High-distance codes with transversal Clifford and T-gates

• URL: http://arxiv.org/abs/2408.12752v2
• Date: Thu, 21 Nov 2024 18:23:33 GMT
• Title: High-distance codes with transversal Clifford and T-gates
• Authors: Shubham P. Jain, Victor V. Albert,
• Abstract summary: We build qubit stabilizer codes that admit fault-tolerant implementations of logical gates with the fewest number of physical qubits. To our knowledge, our doubly even and triorthogonal families are the shortest qubit stabilizer codes of the same distance that can realize their respective gates.
• Score: 0.6138671548064355
• License:
• Abstract: The non-local interactions in several quantum devices allow for the realization of more compact quantum encodings while retaining the same degree of protection against noise. Anticipating that short to medium-length codes will soon be realizable, it is important to construct stabilizer codes that, for a given code distance, admit fault-tolerant implementations of logical gates with the fewest number of physical qubits. We extract high-distance doubly even codes from the quantum quadratic-residue code family that admit a transversal implementation of the single-qubit Clifford group and block transversal implementation of the full Clifford group. Applying a doubling procedure [arXiv:1509.03239] to such codes yields a family of high-distance weak triply even codes which admit a transversal implementation of the logical $\texttt{T}$-gate. Relaxing the triply even property, we also obtain a family of triorthogonal codes which requires an even lower overhead at the cost of additional Clifford gates to achieve the same logical operation. To our knowledge, our doubly even and triorthogonal families are the shortest qubit stabilizer codes of the same distance that can realize their respective gates.

Related papers

• Targeted Clifford logical gates for hypergraph product codes [61.269295538188636]
  We construct explicit targeted logical gates for hypergraph product codes. As a concrete example, we give logical circuits for the $[[18,2,3]]$ toric code.
  arXiv  Detail & Related papers  (2024-11-26T02:32:44Z)
• Measurement-free code-switching for low overhead quantum computation using permutation invariant codes [6.281229317487581]
  We present a measurement-free code-switching protocol for universal quantum computation. The novel non-Clifford gates enabled by this code-switching protocol enable implementation of a universal gate set more efficient than the Clifford$+T$ gate set.
  arXiv  Detail & Related papers  (2024-11-20T09:16:07Z)
• Transversal non-Clifford gates for quantum LDPC codes on sheaves [1.0878040851638]
  A major goal in quantum computing is to build a fault-tolerant quantum computer. One approach involves quantum low-density parity-check (qLDPC) codes that support non-Clifford gates.
  arXiv  Detail & Related papers  (2024-10-18T17:31:19Z)
• 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)
• 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)
• Improved rate-distance trade-offs for quantum codes with restricted connectivity [34.95121779484252]
  We study how the connectivity graph associated with a quantum code constrains the code parameters. We establish a tighter dimension-distance trade-off as a function of the size of separators in the connectivity graph.
  arXiv  Detail & Related papers  (2023-07-06T20:38:34Z)
• 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)
• Hierarchical memories: Simulating quantum LDPC codes with local gates [0.05156484100374058]
  Constant-rate low-density parity-check (LDPC) codes are promising candidates for constructing efficient fault-tolerant quantum memories. We construct a new family of hierarchical codes, that encode a number of logical qubits K = Omega(N/log(N)2. Under conservative assumptions, we find that the hierarchical code outperforms the basic encoding where all logical qubits are encoded in the surface code.
  arXiv  Detail & Related papers  (2023-03-08T18:48:12Z)
• Transversal Injection: A method for direct encoding of ancilla states for non-Clifford gates using stabiliser codes [55.90903601048249]
  We introduce a protocol to potentially reduce this overhead for non-Clifford gates. Preliminary results hint at high quality fidelities at larger distances.
  arXiv  Detail & Related papers  (2022-11-18T06:03:10Z)
• 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)
• The cost of universality: A comparative study of the overhead of state distillation and code switching with color codes [63.62764375279861]
  We compare two leading FT implementations of the T gate in 2D color codes under circuit noise. We find a circuit noise threshold of 0.07(1)% for the T gate via code switching, almost an order of magnitude below that achievable by state distillation in the same setting.
  arXiv  Detail & Related papers  (2021-01-06T19:00:01Z)

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.