A Family of Quantum Codes with Exotic Transversal Gates

Abstract

Recently an algorithm has been constructed that shows the binary icosahedral group $\2I$ together with a $T$-like gate forms the most efficient single-qubit universal gate set. To carry out the algorithm fault tolerantly requires a code that implements $\ico$ transversally. However, no such code has ever been demonstrated in the literature. We fill this void by constructing a family of distance $d=3$ codes that all implement $2I$ transversally. A surprising feature of this family is that the codes can be deduced entirely from symmetry considerations that only $2I$ affords.

arXiv commentSignificant Revision. Modernized tables. Updated proof techniques

References in corpus (24)

• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• A Theory of Quantum Error-Correcting Codes
• Restrictions on Transversal Encoded Quantum Gate Sets
• Stim: a fast stabilizer circuit simulator
• Simulation of quantum circuits by low-rank stabilizer decompositions
• Detecting multiparticle entanglement of Dicke states
• Magic State Distillation: Not as Costly as You Think
• Efficient Discrete Approximations of Quantum Gates
• Codeword Stabilized Quantum Codes
• Robustness of Magic and Symmetries of the Stabiliser Polytope
• A nonadditive quantum code
• Using concatenated quantum codes for universal fault-tolerant quantum gates
• All entangled pure states violate a single Bell's inequality
• Permutation-invariant quantum codes
• Dual pairing of symmetry groups and dynamical groups in physics
• Preparation of Dicke States in an Ion Chain
• Diagonal gates in the Clifford hierarchy
• Encoding a qubit in a spin
• Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count
• Nonadditive quantum error-correcting code
• The disjointness of stabilizer codes and limitations on fault-tolerant logical gates
• Semi-Clifford operations, structure of ${\mathcal{C}}_k$ hierarchy, and gate complexity for fault-tolerant quantum computation
• Super-Golden-Gates for PU(2)
• A Non-Commuting Stabilizer Formalism

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• Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates
• Class of codes correcting absorptions and emissions
• Classification of the Subgroups of the Two-Qubit Clifford Group
• Semidefinite programming bounds on the size of entanglement-assisted codeword stabilized quantum codes
• Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry
• State-adaptive quantum error correction and fault-tolerant quantum computing
• Bosonic quantum Fourier codes
• Measurement-free code-switching for low overhead quantum computation using permutation invariant codes
• Tessellation codes: encoded quantum gates by geometric rotation
• Experimental measurement and a physical interpretation of quantum shadow enumerators
• Flexible Fault Tolerant Gate Gadgets
• Synthesis of Single Qutrit Circuits from Clifford+R