A logical qubit is a quantum bit whose information is protected from errors by being encoded redundantly across many physical qubits using quantum error correction codes. While a single physical qubit might have an error rate of 0.1% per operation, a logical qubit encoded in a distance-d surface code with physical error rates below threshold can achieve error rates that decrease exponentially with d, potentially reaching 10^-10 or better for practical computations.
The overhead for logical qubits is substantial. A surface code at distance d requires approximately 2d² physical qubits (including data and syndrome qubits). To achieve the error rates needed for algorithms like Shor's factoring (approximately 10^-12 per logical gate for factoring 2,048-bit RSA), distances of 20-30 may be required, translating to 800-1,800 physical qubits per logical qubit. A useful fault-tolerant quantum computer running Shor's algorithm might need 2,000-4,000 logical qubits, requiring millions of physical qubits.
The creation of high-quality logical qubits is the central milestone on the roadmap to practical quantum computing. Google's 2024 Willow result showed a surface-code logical qubit whose error rate decreased as code distance increased — the first demonstration of below-threshold error correction with a superconducting processor. Microsoft, Quantinuum, QuEra, and others have also demonstrated logical qubit operations. The transition from NISQ computing (operating on noisy physical qubits) to fault-tolerant computing (operating on clean logical qubits) is the defining challenge of the current era.