Exact quantum circuits for a Heisenberg triangle with Dzyaloshinskii-Moriya interaction

Abstract

We construct exact quantum circuits for a three-spin Heisenberg triangle with Dzyaloshinskii-Moriya (DM) interaction. A change of basis reduces pure DM evolution with arbitrary couplings to two single-qubit rotations, giving a circuit with at most 8 CNOT gates. When $J$ and $D$ are each the same on all bonds, one fixed basis gives a circuit with at most 10 CNOT gates for any real $J,D,t$. Local spin rotations extend the construction to a family with unequal DM couplings and nonzero exchange, requiring at most 14 CNOT gates. All circuits act on the full Hilbert space. An alternative pure DM implementation uses five two-qubit DM gates: analytically retuning the angles of one symmetric Strang step makes it exact. A 12-spin kagome example illustrates the use of these exact local blocks in lattice simulation.