Diagonal gates in the Clifford hierarchy

Abstract

The Clifford hierarchy is a set of gates that appears in the theory of fault-tolerant quantum computation, but its precise structure remains elusive. We give a complete characterization of the diagonal gates in the Clifford hierarchy for prime-dimensional qudits. They turn out to be $p^m$-th roots of unity raised to polynomial functions of the basis state to which they are applied, and we determine which level of the Clifford hierarchy a given gate sits in based on $m$ and the degree of the polynomial.

arXiv comment8 pages

References in corpus (7)

• Quantum Teleportation is a Universal Computational Primitive
• Restrictions on Transversal Encoded Quantum Gate Sets
• Classification of topologically protected gates for local stabilizer codes
• Qudit versions of the qubit "pi-over-eight" gate
• Semi-Clifford operations, structure of ${\mathcal{C}}_k$ hierarchy, and gate complexity for fault-tolerant quantum computation
• Maximum nonlocality and minimum uncertainty using magic states
• Order 3 Symmetry in the Clifford Hierarchy

Cited by in corpus (39)

• Many-body quantum magic
• ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity
• Towards low overhead magic state distillation
• Resource theory of quantum scrambling
• On Optimality of CSS Codes for Transversal $T$
• Efficient classical simulation of Clifford circuits with nonstabilizer input states
• Normal form for single-qutrit Clifford+T operators and synthesis of single-qutrit gates
• Universal MBQC with generalised parity-phase interactions and Pauli measurements
• Unifying the Clifford Hierarchy via Symmetric Matrices over Rings
• A Family of Quantum Codes with Exotic Transversal Gates
• Magic State Distillation with the Ternary Golay Code
• Efficient quantum gate teleportation in higher dimensions
• Un-Weyl-ing the Clifford Hierarchy
• Transversal Diagonal Logical Operators for Stabiliser Codes
• Quantum Pin Codes
• Classical Coding Problem from Transversal $T$ Gates
• Designing the Quantum Channels Induced by Diagonal Gates
• Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers
• On Groups in the Qubit Clifford Hierarchy
• Hierarchies of resources for measurement-based quantum computation
• Accelerating Quantum Algorithms with Precomputation
• The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification
• Climbing the Diagonal Clifford Hierarchy
• Fault-tolerant compiling of classically hard IQP circuits on hypercubes
• Building Qutrit Diagonal Gates from Phase Gadgets
• Clifford operations and homological codes for rotors and oscillators
• Geometric structure and transversal logic of quantum Reed-Muller codes
• Letting the tiger out of its cage: bosonic coding without concatenation
• Scaling W state circuits in the qudit Clifford hierarchy
• Synthesis and Arithmetic of Single Qutrit Circuits
• Classical Coding Approaches to Quantum Applications
• Low Overhead Qutrit Magic State Distillation
• No quantum solutions to linear constraint systems in odd dimension from Pauli group and diagonal Cliffords
• Quantum Theory from Principles, Quantum Software from Diagrams
• Instantaneous Quantum Polynomial-Time Sampling and Verifiable Quantum Advantage: Stabilizer Scheme and Classical Security
• Characterization of non-adaptive Clifford channels
• Scalable Spider Nests (...Or How to Graphically Grok Transversal Non-Clifford Gates)
• The Multiqubit Elegant Joint Measurement
• Symmetric Localizable Multipartite Quantum Measurements from Pauli Orbits