 2 + abs(b) ** 2\n    if nrm < 1e-12:\n        return 'zero-state'\n    cross = (a * b).conjugate()\n    bx = 2 * cross.real / nrm\n    by = 2 * cross.imag / nrm\n    bz = (abs(a) ** 2 - abs(b) ** 2) / nrm\n    return [round(v, 6) + 0.0 for v in (bx, by, bz)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['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']]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"0ca175b9abb8d42b6cb2a42e0e6322cc2fd8e5c8b8f055c15af462fa05d4a842","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x):\n    (ar, ai), (br, bi) = x\n    a, b = complex(ar, ai), complex(br, bi)\n    nrm = abs(a) ** 2 + abs(b) ** 2\n    if nrm < 1e-12:\n        return 'zero-state'\n    cross = a * b.conjugate()\n    bx = 2 * cross.real / nrm\n    by = 2 * cross.imag / nrm\n    bz = (abs(a) ** 2 - abs(b) ** 2) / nrm\n    return [round(v, 6) + 0.0 for v in (bx, by, bz)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['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']]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"fixed":{"sha256":"893ebbded1a611d1a61b679e57073d4b00d3f354c5a3d9a82f3dfe9ec9e1a1c0","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nN = 1\nobservations = []\ndef solve(x):\n    (ar, ai), (br, bi) = x\n    a, b = complex(ar, ai), complex(br, bi)\n    nrm = abs(a) ** 2 + abs(b) ** 2\n    if nrm < 1e-12:\n        return 'zero-state'\n    cross = a.conjugate() * b\n    bx = 2 * cross.real / nrm\n    by = 2 * cross.imag / nrm\n    bz = (abs(a) ** 2 - abs(b) ** 2) / nrm\n    return [round(v, 6) + 0.0 for v in (bx, by, bz)]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['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']]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"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.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-quantum_circuit_simulation-bloch-coordinates-coherence-conjugation-side","generated_at":"2026-09-29T14:51:32.457716+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Bloch coordinates feed visualizers and single-qubit tomography checks; sign or scale slips put states on the wrong axis.","repair":"Use conj(a) * b for the coherence.","root_cause":"The coherence term is a * conj(b) instead of conj(a) * b, flipping the sign of the y component.","sha256":"612a990b04cc6c219766aec5de53dde0380652dfa8dff8a0ffd06b365f8d16cf","title":"Bloch vector conjugates the |1> amplitude · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":40.637,"exit_code":1,"observations":[{"actual":[0.0,-1.0,0.0],"check":"regression: plus i","expected":[0.0,1.0,0.0],"passed":false},{"actual":[0.0,1.0,0.0],"check":"regression: minus i","expected":[0.0,-1.0,0.0],"passed":false},{"actual":[0.0,-0.96,-0.28],"check":"regression: unnormalized","expected":[0.0,0.96,-0.28],"passed":false},{"actual":[0.0,0.0,1.0],"check":"control: ket zero","expected":[0.0,0.0,1.0],"passed":true},{"actual":[0.0,0.0,-1.0],"check":"control: ket one","expected":[0.0,0.0,-1.0],"passed":true},{"actual":[1.0,0.0,0.0],"check":"control: plus","expected":[1.0,0.0,0.0],"passed":true},{"actual":"zero-state","check":"control: zero vector","expected":"zero-state","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: plus i\", \"actual\": [0.0, -1.0, 0.0], \"expected\": [0.0, 1.0, 0.0], \"passed\": false}, {\"check\": \"regression: minus i\", \"actual\": [0.0, 1.0, 0.0], \"expected\": [0.0, -1.0, 0.0], \"passed\": false}, {\"check\": \"regression: unnormalized\", \"actual\": [0.0, -0.96, -0.28], \"expected\": [0.0, 0.96, -0.28], \"passed\": false}, {\"check\": \"control: ket