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arXiv:1709.04471 · PRX Quantum

Error Correction of Quantum Reference Frame Information

Patrick M. Hayden, Sepehr Nezami, Sandu Popescu, Grant Salton

Published Feb 18, 2021• 41 citations
arXivHTMLPDF

Abstract

An investigation of group covariant quantum error correction indicates that the Eastin-Knill theorem, a stumbling block on the path to fault-tolerant quantum computation, may be circumvented by using continuous variables.
DOI
10.1103/prxquantum.2.010326

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Error correcting codes with a universal set of transversal gates are a desideratum for quantum computing. Such codes, however, are ruled out by the Eastin-Knill theorem. Moreover, the theorem also rules out codes which are covariant with respect to the action of transversal unitary operations forming continuous symmetries. In this work, starting from an arbitrary code, we construct approximate codes which are covariant with respect to the entire group of local unitary gates in dimension $d\phantom{\rule{4pt}{0ex}}(<\ensuremath{\infty})$, using quantum reference frames. We show that our codes are capable of efficiently correcting different types of erasure errors. When only a small fraction of the nnn qudits upon which the code is built are erased, our covariant code has an error that scales as 1/n21/{n}^{2}1/n2, which is reminiscent of the Heisenberg limit of quantum metrology. When every qudit has a chance of being erased, our covariant code has an error that scales as 1/n1/n1/n. We show that the error scaling is optimal in both cases. Our approach has implications for fault-tolerant quantum computing, reference frame error correction, and the AdS-CFT duality.
Following the introduction of the task of reference frame error correction \cite{hayden2017error}, 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.