Genons, Double Covers and Fault-tolerant Clifford Gates

• URL: http://arxiv.org/abs/2406.09951v1
• Date: Fri, 14 Jun 2024 11:57:51 GMT
• Title: Genons, Double Covers and Fault-tolerant Clifford Gates
• Authors: Simon Burton, Elijah Durso-Sabina, Natalie C. Brown,
• Abstract summary: We show a construction that produces symplectic double code with naturally occurring fault-tolerant logical Clifford gates. We demonstrate this experimentally on Quantinuum's H1-1 trapped-ion quantum computer.
• Score: 2.5866180357107242
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A great deal of work has been done developing quantum codes with varying overhead and connectivity constraints. However, given the such an abundance of codes, there is a surprising shortage of fault-tolerant logical gates supported therein. We define a construction, such that given an input $[[n,k,d]]$ code, yields a $[[2n,2k,\ge d]]$ symplectic double code with naturally occurring fault-tolerant logical Clifford gates. As applied to 2-dimensional $D(\mathbb{Z}_2)$-topological codes with genons (twists) and domain walls, we find the symplectic double is genon free, and of possibly higher genus. Braiding of genons on the original code becomes Dehn twists on the symplectic double. Such topological operations are particularly suited for architectures with all-to-all connectivity, and we demonstrate this experimentally on Quantinuum's H1-1 trapped-ion quantum computer.

Related papers

• Low-Overhead Entangling Gates from Generalised Dehn Twists [7.059472280274009]
  We generalise the implementation of logical quantum gates via Dehn twists from topological codes to the hypergraph and balanced products of cyclic codes. These generalised Dehn twists implement logical entangling gates with no additional qubit overhead and $mathcalO(d)$ time overhead.
  arXiv  Detail & Related papers  (2024-11-05T17:55:46Z)
• Geometric structure and transversal logic of quantum Reed-Muller codes [51.11215560140181]
  In this paper, we aim to characterize the gates of quantum Reed-Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. A set of stabilizer generators for a RM code can be described via $X$ and $Z$ operators acting on subcubes of particular dimensions.
  arXiv  Detail & Related papers  (2024-10-10T04:07:24Z)
• Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates [23.22566380210149]
  We construct quantum codes that support $CCZ$ gates over qudits of arbitrary prime power dimension $q$. The only previously known construction with such linear dimension and distance required a growing alphabet size $q$.
  arXiv  Detail & Related papers  (2024-08-17T16:54:51Z)
• Logical Operators and Fold-Transversal Gates of Bivariate Bicycle Codes [1.8416014644193066]
  Quantum low-density parity-check (qLDPC) codes offer a promising route to scalable fault-tolerant quantum computation with constant overhead. Recent advancements have shown that qLDPC codes can outperform the quantum memory capability of surface codes even with near-term hardware.
  arXiv  Detail & Related papers  (2024-07-04T14:49:35Z)
• Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries [1.3194391758295114]
  We study fault-tolerant quantum computing for families of homological quantum low-density parity-check codes defined on 3-manifolds with constant or almost-constant encoding rate. We have developed a generic formalism to compute the triple intersection invariants for 3-manifolds.
  arXiv  Detail & Related papers  (2023-10-25T20:33:59Z)
• A Family of Quantum Codes with Exotic Transversal Gates [0.0]
  An algorithm shows the binary icosahedral group $2I$ together with a $T$-like gate forms the most efficient single-qubit gate set. To carry out the algorithm fault tolerantly requires a code that implements $ico$ly. We fill this void by constructing a family of distanced = 3$ codes that all implement $2I$ly.
  arXiv  Detail & Related papers  (2023-05-11T17:58:29Z)
• Homological Quantum Rotor Codes: Logical Qubits from Torsion [51.9157257936691]
  homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code. We show that the $0$-$pi$-qubit as well as Kitaev's current-mirror qubit are indeed small examples of such codes.
  arXiv  Detail & Related papers  (2023-03-24T00:29:15Z)
• Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
  We study quantum Ordered Binary Decision Diagrams($OBDD$) model. We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables. We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
  arXiv  Detail & Related papers  (2022-04-22T12:37:56Z)
• Morphing quantum codes [77.34726150561087]
  We morph the 15-qubit Reed-Muller code to obtain the smallest known stabilizer code with a fault-tolerant logical $T$ gate. We construct a family of hybrid color-toric codes by morphing the color code.
  arXiv  Detail & Related papers  (2021-12-02T17:43:00Z)
• Finding the disjointness of stabilizer codes is NP-complete [77.34726150561087]
  We show that the problem of calculating the $c-disjointness, or even approximating it to within a constant multiplicative factor, is NP-complete. We provide bounds on the disjointness for various code families, including the CSS codes,$d codes and hypergraph codes. Our results indicate that finding fault-tolerant logical gates for generic quantum error-correcting codes is a computationally challenging task.
  arXiv  Detail & Related papers  (2021-08-10T15:00:20Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.