Holography, cellulations and error correcting codes

Quantum error correction codes associated with the hyperbolic plane have been explored extensively in the context of the AdS3/CFT2 correspondence. In this paper we initiate a systematic study of codes associated with holographic geometries in higher dimensions, relating cellulations of the spatial sections of the geometries to stabiliser codes. We construct analogues of the HaPPY code for three-dimensional hyperbolic space (AdS4), using both absolutely maximally entangled (AME) and non-AME codes. These codes are based on uniform regular tessellations of hyperbolic space but we note that AME codes that preserve the discrete symmetry of the polytope of the tessellation do not exist above two dimensions. We also explore different constructions of stabiliser codes for hyperbolic spaces in which the logical information is associated with the boundary and discuss their potential interpretation. We explain how our codes could be applied to interesting classes of holographic dualities based on gravity-scalar theories (such as JT gravity) through toroidal reductions of hyperbolic spaces. ar X iv :2 11 2. 12 46 8v 1 [ he pth ] 2 3 D ec 2 02 1

[1]  Brad Lackey,et al.  Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks , 2021, PRX Quantum.

[2]  T. Osborne,et al.  Holographic networks for (1+1)-dimensional de Sitter spacetime , 2021, 2102.09223.

[3]  J. Eisert,et al.  Holographic tensor network models and quantum error correction: a topical review , 2021, Quantum Science and Technology.

[4]  Guoying Zhao,et al.  Hyperbolic Deep Neural Networks: A Survey , 2021, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[5]  Nathan A. McMahon,et al.  Decoding holographic codes with an integer optimization decoder , 2020 .

[6]  Brad Lackey,et al.  Approximate Bacon-Shor code and holography , 2020, Journal of High Energy Physics.

[7]  A. Shapere,et al.  Quantum stabilizer codes, lattices, and CFTs , 2020, Journal of High Energy Physics.

[8]  Elliott Gesteau,et al.  The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics , 2020, 2005.05971.

[9]  J. Eisert,et al.  Tensor network models of AdS/qCFT , 2020, Quantum.

[10]  Vivien Londe,et al.  Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance , 2020, IEEE Transactions on Information Theory.

[11]  J. Eisert,et al.  Central charges of aperiodic holographic tensor-network models , 2019, Physical Review A.

[12]  M. Farkas,et al.  Quantum error-correction codes and absolutely maximally entangled states , 2019, Physical Review A.

[13]  Markus Grassl,et al.  Quantum Codes of Maximal Distance and Highly Entangled Subspaces , 2019, Quantum.

[14]  J. Eisert,et al.  Majorana dimers and holographic quantum error-correcting codes , 2019, Physical Review Research.

[15]  T. Stace.  Calderbank-Steane-Shor Holographic Quantum Error Correcting Codes , 2019 .

[16]  T. Cubitt,et al.  Toy models of holographic duality between local Hamiltonians , 2018, Journal of High Energy Physics.

[17]  F. Flicker,et al.  Conformal Quasicrystals and Holography , 2018, Physical Review X.

[18]  X. Qi,et al.  Space-time random tensor networks and holographic duality , 2018, 1801.05289.

[19]  D. Browne,et al.  Three-dimensional surface codes: Transversal gates and fault-tolerant architectures , 2018, Physical Review A.

[20]  Anthony Leverrier,et al.  Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds , 2017, Quantum Information and Computation.

[21]  Felix Huber,et al.  Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity , 2017, ArXiv.

[22]  M. Taylor.  Generalized conformal structure, dilaton gravity and SYK , 2017, 1706.07812.

[23]  Douwe Kiela,et al.  Poincaré Embeddings for Learning Hierarchical Representations , 2017, NIPS.

[24]  B. Terhal,et al.  Hyperbolic and semi-hyperbolic surface codes for quantum storage , 2017, 1703.00590.

[25]  John Preskill,et al.  Code properties from holographic geometries , 2016, 1612.00017.

[26]  D. Marolf,et al.  Living on the edge: a toy model for holographic reconstruction of algebras with centers , 2016, 1611.05841.

[27]  Barbara M. Terhal,et al.  Local decoders for the 2D and 4D toric code , 2016, Quantum Inf. Comput..

[28]  O. Gühne,et al.  Absolutely Maximally Entangled States of Seven Qubits Do Not Exist. , 2016, Physical Review Letters.

[29]  Ling-Yan Hung,et al.  Inspecting Non-Perturbative Contributions to the Entanglement Entropy via Wavefunctions , 2016, Entropy.

[30]  Ginestra Bianconi,et al.  Emergent Hyperbolic Network Geometry , 2016, Scientific Reports.

[31]  J. Maldacena,et al.  Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space , 2016, 1606.01857.

[32]  Arpan Bhattacharyya,et al.  Exploring the tensor networks/AdS correspondence , 2016, Journal of High Energy Physics.

