Order 3 Symmetry in the Clifford Hierarchy

Abstract

We investigate the action of the first three levels of the Clifford hierarchy on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop vectors splits into three Clifford orbits. These vectors form configurations with so-called Zauner subspaces, eigenspaces of order 3 elements of the Clifford group highly relevant to the SIC problem. We identify Alltop vectors as the magic states that appear in the context of fault-tolerant universal quantum computing, wherein the appearance of distinct Clifford orbits implies a surprising inequivalence between some magic states.

arXiv comment20 pages, 2 figures. Published version

References in corpus (20)

• Quantum Teleportation is a Universal Computational Primitive
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Quantum Computing with Very Noisy Devices
• Symmetric Informationally Complete Quantum Measurements
• A Theory of Fault-Tolerant Quantum Computation
• A new proof for the existence of mutually unbiased bases
• The Resource Theory of Stabilizer Computation
• SIC-POVMs and the Extended Clifford Group
• Enhanced fault-tolerant quantum computing in $d$-level systems
• Equiangular lines, mutually unbiased bases, and spin models
• Magic state distillation in all prime dimensions using quantum Reed-Muller codes
• Qudit versions of the qubit "pi-over-eight" gate
• Qutrit Magic State Distillation
• SIC~POVMs and Clifford groups in prime dimensions
• On SIC-POVMs in Prime Dimensions
• Properties of the extended Clifford group with applications to SIC-POVMs and MUBs
• Abstract configurations in algebraic geometry
• Linear Dependencies in Weyl-Heisenberg Orbits
• SIC-POVMs and MUBs: Geometrical Relationships in Prime Dimension
• Orbits of mutually unbiased bases

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• Maximum nonlocality and minimum uncertainty using magic states
• Designing the Quantum Channels Induced by Diagonal Gates
• States that are far from being stabilizer states
• On Groups in the Qubit Clifford Hierarchy
• Climbing the Diagonal Clifford Hierarchy
• Classical codes in quantum state space
• Classical Coding Approaches to Quantum Applications
• No quantum solutions to linear constraint systems in odd dimension from Pauli group and diagonal Cliffords
• SICs and the Triangle Group $(3,3,3)$