Qudit versions of the qubit "pi-over-eight" gate

Abstract

When visualised as an operation on the Bloch sphere, the qubit "pi-over-eight" gate corresponds to one-eighth of a complete rotation about the vertical axis. This simple gate often plays an important role in quantum information theory, typically in situations for which Pauli and Clifford gates are insufficient. Most notably, when it supplements the set of Clifford gates then universal quantum computation can be achieved. The "pi-over-eight" gate is the simplest example of an operation from the third level of the Clifford hierarchy (i.e., it maps Pauli operations to Clifford operations under conjugation). Here we derive explicit expressions for all qudit (d-level, where d is prime) versions of this gate and analyze the resulting group structure that is generated by these diagonal gates. This group structure differs depending on whether the dimensionality of the qudit is two, three or greater than three. We then discuss the geometrical relationship of these gates (and associated states) with respect to Clifford gates and stabilizer states. We present evidence that these gates are maximally robust to depolarizing and phase damping noise, in complete analogy with the qubit case. Motivated by this and other similarities we conjecture that these gates could be useful for the task of qudit magic-state distillation and, by extension, fault-tolerant quantum computing. Very recent, independent work by Campbell, Anwar and Browne confirms the correctness of this intuition, and we build upon their work to characterize noise regimes for which noisy implementations of these gates can (or provably cannot) supplement Clifford gates to enable universal quantum computation.

Version 2 changed to reflect improved distillation routines in arXiv:1205.3104v2. Minor typos fixed. 12 Pages,2 Figures,3 Tables

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• Qutrit and Ququint Magic States
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• Magic State Distillation with the Ternary Golay Code
• Graph-theoretic strengths of contextuality
• Stabilizer rank and higher-order Fourier analysis
• Towers of generalized divisible quantum codes
• Magic informationally complete POVMs with permutations
• Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits
• The magic of universal quantum computing with permutations
• Orbits of mutually unbiased bases
• Universal quantum computing with parafermions assisted by a half fluxon
• States that are far from being stabilizer states
• Hybrid quantum computing with ancillas
• Improved Simulation of Quantum Circuits by Fewer Gaussian Eliminations
• Tight bounds for antidistinguishability and circulant sets of pure quantum states
• Contextual bound states for qudit magic state distillation
• Pauli-based model of quantum computation with higher-dimensional systems
• Machine-learning-inspired quantum optimal control of nonadiabatic geometric quantum computation via reverse engineering
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• Qudit Shadow Estimation Based on the Clifford Group and the Power of a Single Magic Gate
• Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates
• Building Qutrit Diagonal Gates from Phase Gadgets
• Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits
• Scaling W state circuits in the qudit Clifford hierarchy
• Complete ZX-calculi for the stabiliser fragment in odd prime dimensions
• Synthesis and Arithmetic of Single Qutrit Circuits
• Superselection rules and bosonic quantum computational resources
• Improved Quantum Ternary Arithmetics
• Ab Initio Polaritonic Chemistry on Diverse Quantum Computing Platforms: Qubit, Qudit, and Hybrid Qubit-Qumode Architectures
• No quantum solutions to linear constraint systems in odd dimension from Pauli group and diagonal Cliffords
• Low Overhead Qutrit Magic State Distillation
• Robust Control and Entanglement of Qudits in Neutral Atom Arrays
• Clifford operators in SU(N)1; N not odd prime
• Nonlinearity of the Fidelity in Open Qudit Systems: Gate and Noise Dependence in High-dimensional Quantum Computing
• SU(N)1 Chern-Simons theory, the Clifford group, and Entropy Cone
• Blind quantum computing with different qudit resource state architectures
• Efficient Implementation of a Single-Qutrit Gate Set via Coherent Control
• Synthesis of Single Qutrit Circuits from Clifford+R
• Constructing all qutrit controlled Clifford+T gates in Clifford+T
• Qudit Clauser-Horne-Shimony-Holt Inequality and Nonlocality from Wigner Negativity
• Hypergraph States in SU(N)1, N odd prime, Chern-Simons Theory