Limits of noisy quantum metrology with restricted quantum controls

Understand

The Heisenberg limit (HL, with estimation error scales as $1/n$) and the standard quantum limit (SQL, $\propto 1/\sqrt{n}$) are two fundamental limits in estimating an unknown parameter in $n$ copies of quantum channels and are achievable with full quantum controls, e.g., quantum error correction (QEC).

• It is unknown though, whether these limits are still achievable in restricted quantum devices when QEC is unavailable, e.g., with only unitary controls or bounded system sizes.
• In this Letter, we discover various new limits for estimating qubit channels under restrictive controls.
• The HL is shown to be unachievable in various cases, indicating the necessity of QEC in achieving the HL.