Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames

Understand

Following the introduction of the task of reference frame error correction, we show how, by using reference frame alignment with clocks, one can add a continuous Abelian group of transversal logical gates to any error-correcting code.

• With this we further explore a way of circumventing the no-go theorem of Eastin and Knill, which states that if local errors are correctable, the group of transversal gates must be of finite order.
• We are able to do this by introducing a small error on the decoding procedure that decreases with the dimension of the frames used.
• Furthermore, we show that there is a direct relationship between how small this error can be and how accurate quantum clocks can be: the more accurate the clock, the smaller the error; and the no-go theorem would be violated if time could be measured perfectly in quantum mechanics.