zero\", \"actual\": [0.0, 0.0, 1.0], \"expected\": [0.0, 0.0, 1.0], \"passed\": true}, {\"check\": \"control: ket one\", \"actual\": [0.0, 0.0, -1.0], \"expected\": [0.0, 0.0, -1.0], \"passed\": true}, {\"check\": \"control: plus\", \"actual\": [1.0, 0.0, 0.0], \"expected\": [1.0, 0.0, 0.0], \"passed\": true}, {\"check\": \"control: zero vector\", \"actual\": \"zero-state\", \"expected\": \"zero-state\", \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":38.487,"exit_code":1,"observations":[{"actual":[0.0,-1.0,0.0],"check":"regression: plus i","expected":[0.0,1.0,0.0],"passed":false},{"actual":[0.0,1.0,0.0],"check":"regression: minus i","expected":[0.0,-1.0,0.0],"passed":false},{"actual":[0.0,-0.96,-0.28],"check":"regression: unnormalized","expected":[0.0,0.96,-0.28],"passed":false},{"actual":[0.0,0.0,1.0],"check":"control: ket zero","expected":[0.0,0.0,1.0],"passed":true},{"actual":[0.0,0.0,-1.0],"check":"control: ket one","expected":[0.0,0.0,-1.0],"passed":true},{"actual":[1.0,0.0,0.0],"check":"control: plus","expected":[1.0,0.0,0.0],"passed":true},{"actual":"zero-state","check":"control: zero vector","expected":"zero-state","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: plus i\", \"actual\": [0.0, -1.0, 0.0], \"expected\": [0.0, 1.0, 0.0], \"passed\": false}, {\"check\": \"regression: minus i\", \"actual\": [0.0, 1.0, 0.0], \"expected\": [0.0, -1.0, 0.0], \"passed\": false}, {\"check\": \"regression: unnormalized\", \"actual\": [0.0, -0.96, -0.28], \"expected\": [0.0, 0.96, -0.28], \"passed\": false}, {\"check\": \"control: ket zero\", \"actual\": [0.0, 0.0, 1.0], \"expected\": [0.0, 0.0, 1.0], \"passed\": true}, {\"check\": \"control: ket one\", \"actual\": [0.0, 0.0, -1.0], \"expected\": [0.0, 0.0, -1.0], \"passed\": true}, {\"check\": \"control: plus\", \"actual\": [1.0, 0.0, 0.0], \"expected\": [1.0, 0.0, 0.0], \"passed\": true}, {\"check\": \"control: zero vector\", \"actual\": \"zero-state\", \"expected\": \"zero-state\", \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":40.565,"exit_code":0,"observations":[{"actual":[0.0,1.0,0.0],"check":"regression: plus i","expected":[0.0,1.0,0.0],"passed":true},{"actual":[0.0,-1.0,0.0],"check":"regression: minus i","expected":[0.0,-1.0,0.0],"passed":true},{"actual":[0.0,0.96,-0.28],"check":"regression: unnormalized","expected":[0.0,0.96,-0.28],"passed":true},{"actual":[0.0,0.0,1.0],"check":"control: ket zero","expected":[0.0,0.0,1.0],"passed":true},{"actual":[0.0,0.0,-1.0],"check":"control: ket one","expected":[0.0,0.0,-1.0],"passed":true},{"actual":[1.0,0.0,0.0],"check":"control: plus","expected":[1.0,0.0,0.0],"passed":true},{"actual":"zero-state","check":"control: zero vector","expected":"zero-state","passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: plus i\", \"actual\": [0.0, 1.0, 0.0], \"expected\": [0.0, 1.0, 0.0], \"passed\": true}, {\"check\": \"regression: minus i\", \"actual\": [0.0, -1.0, 0.0], \"expected\": [0.0, -1.0, 0.0], \"passed\": true}, {\"check\": \"regression: unnormalized\", \"actual\": [0.0, 0.96, -0.28], \"expected\": [0.0, 0.96, -0.28], \"passed\": true}, {\"check\": \"control: ket zero\", \"actual\": [0.0, 0.0, 1.0], \"expected\": [0.0, 0.0, 1.0], \"passed\": true}, {\"check\": \"control: ket one\", \"actual\": [0.0, 0.0, -1.0], \"expected\": [0.0, 0.0, -1.0], \"passed\": true}, {\"check\": \"control: plus\", \"actual\": [1.0, 0.0, 0.0], \"expected\": [1.0, 0.0, 0.0], \"passed\": true}, {\"check\": \"control: zero vector\", \"actual\": \"zero-state\", \"expected\": \"zero-state\", \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}