FA-90941 / Quantum circuit simulation / Open access

Pauli expectation omits complex conjugation of the bra · case 01

Complex states return expectations that differ from <psi|P|psi>, e.g. <X> on |+i> is nonzero.

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

ROOT CAUSE

The accumulator multiplies psi[j] rather than conj(psi[j]) with the transformed ket amplitude.

THE FAILURE

The accumulator multiplies psi[j] rather than conj(psi[j]) with the transformed ket amplitude.

Unsuccessful approach: The attempted repair conjugates the bra but also conjugates the Pauli phase, which negates Y contributions.

Case contract

Input [n, amps, pauli]; pauli is an n-character string over I,X,Y,Z whose rightmost character acts on qubit 0 (LSB). Return the real expectation <psi|P|psi>/<psi|psi> rounded to 6 decimals. Errors: "length-mismatch", "bad-pauli" (any character outside uppercase IXYZ).

Why this case matters

Pauli expectation values are the observable layer of variational algorithms; string-order or phase slips bias every energy estimate.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(x):
    n, amps, pauli = x
    if len(pauli) != n:
        return 'length-mismatch'
    if any(c not in 'IXYZ' for c in pauli):
        return 'bad-pauli'
    psi = [complex(a, b) for a, b in amps]
    acc = 0j
    for i, v in enumerate(psi):
        j = i
        phase = 1 + 0j
        for k, c in enumerate(pauli):
            q = n - 1 - k
            bit = (i >> q) & 1
            if c in 'XY':
                j ^= 1 << q
            if c == 'Z' and bit:
                phase = -phase
            elif c == 'Y':
                phase *= 1j if bit == 0 else -1j
        acc += psi[j] * phase * v
    norm = sum(abs(v) ** 2 for v in psi)
    return round((acc / norm).real, 6) + 0.0
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: Y on |+i>', [1, [[0.707107, 0], [0, 0.707107]], 'Y'], 1.0], ['regression: unnormalized Z', [1, [[3, 0], [0, 4]], 'Z'], -0.28], ['regression: random expectation 4', [1, [[-0.408, 0.461], [-0.359, 0.326]], 'Y'], 0.105809], ['control: ZI on |01>', [2, [[0, 0], [1, 0], [0, 0], [0, 0]], 'ZI'], 1.0], ['control: IZ on |01>', [2, [[0, 0], [1, 0], [0, 0], [0, 0]], 'IZ'], -1.0], ['control: X on |+>', [1, [[0.707107, 0], [0.707107, 0]], 'X'], 1.0], ['control: short string', [2, [[1, 0], [0, 0], [0, 0], [0, 0]], 'Z'], 'length-mismatch']], [['regression: random expectation 2', [1, [[0.489, 0.112], [-0.446, 0.868]], 'X'], -0.200793], ['regression: random expectation 3', [1, [[-0.119, -0.141], [0.0, 0.0]], 'Z'], 1.0], ['regression: random expectation 13', [2, [[-0.447, -0.821], [0.189, -0.56], [0.105, 0.959], [0.741, -0.692]], 'XY'], 0.425941], ['control: long string', [1, [[1, 0], [0, 0]], 'ZZ'], 'length-mismatch'], ['control: short X on bell', [2, [[0.707107, 0], [0, 0], [0, 0], [0.707107, 0]], 'X'], 'length-mismatch'], ['control: short Y on three qubits', [3, [[0.5, 0], [0, 0.5], [0, 0], [0.5, 0], [0, 