Error-corrected Hadamard gate simulated at the circuit level

György P. Gehér; Campbell McLauchlan; Earl T. Campbell; Alexandra E. Moylett; Ophelia Crawford

doi:10.22331/q-2024-07-02-1394

Error-corrected Hadamard gate simulated at the circuit level

György P. Gehér¹, Campbell McLauchlan^1,2, Earl T. Campbell^1,3, Alexandra E. Moylett¹, and Ophelia Crawford¹

¹Riverlane, Cambridge, CB2 3BZ, UK
²DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, UK
³Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK

Published:	2024-07-02, volume 8, page 1394
Eprint:	arXiv:2312.11605v2
Doi:	https://doi.org/10.22331/q-2024-07-02-1394
Citation:	Quantum 8, 1394 (2024).

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Abstract

We simulate the logical Hadamard gate in the surface code under a circuit-level noise model, compiling it to a physical circuit on square-grid connectivity hardware. Our paper is the first to do this for a logical unitary gate on a quantum error-correction code. We consider two proposals, both via patch-deformation: one that applies a transversal Hadamard gate (i.e. a domain wall through time) to interchange the logical $X$ and $Z$ strings, and another that applies a domain wall through space to achieve this interchange. We explain in detail why they perform the logical Hadamard gate by tracking how the stabilisers and the logical operators are transformed in each quantum error-correction round. We optimise the physical circuits and evaluate their logical failure probabilities, which we find to be comparable to those of a quantum memory experiment for the same number of quantum error-correction rounds. We present syndrome-extraction circuits that maintain the same effective distance under circuit-level noise as under phenomenological noise. We also explain how a $SWAP$-quantum error-correction round (required to return the patch to its initial position) can be compiled to only four two-qubit gate layers. This can be applied to more general scenarios and, as a byproduct, explains from first principles how the "stepping" circuits of the recent Google paper [1] can be constructed.

Featured image: 3D schematic diagrams of decoding graphs for logical Hadamard gate on rotated surface code.

Popular summary

Quantum computation promises to have a transformative impact on science and technology. However, in order to achieve advantage beyond classical computing, we require quantum error correction (QEC). We must be able to store and operate on logical quantum states in a noise-resilient manner. It is therefore very important to analyse how well such an encoded state is predicted to be preserved by a QEC code. This question has been researched extensively for the surface code, the leading proposed architecture for error-corrected quantum computation. However, the implementation of logical gates has not been examined in detail under realistic “circuit-level" noise. In this paper, we present a detailed step-by-step guide for implementing a logical gate called the Hadamard gate on the (so-called rotated) surface code, and perform the first circuit-level noise simulations for this gate. The procedure requires moving the corners of the surface code patch using measurements, and also requires two layers of “SWAP'' gates to return the patch to its original position. We construct optimal circuits for each step. We find that the logical Hadamard gate has a very similar QEC performance to simply storing the state in a quantum memory for the same length of time, thereby providing a promising benchmark for this crucial element of quantum computation.

► BibTeX data

@article{Geher2024errorcorrected,
  doi = {10.22331/q-2024-07-02-1394},
  url = {https://doi.org/10.22331/q-2024-07-02-1394},
  title = {Error-corrected {H}adamard gate simulated at the circuit level},
  author = {Geh{\'{e}}r, Gy{\"{o}}rgy P. and McLauchlan, Campbell and Campbell, Earl T. and Moylett, Alexandra E. and Crawford, Ophelia},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1394},
  month = jul,
  year = {2024}
}

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The above citations are from Crossref's cited-by service (last updated successfully 2026-08-12 00:48:38) and SAO/NASA ADS (last updated successfully 2026-08-12 00:48:40). The list may be incomplete as not all publishers provide suitable and complete citation data.

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