Far from Perfect: Quantum Error Correction with (Hyperinvariant) Evenbly Codes
1QuTech, Delft University of Technology, 2628 CJ Delft, The Netherlands
2Quantum and Computer Engineering Department, Delft University of Technology, 2628 CD Delft, The Netherlands
3ARC Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD, 4072, Australia
4Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan
5Department of Physics, Freie Universität Berlin, 14195 Berlin, Germany
| Published: | 2025-08-08, volume 9, page 1826 |
| Editor: | Felix Huber |
| Eprint: | arXiv:2407.11926v2 |
| Doi: | https://doi.org/10.22331/q-2025-08-08-1826 |
| Citation: | Quantum 9, 1826 (2025). |
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Abstract
We introduce a new class of qubit codes that we call Evenbly codes, building on a previous proposal of hyperinvariant tensor networks. Its tensor network description consists of local, non-perfect tensors describing CSS codes interspersed with Hadamard gates, placed on a hyperbolic $\{p,q\}$ geometry with even $q\geq 4$, yielding an infinitely large class of subsystem codes. We construct an example for a $\{5,4\}$ manifold and describe strategies of logical gauge fixing that lead to different rates $k/n$ and distances $d$, which we calculate analytically, finding distances which range from $d=2$ to $d \sim n^{2/3}$. Investigating threshold performance under erasure, depolarizing, and pure Pauli noise channels, we find that the code exhibits a depolarizing noise threshold of about 19.1% in the code-capacity model and 50% for pure Pauli and erasure channels under suitable gauges. We also test a constant-rate version with $k/n = 0.125$, finding excellent error resilience (about 40%) under the erasure channel. Recovery rates for these and other settings are studied both under an optimal decoder as well as a more efficient but non-optimal greedy decoder. We also consider generalizations beyond the CSS tensor construction, compute error rates and thresholds for other hyperbolic geometries, and discuss the relationship to holographic bulk/boundary dualities. Our work indicates that Evenbly codes may show promise for practical quantum computing applications.
Featured image: An example of gauge-fixing a maximum-rate Evenbly code (left) into a constant-rate code under $X$ gauge (right), which turns the whole code into simple copies of the seed code (gray disk with red dot). Other gauges lead to a more complicated embedding of the (logical) bulk qubits into the (physical) boundary qubits.
Popular summary
Our work introduces a new subclass of QEC codes – Evenbly codes – that fall into the class of hyper-invariant holographic codes but are built from simple qubit CSS codes and Hadamard gates, a setting that is more suitable to near-term applications than the original proposal. Evenbly codes are subsystem codes, meaning that a subset of the logical qubits can be gauge-fixed to improve the QEC properties of the others. The code rate can be modified from asymptotically zero to constant values, with code distances of up to approximately $d \sim n^{2/3}$ in some instances, competitive with or better than many existing codes. We also study the resilience of these codes under suitable gauge fixing with regards to various types of noise (erasure, depolarizing, Pauli errors). There we find that both zero-rate versions, where only a single logical qubit remains gauge-free while the number $n$ of physical qubits is scaled up, as well as constant-rate versions exhibit excellent code capacity error thresholds under suitable gauges.
Evenbly codes thus represent a highly-tunable class of qubit codes that appear to perform well against various types of errors while also modeling interesting physics of holographic bulk-boundary dualities.
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