Quantum sensors leverage quantum mechanics to create some of the most sensitive instruments allowed by the laws of physics. However, current sensors are noisy and lose sensitivity and coherence due to environmental interactions. The solution to these noisy sensors is quantum error correction, which can restore coherence and allow for quantum sensors to obtain the “Heisenberg Limit”, or the highest sensitivity allowed by quantum mechanics.

However, when viewing current literature on quantum error corrected sensors, one would find an interesting pattern. No current error corrected sensors use codes that can correct arbitrary single qubit errors. This means that current error corrected sensors are still sensitive to certain types of errors. Why is this the case?

This pattern in the modern literature is due to the Eastin-Knill theorem, which states that “No error correcting code can have a continuous symmetry which acts transversally on physical qubits”. In simpler words, this means that it turns out to be quite difficult to design the signal operator for stronger error correcting codes, and that modern error corrected sensors must leave a degree of freedom uncorrected to allow for a continuous symmetry for the signal to act on.

In the paper below, I show that a diagonal unitary operator can be created from commuting controlled-phase gates, and attempt to use this operator to create a signal for the Steane code. Although the approach taken in this paper is unsuccessful in the Steane code, it may find more success in larger codes such as the Shor, d=3 Surface, [15,1,3], or [11,1,5] codes. However, I do not have the computational resources to test these codes in a reasonable amount of time. Below is a link to the arxiv preprint, explaining the idea and my application of it to the Steane code. Maybe further exploration will result in a better understanding of the structure and underlying difficulty of creating continuous logical operators.