0], [0, 0], [0, 0], [0.5, 0]], 'YZ'], 'length-mismatch'], ['control: short Z on excited high qubit', [2, [[0, 0], [0, 0], [1, 0], [0, 0]], 'Z'], 'length-mismatch']], [['regression: random expectation 5', [3, [[0.616, -0.07], [-0.126, -0.492], [-0.385, 0.046], [-0.714, -0.852], [0.968, -0.035], [-0.129, -0.794], [-0.299, 0.172], [0.772, -0.916]], 'YYZ'], -0.229604], ['regression: random expectation 6', [2, [[0.423, -0.627], [-0.829, 0.103], [0.962, -0.142], [-0.494, -0.183]], 'XX'], -0.7271], ['regression: random expectation 15', [2, [[0.599, 0.301], [1.478, -0.323], [-0.934, -1.434], [0.207, -0.762]], 'YZ'], 0.153104], ['control: short ZX', [3, [[0.6, 0], [0.8, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0]], 'ZX'], 'length-mismatch'], ['control: lowercase pauli', [1, [[1, 0], [0, 0]], 'z'], 'bad-pauli'], ['control: random expectation 1', [1, [[0.0, 0.0], [1.97, 0.65]], 'X'], 0.0], ['control: random expectation 22', [1, [[0.5, 0.0], [0.0, 0.0]], 'Z'], 1.0]], [['regression: random expectation 8', [1, [[0.52, 0.586], [-0.138, -0.781]], 'I'], 1.0], ['regression: random expectation 9', [3, [[0.685, -0.091], [1.56, 1.459], [-1.17, 0.687], [-0.305, 1.609], [-0.876, -1.673], [-0.535, 0.008], [-1.818, 1.331], [1.693, 1.162]], 'IXI'], -0.046053], ['regression: random expectation 19', [3, [[-1.467, 1.96], [1.04, 1.59], [0.568, -0.807], [-1.506, -0.905], [-0.169, 1.361], [-1.64, -0.753], [1.853, -1.891], [0.222, 0.162]], 'ZIY'], -0.70921], ['control: random expectation 45', [1, [[0.0, 0.0], [-0.122, 1.228]], 'Y'], 0.0], ['control: ZI on |01>', [2, [[0, 0], [1, 0], [0, 0], [0, 0]], 'ZI'], 1.0], ['control: IZ on |01>', [2, [[0, 0], [1, 0], [0, 0], [0, 0]], 'IZ'], -1.0], ['control: X on |+>', [1, [[0.707107, 0], [0.707107, 0]], 'X'], 1.0]], [['regression: random expectation 11', [1, [[-0.422, -1.932], [-1.435, 1.833]], 'Z'], -0.161676], ['regression: random expectation 12', [3, [[1.741, 0.945], [0.0, 0.0], [0.705, -0.271], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [-1.277, -0.575], [-0.732, 0.787]], 'ZZX'], 0.12672], ['regression: random expectation 26', [2, [[0.234, -0.739], [-0.627, 0.79], [0.0, 0.0], [0.107, -0.508]], 'YI'], 0.247917], ['control: short string', [2, [[1, 0], [0, 0], [0, 0], [0, 0]], 'Z'], 'length-mismatch'], ['control: long string', [1, [[1, 0], [0, 0]], 'ZZ'], 'length-mismatch'], ['control: short X on bell', [2, [[0.707107, 0], [0, 0], [0, 0], [0.707107, 0]], 'X'], 'length-mismatch'], ['control: short Y on three qubits', [3, [[0.5, 0], [0, 0.5], [0, 0], [0.5, 0], [0, 0], [0, 0], [0, 0], [0.5, 0]], 'YZ'], 'length-mismatch']]]
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: Y on \|+i>	`0.0`	`1.0`	Failed
regression: unnormalized Z	`1.0`	`-0.28`	Failed
regression: random expectation 4	`0.0`	`0.105809`	Failed
control: ZI on \|01>	`1.0`	`1.0`	Passed
control: IZ on \|01>	`-1.0`	`-1.0`	Passed
control: X on \|+>	`1.0`	`1.0`	Passed
control: short string	`length-mismatch`	`length-mismatch`	Passed

