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

TL;DR

This paper demonstrates how to implement continuous Abelian groups of transversal logical gates in quantum error-correcting codes using finite clock reference frames, challenging existing no-go theorems by introducing small, controllable errors.

Contribution

It introduces a method to add continuous transversal gates to quantum codes via reference frame alignment with clocks, circumventing the Eastin-Knill no-go theorem.

Findings

The scheme allows continuous logical gates in quantum error correction.

The accuracy of quantum clocks directly affects the error size.

The approach extends to unknown error locations using majority voting.

Abstract

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.

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1 Introduction and Overview of Results

In order to build a functional universal quantum computer, full fault-tolerance must be achieved. The idea behind fault-tolerance is that the errors that occur at particular points during the computation do not propagate or amplify along the whole computation to the point of being uncorrectable. Due to fundamental physical constraints such as no-cloning, achieving this is a notoriously challenging task, with a number of different requirements on how to prepare, manipulate, and protect the quantum states with error-correcting codes. One of the most desirable features of the codes used in fault-tolerant computation is the ability to apply logical gates transversally, which one can implement while still being able to correct for local errors.

The framework for error correction is based on considering three spaces — a logical $\mathcal{H}_{\textup{L}}$, physical $\mathcal{H}_{\textup{P}}$ and a code $\mathcal{H}_{\textup{Co}}\subseteq\mathcal{H}_{\textup{P}}$ space.111Some authors use the convention of not considering the code space explicitly, in which case one sets $\mathcal{H}_{\textup{Co}}=\mathcal{H}_{\textup{P}}$. Logical states $\rho_{\textup{L}}$ containing quantum information are encoded via an encoding map $\mathcal{E}:\mathcal{B}(\mathcal{H}_{\textup{L}})\rightarrow\mathcal{B}(\mathcal{H}_{\textup{Co}})$ onto the code space, which is a subspace of some larger physical space where errors — represented via error maps $\{\mathfrak{E}_{j}\}_{j}:\mathcal{B}(\mathcal{H}_{\textup{Co}})\rightarrow\mathcal{B}(\mathcal{H}_{\textup{P}})$ — can occur. Decoding maps $\{\mathcal{D}_{j}\}_{j}:\mathcal{B}(\mathcal{H}_{\textup{P}})\rightarrow\mathcal{B}(\mathcal{H}_{\textup{L}})$ can then retrieve the information while correcting for errors; outputting the logical state $\rho_{\textup{L}}$. That is:

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