Many-body strategies for multiqubit gates: Quantum control through Krawtchouk-chain dynamics

We propose a strategy for engineering multiqubit quantum gates. As a first step, it employs an eigengate to map states in the computational basis to eigenstates of a suitable many-body Hamiltonian. The second step employs resonant driving to enforce a transition between a single pair of eigenstates, leaving all others unchanged. The procedure is completed by mapping back to the computational basis. We demonstrate the strategy for the case of a linear array with an even number $N$ of qubits, with specific $XX+YY$ couplings between nearest neighbors. For this so-called Krawtchouk chain, a two-body driving term leads to the ${\mathrm{iSWAP}}_N$ gate, which we numerically test for $N=4$ and 6.

Figure 1

Protocols for the proposed ${\mathrm{iSWAP}}_N$ gates. (a) The $N=6$ qubit chain (spheres) evolving under the Krawtchouk Hamiltonian (solid lines). The driving Hamiltonian $H_D^{(1,-)}$ is depicted as the corkscrew line. (b) Field strengths as a function of time. (c) The spectrum of $H^K$ for $N=4$, with the resonant transition depicted as the curvy line.

Figure 2

Fidelities of the resonant driving part of the ${\mathrm{iSWAP}}_4$ and ${\mathrm{iSWAP}}_6$ protocols, including the halfway inversion. The thick lines indicate the errors in the ideal case, thin lines under various values of noise $\varepsilon$. As in the prototypical example Eq. (1), the errors fall off like ${\tau }_D^{-2}$ (dashed), until the noise $\varepsilon$ becomes the leading source of errors.

Figure 3

Numerical results of the trace error compared to the analytical eigengate for various noise amplitudes, for $N=2,4,8$, and 12. The error scales roughly as $\mathcal{E}\propto {\varepsilon }^2$ until the error saturates. Results are averages of at least 110 runs.