[33]  J. Maldacena,et al.  Remarks on the Sachdev-Ye-Kitaev model , 2016, 1604.07818.

[34]  P. Hayden,et al.  Holographic duality from random tensor networks , 2016, 1601.01694.

[35]  J. Latorre,et al.  Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices , 2015, 1506.08857.

[36]  J. Preskill,et al.  Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence , 2015, 1503.06237.

[37]  Martin Rötteler,et al.  Quantum MDS codes over small fields , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[38]  Damian Markham,et al.  Scheme for constructing graphs associated with stabilizer quantum codes , 2014, 1407.2777.

[39]  A. Lubotzky,et al.  Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds , 2013, 1310.5555.

[40]  W. Cui,et al.  Absolutely Maximally Entangled States: Existence and Applications , 2013, 1306.2536.

[41]  Nicolas Delfosse,et al.  Tradeoffs for reliable quantum information storage in surface codes and color codes , 2013, 2013 IEEE International Symposium on Information Theory.

[42]  J. Latorre,et al.  Absolute maximal entanglement and quantum secret sharing , 2012, 1204.2289.

[43]  F. Verstraete,et al.  Entanglement renormalization for quantum fields in real space. , 2011, Physical review letters.

[44]  J I Cirac,et al.  Continuous matrix product states for quantum fields. , 2010, Physical review letters.

[45]  Kostas Skenderis,et al.  Holography and Wormholes in 2+1 Dimensions , 2009, Communications in Mathematical Physics.

[46]  B. Terhal,et al.  Tradeoffs for reliable quantum information storage in 2D systems , 2009, Quantum Cryptography and Computing.

[47]  Gilles Zémor,et al.  On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction , 2009, IWCC.

[48]  I. Kanitscheider,et al.  Universal hydrodynamics of non-conformal branes , 2009, 0901.1487.

[49]  Michal Horodecki,et al.  On Thermal Stability of Topological Qubit in Kitaev's 4D Model , 2008, Open Syst. Inf. Dyn..

[50]  I. Kanitscheider,et al.  Precision holography for non-conformal branes , 2008, Journal of High Energy Physics.

[51]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[52]  S. Beigi,et al.  Graph States Under the Action of Local Clifford Group in Non-Binary Case , 2006, quant-ph/0610267.

[53]  T. Takayanagi,et al.  Aspects of Holographic Entanglement Entropy , 2006, hep-th/0605073.

[54]  T. Matsui,et al.  Self-duality and phase structure of the 4D random-plaquette Z2 gauge model , 2004, hep-th/0409076.

[55]  Bart De Moor,et al.  Graphical description of the action of local Clifford transformations on graph states , 2003, quant-ph/0308151.

[56]  J. Eisert,et al.  Multiparty entanglement in graph states , 2003, quant-ph/0307130.

[57]  D. Schlingemann.  Stabilizer codes can be realized as graph codes , 2001, Quantum Inf. Comput..

[58]  J. Preskill,et al.  Topological quantum memory , 2001, quant-ph/0110143.

[59]  B. Sanders,et al.  Quantum encodings in spin systems and harmonic oscillators , 2001, quant-ph/0109066.

[60]  J. Ratcliffe,et al.  On the Davis hyperbolic 4-manifold , 2001 .

[61]  Dirk Schlingemann,et al.  Quantum error-correcting codes associated with graphs , 2000, ArXiv.

[62]  J. Preskill,et al.  Encoding a qubit in an oscillator , 2000, quant-ph/0008040.

[63]  K. Krasnov.  Holography and Riemann surfaces , 2000, hep-th/0005106.

[64]  E. Knill,et al.  Nonbinary quantum stabilizer codes , 2000, IEEE Trans. Inf. Theory.

[65]  A. Kitaev,et al.  Quantum codes on a lattice with boundary , 1998, quant-ph/9811052.

[66]  M. Freedman,et al.  Projective Plane and Planar Quantum Codes , 1998, Found. Comput. Math..

[67]  J. Maldacena,et al.  Supergravity and The Large N Limit of Theories With Sixteen Supercharges , 1998, hep-th/9802042.

[68]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[69]  E. Rains.  Quantum Weight Enumerators , 1996, IEEE Trans. Inf. Theory.

[70]  E. Knill,et al.  Theory of quantum error-correcting codes , 1996, quant-ph/9604034.

[71]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[72]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[73]  H. Zassenhaus,et al.  The Pauli matrices in n dimensions and finest gradings of simple Lie algebras , 1988 .

[74]  Aleksander Kubica,et al.  The ABCs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-Tolerant Quantum Computation and Quantum Phases Of Matter , 2018 .

[75]  B. Palomo,et al.  Single-Shot Fault-Tolerant Quantum Error Correction , 2014, 1404.5504.