SHA-256 / 7f52465d6bf57053eecbebc1944511bbf91d5ad4a14c3d33fa4f20ecae9c3d86

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, amps, pauli = x
    if len(pauli) != n:
        return 'length-mismatch'
    if any(c not in 'IXYZ' for c in pauli):
        return 'bad-pauli'
    psi = [complex(a, b) for a, b in amps]
    acc = 0j
    for i, v in enumerate(psi):
        j = i
        phase = 1 + 0j
        for k, c in enumerate(pauli):
            q = n - 1 - k
            bit = (i >> q) & 1
            if c in 'XY':
                j ^= 1 << q
            if c == 'Z' and bit:
                phase = -phase
            elif c == 'Y':
                phase *= 1j if bit == 0 else -1j
        acc += psi[j].conjugate() * phase.conjugate() * v
    norm = sum(abs(v) ** 2 for v in psi)
    return round((acc / norm).real, 6) + 0.0
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: Y on |+i>', [1, [[0.707107, 0], [0, 0.707107]], 'Y'], 1.0], ['regression: unnormalized Z', [1, [[3, 0], [0, 4]], 'Z'], -0.28], ['regression: random expectation 4', [1, [[-0.408, 0.461], [-0.359, 0.326]], 'Y'], 0.105809], ['control: ZI on |01>', [2, [[0, 0], [1, 0], [0, 0], [0, 0]], 'ZI'], 1.0], ['control: IZ on |01>', [2, [[0, 0], [1, 0], [0, 0], [0, 0]], 'IZ'], -1.0], ['control: X on |+>', [1, [[0.707107, 0], [0.707107, 0]], 'X'], 1.0], ['control: short string', [2, [[1, 0], [0, 0], [0, 0], [0, 0]], 'Z'], 'length-mismatch']], [['regression: random expectation 2', [1, [[0.489, 0.112], [-0.446, 0.868]], 'X'], -0.200793], ['regression: random expectation 3', [1, [[-0.119, -0.141], [0.0, 0.0]], 'Z'], 1.0], ['regression: random expectation 13', [2, [[-0.447, -0.821], [0.189, -0.56], [0.105, 0.959], [0.741, -0.692]], 'XY'], 0.425941], ['control: long string', [1, [[1, 0], [0, 0]], 'ZZ'], 'length-mismatch'], ['control: short X on bell', [2, [[0.707107, 0], [0, 0], [0, 0], [0.707107, 0]], 'X'], 'length-mismatch'], ['control: short Y on three qubits', [3, [[0.5, 0], [0, 0.5], [0, 0], [0.5, 0], [0, 0], [0, 0], [0, 0], [0.5, 0]], 'YZ'], 'length-mismatch'], ['control: short Z on excited high qubit', [2, [[0, 0], [0, 0], [1, 0], [0, 0]], 'Z'], 'length-mismatch']], [['regression: random expectation 5', [3, [[0.616, -0.07], [-0.126, -0.492], [-0.385, 0.046], [-0.714, -0.852], [0.968, -0.035], [-0.129, -0.794], [-0.299, 0.172], [0.772, -0.916]], 'YYZ'], -0.229604], ['regression: random expectation 6', [2, [[0.423, -0.627], [-0.829, 0.103], [0.962, -0.142], [-0.494, -0.183]], 'XX'], -0.7271], ['regression: random expectation 15', [2, [[0.599, 0.301], [1.478, -0.323], [-0.934, -1.434], [0.207, -0.762]], 'YZ'], 0.153104], ['control: short ZX', [3, [[0.6, 0], [0.8, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0]], 'ZX'], 'length-mismatch'], ['control: lowercase pauli', [1, [[1, 0], [0, 0]], 'z'], 'bad-pauli'], ['control: random expectation 1', [1, [[0.0, 0.0], [1.97, 0.65]], 'X'], 0.0], ['control: random expectation 22', [1, [[0.5, 0.0], [0.0, 0.0]], 'Z'], 1.0]], [['regression: random expectation 8', [1, [[0.52, 0.586], [-0.138, -0.781]], 'I'], 1.0], ['regression: random expectation 9', [3, [[0.685, -0.091], [1.56, 1.459], [-1.17, 0.687], [-0.305, 1.609], [-0.876, -1.673], [-0.535, 0.008], [-1.818, 1.331], [1.693, 1.162]], 'IXI'], -0.046053], ['regression: random expectation 19', [3, [[-1.467, 1.96], [1.04, 1.59], [0.568, -0.807], [-1.506, -0.905], [-0.169, 1.361], [-1.64, -0.753], [1.853, -1.891], [0.222, 0.162]], 'ZIY'], -0.70921], ['control: random expectation 45', [1, [[0.0, 0.0], [-0.122, 1.228]], 'Y'], 0.0], ['control: ZI on |01>', [2, [[0, 0], [1, 0], [0, 0], [0, 0]], 'ZI'], 1.0], ['control: IZ on |01>', [2, [[0, 0], [1, 0], [0, 0], [0, 0]], 'IZ'], -1.0], ['control: X on |+>', [1, [[0.707107, 0], [0.707107, 0]], 'X'], 1.0]], [['regression: random expectation 11', [1, [[-0.422, -1.932], [-1.435, 1.833]], 'Z'], -0.161676], ['regression: random expectation 12', [3, [[1.741, 0.945], [0.0, 0.0], [0.705, -0.271], [0.0, 0.0], [0.0, 0.0], [0.0, 0.0], [-1.277, -0.575], [-0.732, 0.787]], 'ZZX'], 0.12672], ['regression: random expectation 26', [2, [[0.234, -0.739], [-0.627, 0.79], [0.0, 0.0], [0.107, -0.508]], 'YI'], 0.247917], ['control: short string', [2, [[1, 0], [0, 0], [0, 0], [0, 0]], 'Z'], 'length-mismatch'], ['control: long string', [1, [[1, 0], [0, 0]], 'ZZ'], 'length-mismatch'], ['control: short X on bell', [2, [[0.707107, 0], [0, 0], [0, 0], [0.707107, 0]], 'X'], 'length-mismatch'], ['control: short Y on three qubits', [3, [[0.5, 0], [0, 0.5], [0, 0], [0.5, 0], [0, 0], [0, 0], [0, 0], [0.5, 0]], 'YZ'], 'length-mismatch']]]
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: Y on \|+i>	`-1.0`	`1.0`	Failed
regression: unnormalized Z	`-0.28`	`-0.28`	Passed
regression: random expectation 4	`-0.105809`	`0.105809`	Failed
control: ZI on \|01>	`1.0`	`1.0`	Passed
control: IZ on \|01>	`-1.0`	`-1.0`	Passed
control: X on \|+>	`1.0`	`1.0`	Passed
control: short string	`length-mismatch`	`length-mismatch`	Passed

SHA-256 / 609bd35ce94afa379776e26a287f59064b94f22326d31ed1791dc02b91b76a11

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 7 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

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 2026-09-29T14:51:31.417146+00:00.

Case digest / 69f463628ab8c83b9ef964951fdca35e24babebd7677720315c4df287752fde2