Code conversion with the quantum Golay code for a universal transversal gate set

Abstract

The $[[7,1,3]]$ Steane code and $[[23,1,7]]$ quantum Golay code have been identified as good candidates for fault-tolerant quantum computing via code concatenation. These two codes have transversal implementations of all Clifford gates, but require some other scheme for fault-tolerant $T$ gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault-tolerantly without the use of magic state distillation. Analogously to how the $[[7,1,3]]$ Steane code can be fault-tolerantly converted to and from the $[[15,1,3]]$ Reed-Muller code which has a transversal $T$ gate, the $[[23,1,7]]$ Golay code can be converted to a $[[95,1,7]]$ triorthogonal code with a transversal $T$ gate. A crucial ingredient to this procedure is the $[[49,1,5]]$ triorthogonal code, which can itself be seen as related to the self-dual $[[17,1,5]]$ 2D color code. Additionally, a method for code conversion based on a transversal CNOT between the codes, rather than stabilizer measurements, is described.

11 pages, 3 figures, equivalent to published version

References in corpus (20)

• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Topological Quantum Distillation
• Restrictions on Transversal Encoded Quantum Gate Sets
• Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States
• Magic state distillation with low overhead
• Quantum Low-Density Parity-Check Codes
• Demonstration of quantum error correction and universal gate set on a binomial bosonic logical qubit
• Universal fault-tolerant quantum computation with only transversal gates and error correction
• Magic State Distillation: Not as Costly as You Think
• Topological Computation without Braiding
• Fault-tolerant conversion between the Steane and Reed-Muller quantum codes
• Quantum Error Correction with the Gottesman-Kitaev-Preskill Code
• Using concatenated quantum codes for universal fault-tolerant quantum gates
• Thresholds for universal concatenated quantum codes
• Stacked codes: universal fault-tolerant quantum computation in a two-dimensional layout
• Partitioning qubits in hypergraph product codes to implement logical gates
• Gauge color codes in two dimensions
• Quantum error correction with the color-Gottesman-Kitaev-Preskill code
• Fault-tolerant gates via homological product codes
• Efficient Preparation of Large Block Code Ancilla States for Fault-tolerant Quantum Computation

Cited by in corpus (6)

• Experimental Demonstration of High-Fidelity Logical Magic States from Code Switching
• Code switching revisited: Low-overhead magic state preparation using color codes
• Efficient fault-tolerant code switching via one-way transversal CNOT gates
• Transversal Clifford and T-gate codes of short length and high distance
• Efficient simulation of logical magic state preparation protocols
• Magic tricycles: Efficient magic state generation with finite block-length quantum LDPC codes