Logical Qubits
A logical qubit is an encoded degree of freedom spread across many physical qubits, protected so its state survives local errors.
Encoded degrees of freedom
A logical qubit is not a single physical object but a two-dimensional subspace of a many-qubit Hilbert space, singled out by a stabilizer group. Its logical basis states |0_L> and |1_L> are entangled configurations of the physical qubits. Logical operators X_L and Z_L are Pauli strings that act within this subspace, commuting with every stabilizer yet not themselves stabilizers.
The point of encoding is that the logical information is nonlocal. To change the logical state, an error must apply a full logical operator, which has weight at least the code distance d. Any error of weight below d/2 is detected and corrected, so the encoded qubit is far more stable than any of its constituents.
Logical operations
Computing on logical qubits requires performing logical gates while preserving the encoding. The safest are transversal gates, applied qubit-by-qubit. In the surface code, logical qubits are patches of lattice and gates are done by lattice surgery, merging and splitting patches to enact joint measurements. Non-Clifford gates need magic states.
- Logical qubit = protected 2D subspace defined by stabilizers.
- Logical X_L, Z_L are minimum-weight Paulis acting on that subspace.
- Errors below distance d/2 are corrected; the encoded state persists.
- Logical gates must respect the code; transversality and surgery are the tools.
Physical cost
One logical qubit consumes many physical qubits: order d^2 for a distance-d surface code, plus ancillas for syndrome extraction and factories for magic states. The number needed depends on the target logical error rate and the length of the algorithm. Quantifying this is the subject of the overhead of fault tolerance. As hardware improves, the ratio of physical to logical qubits shrinks, but for the foreseeable future a single logical qubit will cost hundreds to thousands of physical ones.
The logical qubit is the true unit of quantum computation; algorithms are written and resource-estimated in terms of logical qubits and logical gates, not physical hardware.