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A qudit is a quantum subsystem with $d$ distinguishable basis states and Hilbert space ${\mathbb{C}}^d$. A qubit is the special case $d=2$. Tensor products of qudits give a natural setting for local Hamiltonians and finite-dimensional quantum circuits.

A stoquastic Hamiltonian is a real Hermitian Hamiltonian operator with nonpositive off-diagonal entries in a specified basis, usually the computational basis. The property depends on that basis. Permutation matrices in a quantum circuit's propagation terms and $|-⟩⟨-|$ input penalties give a useful family of such operators.

The inversion about the mean for four computational-basis states can be implemented using Hadamard gates, Pauli X gates and one controlled-NOT gate, up to global phase. Let $Q=H\otimes H$ and $P=X\otimes X$. Since $CZ=(I\otimes H)\mathrm{CNOT}(I\otimes H)=I-2|11⟩⟨11|$, one has $PCZP=I-2|00⟩⟨00|$ and therefore $QPCZPQ=I-2|++⟩⟨++|=-D$. This gives an exact quantum circuit representative of the four-entry Grover search algorithm diffusion step modulo its irrelevant overall minus sign.