Selective recoupling and stochastic dynamical decoupling

An embedded selective recoupling method is proposed which is based on the idea of embedding the recently proposed deterministic selective recoupling scheme of Yamaguchi et al. into a stochastic dynamical decoupling method, such as the recently proposed Pauli-random-error-correction scheme. The recoupling scheme enables the implementation of elementary quantum gates in a quantum information processor by partial suppression of the unwanted interactions. The random dynamical decoupling method cancels a significant part of the residual interactions. Thus the time scale of reliable quantum computation is increased significantly. Numerical simulations are presented for a conditional two-qubit swap gate and for a complex iterative quantum algorithm.

Figure 1

Schematic representation of the unitary ${\widehat{W}}_{xy}^{kl}$ quantum gate acting on qubits $k$ and $l$: Free evolution indicates time evolution according to the Hamiltonian ${\widehat{H}}_D$ over a time interval of duration $\tau$.

Figure 2

(Color online) Schematic representation of a conditional two-qubit gate (${\widehat{U}}_{\varphi }$-gate) obtained by recoupling qubits $k$ and $l$ according to Eq. (5): The ${\widehat{W}}^{*}$ gates are obtained by reversing the order of the broadband pulses suppressing the Zeeman term. Residual second-order terms of AHT be eliminated by the restricted randomization step accomplished by random selective $\pi$-pulses ${\widehat{\sigma }}_{r_{i,j}}$. Thereby $r_{i,k}$ has to be equal to $r_{i,l}$ to ensure that the wanted gate action is not disturbed. Still remaining terms are symmetrized by random $\pi ∕2$-pulses ${\widehat{R}}_{{\alpha }_i}(\pi ∕2)=\exp (-i{\widehat{\sigma }}_{{\alpha }_i}\pi ∕4)$, ${\alpha }_i∊\{x,y,z\}$. Either condition (6) or condition (14) has to be fulfilled depending on whether the original super-WHH or the symmetrized super-WHH sequence is used.

Figure 3

(Color online) The fidelity (bottom) of the ${\widehat{U}}_{\pi ∕4}^{12}$-gate on a linear four-qubit chain (top) as a function of the number of repetitions $n_{\mathrm{swhh}}$ of the super-WHH scheme: original super-WHH sequence (diamonds), unsymmetrized embedded scheme (circles), and symmetrized embedded scheme with adapted pulse interval ${\tau }^{\prime }$ according to Eq. (14) (squares). The solid lines represent the fitting functions $\exp (-c∕n_{\mathrm{swhh}}^4)$ and $\exp (-c_{rs}∕n_{\mathrm{swhh}}^5)$ with $c=0.2$ and $c_{rs}=0.41$.

Figure 4

Quantum gates of the SWAP, CNOT, and controlled-phase gate $\left[\mathrm{CP}\right(\phi \left)\right]:{\widehat{R}}_{\pm \alpha }\left(\phi \right)=\exp (\mp i\phi {\widehat{\sigma }}_{\alpha }∕2)$, $\alpha ∊\{x,y,z\}$.

Figure 5

Left: The qubits of the nine-qubit quantum information processor are arranged on a lattice. The lines connecting qubits $i$ and $j$ indicate the values of the coupling constants $J_{ij}$; right: The table shows the two different values of the coupling constants $J_{kl}^{\left(2\right)}$ for each qubit pair $(k,l)$.

Figure 6

(Color online) Upper plot: Fidelity plots of the quantum sawtooth map implemented with the original recoupling scheme of Yamaguchi et al. [6] (lower plots, black) and the corresponding plots of the embedded recoupling scheme (upper plots, red): Dashed curves show the fidelity estimations according to Eqs. (18, 19). Lower plot: Fidelity plots of the embedded recoupling scheme (upper plots, red) and the embedded but unsymmetrized scheme (lower plots, blue).

Figure 7

(Color online) Logarithmic fidelity plots of the quantum sawtooth map with $n_{\mathrm{swhh}}=5$ (left) and $n_{\mathrm{swhh}}=10$ (right): the original scheme of Yamaguchi et al. Ref. 6 (upper curve, black), the embedded scheme (lowest curve, red), and the embedded scheme without symmetrization (middle curve, blue).

Figure 8

(Color online) Husimi distributions of the quantum states resulting from the quantum sawtooth map: (Upper row) The original scheme of Yamaguchi et al. [6], (middle row) the embedded but unsymmetrized approach, (lower row) the embedded and symmetrized scheme. The ${\widehat{U}}_{\pi ∕8}$-gates used in the computations consist of $n_{\mathrm{swhh}}=\{6,8,10\}$ super-WHH sequences (from left to right). These distributions are averaged over $290\le t\le 299$ numbers $t$ of iterations of the sawtooth map.