Mark Howard, National University of Ireland, Maynooth
Abstract
When visualized as an operation on the Bloch sphere, the qubit
π/8 gate corresponds to 1/8 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, if it supplements the set of
Clifford gates, then universal quantum computation can be achieved. The
π/8 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 conjectured that these gates could be useful for
the task of qudit magic-state distillation and, by extension,
fault-tolerant quantum computing. Very recently, independent work by
Campbell et al. confirmed 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. Finally, turning
our attention to nonlocality, we show that these gates achieve the
optimal quantum strategy for the generalized qudit CHSH inequality.