Fig. 2: Grover’s algorithm implemented with a single qudit.
From: Efficient implementation of a quantum algorithm with a trapped ion qudit
a The qudit is first prepared in the \(\left|0\right\rangle\) state, followed by a Hadamard transformation that generates the equal superposition state \(\left|s\right\rangle\). For each iteration, the oracle O is applied to mark the target state, and the reflection operator \(2\left|s\right\rangle \left\langle s\right|-I\) amplifies the population of the marked state. The phase factor of eiπ/d is needed if d is even; otherwise, the reflection operator cannot be implemented using SU(d) operations due to its negative determinant. A comparison of two circuits implementing the phase oracle with a d = 8 qudit (b), which consists of three displacement pulses \(D({{{\boldsymbol{\varphi }}}},\theta )\equiv {e}^{-i\theta {H}_{rot}({{{\boldsymbol{\varphi }}}})}\) where Hrot is defined in (5), and with three qubits (c), which requires a three-qubit Toffoli gate. In (b), the displacement pulse parameters {φi, θi} are given in Supplementary Information and are found using a gradient-descent-based numerical optimization method (see “Method”s).
