Hamiltonian Engineering for High Fidelity Quantum Operations

High-fidelity gates and operations are crucial to almost every aspect of quantum information processing. In recent experiments~[1], fidelity is mostly limited by unwanted couplings with states living out of the logical subspace. This results in both leakage and phase errors. Here, we present a general method to deal simultaneously with both these issues and improve the fidelity of quantum gates and operations. Our method is applicable to a wide variety of systems. As an example, we can correct gates for superconducting qubits~[1], improve coherent state transfer between a single NV centre electronic spin and a single nitrogen nuclear spin~[2], improve control over a nuclear spin ensemble~[3], etc. Our method is intimately linked to the Magnus expansion. By modifying the Magnus expansion of an initially given Hamiltonian $H_{\mathrm{i}}$, we find analytically additional control Hamiltonians $H_{\mathrm{ctrl}}$ such that $H_{\mathrm{i}} + H_{\mathrm{ctrl}}$ leads to the desired gate while minimizing both leakage and phase errors. \newline \vskip\baselineskip \noindent [1] Zijun Chen, \emph{et al.}, arXiv:1509.05470. \newline [2] G. D. Fuchs, \emph{et al.}, Nat. Phys. \textbf{7}, 789–793 (2011). \newline [3] Mathieu Munsch, \emph{et al.}, Nat. Nano. 9, 671–675 (2014).