Logical Qubits and the Overhead Bill

A logical qubit is one error-corrected qubit built from many physical qubits, and the exchange rate is steep: at surface-code distance 25, one logical qubit costs about 1,250 physical qubits, so 1,000 logical qubits means roughly 1.25 million physical ones (illustrative scaling arithmetic, not a vendor figure). That multiplication — plus time and classical-processing overhead — is why you must read every vendor "logical qubit" announcement with the demonstrated-vs-roadmap distinction firmly in hand.

What exactly is a logical qubit?

A physical qubit is a real hardware object — a superconducting circuit, a trapped ion — with real error rates. A logical qubit is the protected, abstract qubit that an error-correcting code builds out of many physical qubits: the thing your algorithm actually uses. The code spreads one qubit's worth of information across the group (as entanglement, per the no-cloning chapter), runs parity checks continuously, and fixes physical faults before they corrupt the logical information. The distinction has its own chapter in our processors course: physical vs logical qubits.

You have already built logical qubits in this course: the three-qubit repetition code makes one (partially) protected logical qubit from three physical ones, and the surface code makes one from 2d²−1 physical qubits at distance d. This chapter is about the bill: how many physical qubits, gates, and classical resources one logical qubit really costs — because that arithmetic, more than any single device spec, sets the timeline for useful fault-tolerant computing.