Logical Qubit

A qubit encoded across multiple physical qubits for error protection. The basic unit of fault-tolerant quantum computing.

A logical qubit is quantum information encoded in a way that’s protected from errors. It’s built from multiple physical qubits using quantum error correction.

Physical vs Logical

Encoding

A logical qubit state is a superposition in a larger Hilbert space:

$$|\psi {⟩}_L=\alpha |0{⟩}_L+\beta |1{⟩}_L$$

where $|0{⟩}_L$ and $|1{⟩}_L$ are encoded states spanning multiple physical qubits.

Example: 3-Qubit Code

$$|0{⟩}_L=|000⟩$$ $$|1{⟩}_L=|111⟩$$

3 physical qubits → 1 logical qubit

Surface Code

$$|0{⟩}_L,|1{⟩}_L\text{ encoded in }d\times d\text{ grid}$$

~$d^2$ physical qubits → 1 logical qubit

Logical Operations

Operations on logical qubits are constructed from physical operations:

The Overhead

The “cost” of a logical qubit depends on:

• Error correction code used
• Physical error rate
• Target logical error rate

Quality Metrics

Logical Error Rate

Probability of logical error per round: $$p_L\propto {\left(\frac{p}{p_{th}}\right)}^{(d+1)/2}$$

Lower is better.

Logical Gate Fidelity

How well logical gates are implemented.

Why We Need Them

NISQ devices use physical qubits directly:

• Limited circuit depth
• Errors accumulate
• Results become noise

Logical qubits enable:

• Arbitrary circuit depth
• Reliable computation
• True quantum advantage

Current Status

• Demonstrations of logical qubits (Google, Quantinuum, IBM)
• Error rates not yet better than physical in useful regimes
• Key milestone: Logical qubit outperforming its physical components

See also: Physical Qubit, Quantum Error Correction, Surface Code, Fault-Tolerant Quantum Computing