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. The asymptotic scaling of the error is studied under a number of
scenarios of reference frames and error models. The scheme is also extended to
errors at unknown locations, and we show how to achieve this by simple majority
voting related error correction schemes on the reference frames. In the
Outlook, we discuss our results in relation to the AdS/CFT correspondence and
the Page-Wooters mechanism.Comment: 10+35 pages. Also see related work uploaded to the arXiv on the same
day; arXiv:1902.0771
