Benchmarks & Metrics
Logical Error Rate
How often an error-corrected logical qubit fails, usually measured per round of error correction.
Definition
The logical error rate tells you how reliable an error-corrected qubit is. A physical qubit might fail once in a thousand operations, but a logical qubit, built from many physical ones and constantly checked, should fail far less often. The logical error rate is the number to watch: if it falls as you add more qubits to the code, the machine is working below threshold, and if it does not, error correction is not helping yet.
In depth
A useful quantum algorithm needs logical error rates far below anything achieved today. Breaking RSA-2048 or simulating a large molecule involves billions of logical operations, so the per-operation logical error must be around 10 to the minus 9 to 10 to the minus 12. Current experiments are at about 10 to the minus 3 per cycle. The gap is closed by raising the code distance and by improving physical error rates, since the logical error falls roughly exponentially with distance when the hardware is below threshold.
The milestone results are measured in these terms. In Google's 2024 Willow experiment, a distance-7 surface code memory had a logical error of about 0.143 percent per cycle, and its logical lifetime of about 291 microseconds exceeded that of the best physical qubit on the chip, about 85 microseconds, by roughly a factor of 2.4. This crossing is called break-even. Quantinuum, Harvard and QuEra, and other groups have reported logical error rates below physical ones on trapped-ion and neutral-atom devices, including for logical gates and magic states.
Interpreting the number needs care. Rates can be quoted per cycle, per logical operation, or per second, and a code that runs faster cycles has a different time scale. Logical memory, logical gates, and logical measurement have different error rates, and the rate depends on the decoder used. Comparing results across platforms requires matching these definitions, and ideally reporting the error suppression factor across distances.
Technical Definition
The logical error rate is the probability that a logical qubit has an uncorrected error per error-correction cycle, per logical gate, or per unit time. It is estimated by running a logical memory experiment for many cycles and fitting the decay of the logical state's fidelity. The error suppression factor lambda is the ratio of logical error rates between code distances d and d plus 2.
Visual Explanation: An Analogy
Think of a team of people carrying a fragile vase together, with each person watching the others and fixing slips. How often the vase actually falls, despite all the teamwork, is the logical error rate, and it should drop as the team gets larger and better coordinated.
Key facts
| Item | Detail |
|---|---|
| Willow (2024) | Distance-7 memory: about 0.143 percent logical error per cycle, lifetime about 2.4 times the best physical qubit |
| Error suppression factor | About 2 per distance step on Willow |
| Target for useful algorithms | 10^-9 to 10^-12 per logical operation, depending on the algorithm |
| Break-even | Logical qubit outlives its best physical qubit |
| Depends on | Code, distance, decoder, physical error rate, and cycle time |
Where the idea comes from
Logical error rates were studied theoretically from the beginning of error correction research. Experimental reports began in 2021 with Google's distance-3 and distance-5 surface codes and became central to hardware progress in 2024 and 2025, as several groups showed logical qubits that beat their physical components.
Real-World Use Cases
- Demonstrating that error correction works below threshold.
- Setting requirements for hardware and code distance in resource estimates.
- Comparing platforms and decoders for error-correction performance.
- Tracking progress toward fault-tolerant machines.
How it differs from related ideas
- The physical error rate describes a single qubit or gate, and the logical error rate describes an encoded qubit.
- Logical error per cycle differs from logical error per gate, and both should be reported.
- Break-even compares a logical qubit with its best physical qubit, which is a weaker test than meeting an algorithm's needs.
Common Misconceptions
- A logical error rate that beats a physical qubit does not mean the machine is ready for useful work.
- Lower is not always comparable. Different experiments define the rate differently.
- The logical error rate is not a fixed property of a code. It depends on the physical noise and decoder.
Frequently asked questions
What is the logical error rate?
The probability that an error-corrected logical qubit fails per cycle or per operation.
What value is needed for useful algorithms?
Typically 10 to the minus 9 to 10 to the minus 12 per logical operation, depending on the algorithm.
What did Willow show?
A distance-7 surface code with about 0.143 percent logical error per cycle and a logical lifetime longer than the best physical qubit.
Related Terms
Keep exploring
- Algorithm: Quantum Error-Correcting Codes (Shor Code, Surface Code)
- Hardware: Google Willow
- Hardware: Quantinuum H2