{
  "slug": "2026-09-09-local-quantum-error-correction",
  "surface_code": {
    "X_checks": [
      [
        0,
        1
      ],
      [
        1,
        2,
        4,
        5
      ],
      [
        3,
        4,
        6,
        7
      ],
      [
        7,
        8
      ]
    ],
    "Z_checks": [
      [
        0,
        1,
        3,
        4
      ],
      [
        2,
        5
      ],
      [
        3,
        6
      ],
      [
        4,
        5,
        7,
        8
      ]
    ],
    "logical_X": [
      0,
      3,
      6
    ],
    "logical_Z": [
      0,
      1,
      2
    ],
    "code_distance": 3,
    "enumerated_paulis": 262144,
    "corrected_single_paulis": 27,
    "state_infidelity_max": 0.0,
    "logical_basis_phase_mismatch": 0.0,
    "coherent_rotation_angle": 0.2,
    "coherent_recovery_density_max_error": 6.979547201326647e-18,
    "two_error_example": {
      "errors_one_based": [
        1,
        4
      ],
      "decoder_correction_one_based": [
        7
      ],
      "result": "logical X"
    },
    "results": [
      {
        "physical_depolarizing_probability": 0.0001,
        "logical_I": 0.9999998533770321,
        "logical_X": 7.552637520949402e-08,
        "logical_Z": 7.108518899732874e-08,
        "logical_Y": 1.140367458564122e-11,
        "logical_nonidentity": 1.466229678814084e-07
      },
      {
        "physical_depolarizing_probability": 0.001,
        "logical_I": 0.9999853769878592,
        "logical_X": 7.526418727186486e-06,
        "logical_Z": 7.085223288418476e-06,
        "logical_Y": 1.1370125196790689e-08,
        "logical_nonidentity": 1.4623012140801751e-05
      },
      {
        "physical_depolarizing_probability": 0.01,
        "logical_I": 0.9985765459534502,
        "logical_X": 0.0007268505050503673,
        "logical_Z": 0.0006855643890004174,
        "logical_Y": 1.1039152499042703e-05,
        "logical_nonidentity": 0.0014234540465498274
      },
      {
        "physical_depolarizing_probability": 0.03,
        "logical_I": 0.9879403642985509,
        "logical_X": 0.00605027745562624,
        "logical_Z": 0.005730382228553728,
        "logical_Y": 0.00027897601726873593,
        "logical_nonidentity": 0.012059635701448704
      },
      {
        "physical_depolarizing_probability": 0.1,
        "logical_I": 0.8922256077028909,
        "logical_X": 0.050837336779962425,
        "logical_Z": 0.04878751602337043,
        "logical_Y": 0.008149539493776357,
        "logical_nonidentity": 0.10777439229710922
      },
      {
        "physical_depolarizing_probability": 0.15,
        "logical_I": 0.7935220479999996,
        "logical_X": 0.09321151999999996,
        "logical_Z": 0.09018406399999998,
        "logical_Y": 0.023082367999999992,
        "logical_nonidentity": 0.20647795199999994
      }
    ],
    "limits": "9 data qubits only; perfect stabilizers, no faulty ancilla circuit or measurement, fixed minimum-weight decoder; not a hardware threshold or general surface-code scaling test."
  },
  "history": {
    "data_error_cycle": 3,
    "readout_error_cycle": 5,
    "detection_coordinates_one_based": [
      [
        3,
        2
      ],
      [
        3,
        3
      ],
      [
        5,
        2
      ],
      [
        6,
        2
      ]
    ],
    "limits": "A selected classical X-error history in an ideal repetition code, with one readout-bit error. No active physical correction, no full circuit decoder simulation."
  },
  "noise": {
    "marginal_error": 0.01,
    "burst": 0.001,
    "conditional_error": 0.00900900900900901,
    "rows": [
      {
        "data_qubits": 1,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 0.01,
        "common_burst_logical_error": 0.010000000000000002
      },
      {
        "data_qubits": 3,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 0.00029800000000000003,
        "common_burst_logical_error": 0.001241782322863404
      },
      {
        "data_qubits": 5,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 9.850600000000002e-06,
        "common_burst_logical_error": 0.0010072062467774469
      },
      {
        "data_qubits": 7,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 3.416698e-07,
        "common_burst_logical_error": 0.0010002253825701435
      },
      {
        "data_qubits": 9,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 1.2185368570000002e-08,
        "common_burst_logical_error": 0.0010000072482693717
      },
      {
        "data_qubits": 11,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 4.4254343383480005e-10,
        "common_burst_logical_error": 0.0010000002373787697
      },
      {
        "data_qubits": 13,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 1.6278881392003243e-11,
        "common_burst_logical_error": 0.0010000000078741942
      },
      {
        "data_qubits": 15,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 6.045248493209707e-13,
        "common_burst_logical_error": 0.0010000000002636908
      },
      {
        "data_qubits": 17,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 2.261436267389132e-14,
        "common_burst_logical_error": 0.0010000000000088955
      },
      {
        "data_qubits": 19,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 8.509104732905514e-16,
        "common_burst_logical_error": 0.0010000000000003019
      },
      {
        "data_qubits": 21,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 3.2169401503950556e-17,
        "common_burst_logical_error": 0.0010000000000000102
      },
      {
        "data_qubits": 23,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 1.220988990417036e-18,
        "common_burst_logical_error": 0.0010000000000000005
      },
      {
        "data_qubits": 25,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 4.649673552843913e-20,
        "common_burst_logical_error": 0.001
      },
      {
        "data_qubits": 27,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 1.775684284604094e-21,
        "common_burst_logical_error": 0.001
      },
      {
        "data_qubits": 29,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 6.797899925022196e-23,
        "common_burst_logical_error": 0.001
      },
      {
        "data_qubits": 31,
        "marginal_error": 0.01,
        "burst_probability": 0.001,
        "conditional_independent_error": 0.00900900900900901,
        "independent_logical_error": 2.6080409268757338e-24,
        "common_burst_logical_error": 0.001
      }
    ],
    "limits": "Only bit-flip protection, perfect checks, no burst flag, fixed decoding rule. The common all-qubit flip has zero syndrome; not every correlated error is uncorrectable."
  },
  "channel_ambiguity": {
    "channels": [
      {
        "channel": "independent_X_Z",
        "entanglement_fidelity": 0.8099999999999996,
        "Z+": 0.09999999999999998,
        "Z-": 0.09999999999999998,
        "X+": 0.10000000000000031,
        "X-": 0.10000000000000031,
        "Y+": 0.18000000000000027,
        "Y-": 0.18000000000000027
      },
      {
        "channel": "correlated_Y",
        "entanglement_fidelity": 0.8999999999999997,
        "Z+": 0.09999999999999998,
        "Z-": 0.09999999999999998,
        "X+": 0.10000000000000042,
        "X-": 0.10000000000000042,
        "Y+": 4.440892098500626e-16,
        "Y-": 4.440892098500626e-16
      }
    ],
    "limits": "Two chosen Pauli channels illustrate missing Y information. They do not refute an analysis that explicitly assumes independent logical X/Z errors."
  }
}