Pastawski-Yoshida-Harlow-Preskill (HaPPY) code

Pastawski-Yoshida-Harlow-Preskill (HaPPY) code[1]

Also known as Perfect holographic code.

Description

Holographic code constructed out of a network of hexagonal perfect tensors that tesselates hyperbolic space. The code serves as a minimal model for several aspects of the AdS/CFT holographic duality [2] and potentially a dF/CFT duality [3]. It has been generalized to higher dimensions [4] and to include gauge-like degrees of freedom on the links of the tensor network [5,6]. All boundary global symmetries must be dual to bulk gauge symmetries, and vice versa [7].

The construction below is described for qubits, but straightforward generalizations exist to modular qudits, oscillators, and rotors [8]. Encoding is accomplished using a tensor network of five-qubit encoding isometries, which are six-legged perfect tensors (with five legs corresponding to the physical qubits and one for the encoded logical qubit).

To construct the encoding, one first uniformly tiles the hyperbolic AdS/CFT disc using pentagons and hexagons. Then, one places a 6-legged five-qubit encoding tensor at each hexagon and pentagon, contracting legs between neighboring shapes and leaving one leg uncontracted at each pentagon. This construction forms an encoding isometry from the uncontracted legs in the bulk to the uncontracted legs at the boundary.

The single-qubit HaPPY code has a central pentagon encoding one bulk operator and hexagons tiling all other layers. The pentagon-hexagon HaPPY code has alternating layers of pentagons and hexagons in the tiling. The pentagon HaPPY code (a.k.a. the hyperbolic pentagon code, or HyPeC) consists of a purely pentagonal tiling.

Protection

Protects against erasure errors and Pauli errors on the boundary qubits.

Rate

The pentagon HaPPY code has an asymptotic rate \(\frac{1}{\sqrt{5}} \approx 0.447\). The pentagon-hexagon HaPPY code has a rate of \(0.299\) if the last layer is a pentagon layer and a rate of \(0.088\) if the last layer is a hexagon layer.

Encoding

Heisenberg-picture encoding is done through tensor pushing. Each bulk operator (logical) is pushed to an operator supported on a portion of the boundary region (physical). Pushing all the bulk operators through results in reconstruction of the boundary.

Transversal Gates

For locality-preserving physical gates on the boundary, the set of transversally implementable logical operations in the bulk is strictly contained in the Clifford group [9].

Decoding

Hierarchical recovery model [1].Greedy decoder [1].

Code Capacity Threshold

\(26\%\) for boundary erasure errors on the pentagon-hexagon HaPPY code under the greedy decoder [1].Lower bound of \(1/12 \approx 8.3\%\) for boundary erasure errors on the single-qubit HaPPY code under hierarchical recovery [1]. Numerical evidence indicates the threshold may be closer to \(50\%\).There is no threshold for the pentagon HaPPY code as a constant number of errors (four) can make bulk recovery impossible [1].\(16.3\%\) for boundary Pauli errors on the single-qubit HaPPY code with 3 layers using integer optimization decoder [10].\(50\%\) against biased Pauli noise for single-qubit HaPPY code under tensor-network decoder [11].

Threshold

A single-qubit HaPPY code has a measurement threshold of one [12] (see also [13]).

Notes

Reference [3] discusses the HaPPY code for an AdS_3 space and its relation to a dS_2 braneworld with a conformal boundary.

Parents

Qubit stabilizer code — The HaPPY code is a stabilizer code because it is defined by a contracted network of stabilizer tensors; see Thm. 6 in Ref. [1].
Holographic tensor-network code — The encoding of a HaPPy code is a holographic tensor network consisting of pentagon and hexagon perfect tensors.

Child

Five-qubit perfect code — The five-qubit code is the smallest (i.e., radius-one) single-qubit HaPPY code. The five-qubit encoding isometry tiles various holographic codes because its corresponding encoding isometry tensor is a perfect tensor [1].

Cousins

Majorana stabilizer code — The pentagon HaPPY code Hamiltonian can be expressed in terms of mutually commuting weight-two (two-body) Majorana operators [14].
Perfect-tensor code — The encoding of a HaPPy code is a holographic tensor network consisting of pentagon and hexagon perfect tensors.

References

[1]: F. Pastawski et al., “Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence”, Journal of High Energy Physics 2015, (2015) arXiv:1503.06237 DOI
[2]: T. J. Osborne and D. E. Stiegemann, “Dynamics for holographic codes”, Journal of High Energy Physics 2020, (2020) arXiv:1706.08823 DOI
[3]: J. Cotler and A. Strominger, “The Universe as a Quantum Encoder”, (2022) arXiv:2201.11658
[4]: M. Taylor and C. Woodward, “Holography, cellulations and error correcting codes”, (2023) arXiv:2112.12468
[5]: W. Donnelly et al., “Living on the edge: a toy model for holographic reconstruction of algebras with centers”, Journal of High Energy Physics 2017, (2017) arXiv:1611.05841 DOI
[6]: K. Dolev et al., “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
[7]: D. Harlow and H. Ooguri, “Symmetries in quantum field theory and quantum gravity”, (2019) arXiv:1810.05338
[8]: P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
[9]: S. Cree et al., “Fault-Tolerant Logical Gates in Holographic Stabilizer Codes Are Severely Restricted”, PRX Quantum 2, (2021) arXiv:2103.13404 DOI
[10]: R. J. Harris et al., “Decoding holographic codes with an integer optimization decoder”, Physical Review A 102, (2020) arXiv:2008.10206 DOI
[11]: J. Fan et al., “Overcoming the Zero-Rate Hashing Bound with Holographic Quantum Error Correction”, (2024) arXiv:2408.06232
[12]: D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
[13]: S. Antonini et al., “Holographic measurement and bulk teleportation”, Journal of High Energy Physics 2022, (2022) arXiv:2209.12903 DOI
[14]: A. Jahn et al., “Majorana dimers and holographic quantum error-correcting codes”, Physical Review Research 1, (2019) arXiv:1905.03268 DOI

Page edit log

Victor V. Albert (2022-07-28) — most recent
Victor V. Albert (2021-12-29)
Joel Rajakumar (2021-12-20)

Cite as:

“Pastawski-Yoshida-Harlow-Preskill (HaPPY) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/happy

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/holographic/happy.yml.