Qudit versions of the qubit $\pi /8$ gate

When visualized as an operation on the Bloch sphere, the qubit $\pi /8$ 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 $\pi /8$ 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 conjecture 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.

Figure 1

Geometry of states and operations: Filled circles (vertices) correspond to stabilizer states or Clifford gates, respectively. (a) Polytope of stabilizer states: A single qudit has $p(p+1)$ stabilizer states, the convex hull of which comprises a polytope with $p^{p+1}$ faces. The states $|{\psi }_{U_{\upsilon }}\rangle$ are the farthest outside the polytope of all states $|{\psi }_{U_{\theta }}\rangle$ [defined in Eq. (35)]. For $p=2$, the stabilizer polytope corresponds to an octahedron (as depicted) with six vertices and eight faces. In this dimension, $|{\psi }_{U_{\upsilon }}\rangle =|H\rangle$, which is the magic state introduced by Refs. [14, 15]. Recent results by Campbell et al. [17] show that $|{\psi }_{U_{\upsilon }}\rangle$ is a magic state for all $p$. (b) Polytope of Clifford gates: A schematic picture [the object is actually ${(p^2-1)}^2$ dimensional] of the polytope whose vertices correspond to Clifford gates. It is known with certainty in $p=2$, and with a high degree of plausibility in $p\in \{3,5,7\}$ [16], that the gates $U_{\upsilon }$ are farthest outside the Clifford polytope of all $U\in U\left(p\right)$. Here, each unitary gate, $U_{\upsilon }$ or $C_{\left(F\right|\vec{\chi })}\in \mathcal{C}$ [refer to Eq. (8)], is represented by its Jamiołkowski state [as defined in Eq. (36)].

Figure 2

Dilution of noise in magic-state preparation: (a) Straightforward magic-state preparation, using the superoperator ${\mathcal{E}}_{\left(\right|{\psi }_{U_{\upsilon }}\rangle ,\varepsilon )}$ to prepare an imperfect (depolarized) version of a magic state $|{\psi }_{U_{\upsilon }}\rangle$. (b) Postselected magic-state preparation, using the same superoperator ${\mathcal{E}}_{\left(\right|{\psi }_{U_{\upsilon }}\rangle ,\varepsilon )}$ to create a less imperfect (${\varepsilon }^{\prime }<\varepsilon$) version of the magic state $|{\psi }_{U_{\upsilon }}\rangle$. The Pauli measurement operator $\Pi$ stands for postselection on receiving the outcome $+1$ using the measurement ${\Pi }_{(0,0|1,p-1\left)\right[0]}$ [see Eq. (62)]. The circuit elements to the left of $\Pi$ implement the creation of the Jamiołkowski state ${\varrho }_{\mathcal{E}}$ [see Eq. (38)] describing ${\mathcal{E}}_{\left(\right|{\psi }_{U_{\upsilon }}\rangle ,\varepsilon )}$.