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

Mischa P. Woods1 and Álvaro M. Alhambra2

1Institute for Theoretical Physics, ETH Zurich, Switzerland
2Perimeter Institute for Theoretical Physics, Waterloo, Canada

### Abstract

Following the introduction of the task of $\textit{reference frame error}$ $\textit{correction}$ [1], 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.

Quantum error correction is the field that aims to devise schemes that protect quantum information as it is being processed in computations. This is a monumental task, which needs to overcome fundamental physical constraints unique to quantum physics such as the no-cloning theorem. The codes must then present different features that makes them both scalable and robust. A crucial one is the existence of "transversal gates", which are logical operations that can be implemented by acting separately on the different subsystems within the code.

Previously, a number of no-go theorems heavily restricted the possibility of such transversal gates, establishing that the set of them must in general be finite. Here, we show that one can in fact enhance these sets to continuous or infinite groups by appending a so-called "quantum clock" to the codes. This is only possible if one allows for a small error in the decoding procedure — an idea that had been overlooked in much of the previous literature. We show that the error here in fact quickly vanishes as the size and precision of the clocks increases, so that in principle it can be made as small as required in the different implementations. Quantum error correcting codes have recently found numerous applications in physics beyond their initial purpose, and we explain what our results could imply for these connections.

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[2] Dong-Sheng Wang, Yun-Jiang Wang, Ningping Cao, Bei Zeng, and Raymond Laflamme, "Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes", New Journal of Physics 24 2, 023019 (2022).

[3] Dong‐Sheng Wang, "A comparative study of universal quantum computing models: Toward a physical unification", Quantum Engineering 3 4(2021).

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[5] Dong-Sheng Wang, "A prototype of quantum von Neumann architecture", Communications in Theoretical Physics 74 9, 095103 (2022).

[6] Sam Cree, Kfir Dolev, Vladimir Calvera, and Dominic J. Williamson, "Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted", PRX Quantum 2 3, 030337 (2021).

[7] Rhea Alexander, Si Gvirtz-Chen, and David Jennings, "Infinitesimal reference frames suffice to determine the asymmetry properties of a quantum system", New Journal of Physics 24 5, 053023 (2022).

[8] Sami Boulebnane, Mischa P. Woods, and Joseph M. Renes, "Waveform Estimation from Approximate Quantum Nondemolition Measurements", Physical Review Letters 127 1, 010502 (2021).

[9] Kun Fang and Zi-Wen Liu, "No-Go Theorems for Quantum Resource Purification: New Approach and Channel Theory", PRX Quantum 3 1, 010337 (2022).

[10] Yuxiang Yang, Yin Mo, Joseph M. Renes, Giulio Chiribella, and Mischa P. Woods, "Optimal universal quantum error correction via bounded reference frames", Physical Review Research 4 2, 023107 (2022).

[11] Sisi Zhou, Zi-Wen Liu, and Liang Jiang, "New perspectives on covariant quantum error correction", Quantum 5, 521 (2021).

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[14] Philippe Faist, Sepehr Nezami, Victor V. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, and John Preskill, "Continuous Symmetries and Approximate Quantum Error Correction", Physical Review X 10 4, 041018 (2020).

[15] Bartosz Regula and Ryuji Takagi, "Fundamental limitations on distillation of quantum channel resources", Nature Communications 12 1, 4411 (2021).

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[17] Aleksander Kubica and Rafał Demkowicz-Dobrzański, "Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem", Physical Review Letters 126 15, 150503 (2021).

[18] Dong-Sheng Wang, Guanyu Zhu, Cihan Okay, and Raymond Laflamme, "Quasi-exact quantum computation", Physical Review Research 2 3, 033116 (2020).

[19] Hyukjoon Kwon and M. S. Kim, "Fluctuation Theorems for a Quantum Channel", Physical Review X 9 3, 031029 (2019).

[20] Tamara Kohler and Toby Cubitt, "Toy models of holographic duality between local Hamiltonians", Journal of High Energy Physics 2019 8, 17 (2019).

[21] Philippe Faist, Mischa P. Woods, Victor V. Albert, Joseph M. Renes, Jens Eisert, and John Preskill, "Time-energy uncertainty relation for noisy quantum metrology", arXiv:2207.13707.

[22] Alexey Milekhin, "Quantum error correction and large N", SciPost Physics 11 5, 094 (2021).

[23] Hayata Yamasaki, Kosuke Fukui, Yuki Takeuchi, Seiichiro Tani, and Masato Koashi, "Polylog-overhead highly fault-tolerant measurement-based quantum computation: all-Gaussian implementation with Gottesman-Kitaev-Preskill code", arXiv:2006.05416.

[24] Ning Bao and Newton Cheng, "Eigenstate thermalization hypothesis and approximate quantum error correction", Journal of High Energy Physics 2019 8, 152 (2019).

[25] Mischa P. Woods and Michał Horodecki, "The Resource Theoretic Paradigm of Quantum Thermodynamics with Control", arXiv:1912.05562.

[26] Sami Boulebnane, Mischa P. Woods, and Joseph M. Renes, "Approximate quantum non-demolition measurements", arXiv:1909.05265.

The above citations are from Crossref's cited-by service (last updated successfully 2022-09-24 05:37:22) and SAO/NASA ADS (last updated successfully 2022-09-24 05:37:23). The list may be incomplete as not all publishers provide suitable and complete citation data.