Arbitrarily Accurate Dynamical Control in Open Quantum Systems

We show that open-loop dynamical control techniques may be used to synthesize unitary transformations in open quantum systems in such a way that decoherence is perturbatively compensated for to a desired (in principle arbitrarily high) level of accuracy, which depends only on the strength of the relevant errors and the achievable rate of control modulation. Our constructive and fully analytical solution employs concatenated dynamically corrected gates, and is applicable independently of detailed knowledge of the system-environment interactions and environment dynamics. Explicit implications for boosting quantum gate fidelities in realistic scenarios are addressed.

Figure 1

Fidelity of increasing DCG orders for $Q=\exp [-i(\pi /3)X]$ applied to $|\psi ⟩=(|0⟩+|1⟩)/\sqrt{2}$, and ${\rho }_B$ fully mixed. The fidelity is $f\equiv \mathrm{Tr}\sqrt{\sqrt{{\rho }_a}{\rho }_t\sqrt{{\rho }_a}}$, where ${\rho }_t$ (${\rho }_a$) is the target (actual) final qubit state, and satisfies $1-\Delta \le f\le \sqrt{1-{\Delta }^2}$ [30]. The bath spins couple to the central system spin $\mathbf{S}$ via a Heisenberg interaction $H_{SB}=\sum _{i=1}^5{j}_i\mathbf{S}·{\mathbf{I}}^{(i)}$, with $j_i$ randomly picked in [0, $J$], $J\equiv 10$ (arbitrary units). The bath spins $I^{(j)}$ evolve under a dipolar interaction $H_B=\sum _{i,j}{b}_{ij}(I_X^{(i)}{I}_X^{(j)}+I_Y^{(i)}{I}_Y^{(j)}-2I_Z^{(i)}{I}_Z^{(j)})$, with $b_{ij}$ randomly picked in [0, ${10}^{-2}$]. Primitive gates were implemented using rectangular pulse shapes to allow for numerically exact simulations. For fixed $J$, reducing ${\tau }_{\min }$ may be understood in terms of a finer temporal resolution of the resulting “digitized pulse profile”. Note that at ${log}_{10}({\tau }_{\min }J)=-5.5$ the fidelity for ${\mathrm{DCG}}^{[3]}$ is 13 orders of magnitude better than for ${\mathrm{DCG}}^{[0]}$, in spite of the gate taking nearly 6500 times longer.