Quantum Codes from Twisted Unitary $t$-groups

Abstract

We introduce twisted unitary $t$-groups, a generalization of unitary $t$-groups under a twisting by an irreducible representation. We then apply representation theoretic methods to the Knill-Laflamme error correction conditions to show that twisted unitary $t$-groups automatically correspond to quantum codes with distance $d=t+1$. By construction these codes have many transversal gates, which naturally do not spread errors and thus are useful for fault tolerance.

arXiv commentUpdated to PRL version: changed the title, removed the word "free", added a non-additive codes disclaimer, reorganized the supplemental material

References in corpus (23)

• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• A Theory of Quantum Error-Correcting Codes
• Chaos and complexity by design
• Restrictions on Transversal Encoded Quantum Gate Sets
• Evenly distributed unitaries: on the structure of unitary designs
• Magic State Distillation: Not as Costly as You Think
• Optimizing quantum process tomography with unitary 2-designs
• Unitary designs and codes
• A nonadditive quantum code
• LieART 2.0 -- A Mathematica Application for Lie Algebras and Representation Theory
• Permutation-invariant quantum codes
• Dual pairing of symmetry groups and dynamical groups in physics
• Encoding a qubit in a spin
• Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count
• Nonadditive quantum error-correcting code
• Explicit and spontaneous breaking of SU(3) into its finite subgroups
• Quantum circuits for exact unitary $t$-designs and applications to higher-order randomized benchmarking
• Super-Golden-Gates for PU(2)
• Comments on the classification of the finite subgroups of SU(3)
• Non-malleable encryption of quantum information
• A Family of Quantum Codes with Exotic Transversal Gates
• A family of permutationally invariant quantum codes
• On the explicit constructions of certain unitary $t$-designs

Cited by in corpus (8)

• Quantum error-correcting codes with a covariant encoding
• Covariant Quantum Error-Correcting Codes with Metrological Entanglement Advantage
• Classification of the Subgroups of the Two-Qubit Clifford Group
• Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry
• Bosonic quantum Fourier codes
• Quantum Codes and Irreducible Products of Characters
• Synthesis of Single Qutrit Circuits from Clifford+R
• Flexible Fault Tolerant Gate Gadgets