Qudit versions of the qubitπ/8gate

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 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.

Order 3 symmetry in the Clifford hierarchy

“…In dimensions of the form p = 2 mod 3, every magic state can be converted to an equivalent magic state by means of a Clifford gate, meaning that all these magic states exhibit the same amount of robustness to noise [10] or mana (quantified as a resource [37]). In dimensions p = 1 mod 3, the partitioning of Alltop vectors into three distinct Clifford orbits means this no longer holds true, and magic states from a particular orbit can be preferable to states from the remaining orbits.…”

“…We assume that p > 3, but analogues exist also for p = 3, and for p = 2 where the analogue of M is known as the pi-over-eight gate. Then the analogous abelian subgroups are Z 3 × Z 9 and Z 8 , respectively [10].…”

Qudit color codes and gauge color codes in all spatial dimensions

“…For prime dimensions, the first investigation characterizing all the phase gates from the third level of the CH was performed by Howard and Vala [4]. In general, there is a close correspondence between the order of the polynomial in the exponent of ω and the lowest level of the CH the phase gate belongs to.…”

“…Such a gate is not unique and in the qubit case, the T gate diag (1,e iπ/4 ) is usually chosen for this purpose. For the qudit case, we choose the following particularly convenient definition for the T gate, which is valid in all dimensions except when d = 2,3,6 [4,41],…”

“…Indeed, Knuth has advocated the use of balanced ternary, a three-state classical logic [1]. In the quantum domain, qudits offer a state space with a richer structure than their two-level counterparts and the merits of this for quantum information have been explored in many contexts [2][3][4][5][6][7][8][9][10]. Recent experiments have shown that even large-d quantum systems can be precisely controlled [11,12].…”