Efficient quantum gate teleportation in higher dimensions

Abstract

The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and 'nearly diagonal' semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states. Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting. Our contribution is to leverage the discrete Stone-von Neumann theorem and the symplectic formalism of qudit stabiliser mechanics towards extending results of Zeng-Cheng-Chuang (2008) and Beigi-Shor (2010) to higher dimensions in a uniform manner. We further give a simple algorithm for recursively enumerating all gates of the Clifford hierarchy, a simple algorithm for recognising and diagonalising semi-Clifford gates, and a concise proof of the classification of the diagonal Clifford hierarchy gates due to Cui-Gottesman-Krishna (2016) for the single-qudit case. We generalise the efficient gate teleportation protocols of semi-Clifford gates to the qudit setting and prove that every third level gate of one qudit (of any prime dimension) and of two qutrits can be implemented efficiently. Numerical evidence gathered via the aforementioned algorithms support the conjecture that higher-level gates can be implemented efficiently.

arXiv commentDraft version. Comments, questions, & corrections welcome

References in corpus (7)

• Quantum Teleportation is a Universal Computational Primitive
• Methodology for quantum logic gate constructions
• Hudson's Theorem for finite-dimensional quantum systems
• Diagonal gates in the Clifford hierarchy
• Semi-Clifford operations, structure of ${\mathcal{C}}_k$ hierarchy, and gate complexity for fault-tolerant quantum computation
• Unifying the Clifford Hierarchy via Symmetric Matrices over Rings
• Un-Weyl-ing the Clifford Hierarchy

Cited by in corpus (12)

• On Groups in the Qubit Clifford Hierarchy
• Hierarchies of resources for measurement-based quantum computation
• Pauli-based model of quantum computation with higher-dimensional systems
• The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification
• Scaling W state circuits in the qudit Clifford hierarchy
• Nonlocal transfer of high-dimensional unitary operations
• Fast algorithms for classical specifications of stabiliser states and Clifford gates
• No quantum solutions to linear constraint systems in odd dimension from Pauli group and diagonal Cliffords
• Path superposition activating perfect quantum teleportation ability for separable states
• Quantum advantage in temporally flat measurement-based quantum computation
• Unitary and non-unitary operators leverage perfect and imperfect single qutrit teleportation
• Transversal AND in Quantum Codes