Toward Constructing a Continuous Logical Operator for Error-Corrected Quantum Sensing

Error correction has long been suggested to extend the sensitivity of quantum sensors into Heisenberg Limit. However, operations on logical qubits are only performed through universal gate sets consisting finite-sized gates such as Clifford + T. Although these allow for computation, finite sizes present a problem sensing, since in sensing protocols, Ramsey measurement protocol, signal must act continuously. The difficulty constructing continuous op-erator comes from Eastin-Knill theorem, which prevents sig-nal being both fault-tolerant local errors and transverse. Since error is needed approach Limit noisy environment, it important explore how construct operators. In this paper, protocol design z-rotations proposed applied Steane Code. fault tolerance designed operator investigated using Knill-Laflamme conditions. condi-tions indicate that diagonal unitary constructed cannot be tolerant solely due possibilities X middle qubit. ap-proach demonstrated throughout paper may, however, find success codes with more Shor code, distance 3 surface [15, 1, 3] or larger [11, 5] code.