Logical Qubits and the Overhead Bill
A logical qubit is one error-corrected qubit built from many physical qubits. 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. (This is example scaling math, not a company's figure.) That multiplication, plus extra time and extra normal computing, is why you should read every company's "logical qubit" announcement carefully. Always ask: is it demonstrated, or only on a roadmap?
What exactly is a logical qubit?
A physical qubit is a real piece of hardware, like a superconducting circuit or a trapped ion. It has real error rates. A logical qubit is the protected qubit that an error-correcting code builds out of many physical qubits. It is the thing your algorithm actually uses.
The code spreads one qubit's worth of information across the group, as entanglement (see the no-cloning chapter). It runs parity checks all the time. It fixes physical faults before they damage the logical information. The difference between the two kinds of qubit has its own chapter in our processors course: physical vs logical qubits.
An everyday example: a sports team. The team is what wins or loses a game. It is built from many players, and some players are always tired or hurt. The team still plays well as long as enough players are fine. The picture breaks in one way. Every player on a team does something visible. In a code, no single physical qubit holds the logical value at all.
You have already built logical qubits in this course. The three-qubit repetition code makes one partly protected logical qubit from three physical ones. 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 normal computing resources does one logical qubit really cost? That math, more than any single device spec, sets the timeline for useful fault-tolerant computing. (Fault-tolerant means the machine keeps working correctly even though its parts make errors.)
- How do you compute the physical-qubit bill?
- Where does the distance number come from?
- What does one logical gate look like on three qubits?INTERACTIVE
- Is one physical X a gate or an error?INTERACTIVE
- What else is on the bill besides qubits?
- How should you read a vendor's logical-qubit announcement?
- What do logical-qubit demonstrations look like today?
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