Preparation of decoherence-free cluster states with optical superlattices

We present a protocol to prepare decoherence-free cluster states using ultracold atoms loaded in a two dimensional superlattice. The superlattice geometry leads to an array of $2\times 2$ plaquettes, each of them holding four spin-$1∕2$ particles that can be used for encoding a single logical qubit in the twofold singlet subspace, insensitive to uniform magnetic field fluctuations in any direction. Dynamical manipulation of the supperlattice yields distinct inter- and intraplaquette interactions and permits us to realize one qubit and two qubit gates with high fidelity, leading to the generation of universal cluster states for measurement based quantum computation. Our proposal based on inter- and intraplaquette interactions also opens the path to study polymerized Hamiltonians which support ground states describing arbitrary quantum circuits.

Figure 1

(Color online) The optical superlattice consists of a periodic array of $2\times 2$ plaquettes. The 2D optical trapping potential is created by adding two superlattice potentials $V_x\left(x\right)$ and $V_y\left(y\right)$ illustrated in the bottom and left panels. The intraplaquette coupling is represented by the solid lines and the interplaquette coupling is represented by the dashed lines.

Figure 2

(Color online) Simultaneous coupling between neighboring plaquettes. (a) It is possible to simultaneously implement the controlled-phase gates between neighboring plaquettes along the horizontal direction, if only four sites are involved for each controlled-phase gate (e.g., for the geometric phase approach or the optimal control approach). (b) The optical superlattice (with interplaquette coupling and alternating energy offset for odd and even plaquettes, created by $\lambda ,2\lambda ,4\lambda$) can yield the interplaquette coupling for the geometric phase approach. (c) The spin-dependent optical superlattice (indicated by thin and thick lines of potential profiles) can generate Ising interaction [13] useful for the optimal control approach.

Figure 3

(Color online) Intraplaquette superexchange couplings. The coupling strengths are (a) $J_H=t_H^2∕U$ between horizontal neighbors and (b) $J_V=t_V^2∕U$ between vertical neighbors. They can be changed independently by tuning the barriers between the sites.

Figure 4

(Color online) The Bloch sphere representation for the singlet subspace. The state ∣0⟩ (∣1⟩) is associated with the north (south) pole. (a) The superexchange coupling $J_H$ $\left(J_V\right)$ is associated with the rotation around the ${\vec{n}}_H$ $\left({\vec{n}}_V\right)$ axis. Here ${\vec{n}}_H=(\frac{\sqrt{3}}{2},0,\frac{-1}{2})$ and ${\vec{n}}_V=(0,0,1)$. The sequential rotations around the ${\vec{n}}_H$ and ${\vec{n}}_V$ axes can rotate the Bloch vector from ${\vec{n}}_V$ to ${\vec{n}}_g=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})$ [i.e., state $\frac{1}{\sqrt{2}}(\mid 0⟩+\mid 1⟩)$]. (b) Alternatively combined superexchange coupling (with contributions from both $J_H$ and $J_V$) can implement the rotation around the axis ${\vec{n}}_C$, which rotates the Bloch vector from ${\vec{n}}_V$ to ${\vec{n}}_g$ in one step.

Figure 5

(Color online) Geometric phase approach to controlled-phase gate (step 1). (a) The sites from two neighboring plaquettes are labeled. (b) The intraplaquette trapping potential along the vertical direction is adiabatically tilted. This results in single (or double) occupancy at the lower site if the vertical pair of particles is in the singlet (or triplet) state. (c) Particle number configurations are plotted for four possible of spin states: $S_{2,3}\otimes S_{1^{\prime },4^{\prime }}$, $S_{2,3}\otimes T_{1^{\prime },4^{\prime }}$, $T_{2,3}\otimes S_{1^{\prime },4^{\prime }}$, and $T_{2,3}\otimes T_{1^{\prime },4^{\prime }}$. ($S$ and $T$ indicate singlet and triplet.)

Figure 6

(Color online) Geometric phase approach to controlled-phase gate (step 2). (a) The sites from two neighboring plaquettes are labeled. (b) A defined bias (△) of the interplaquette potential is quickly applied and the interplaquette barrier is lowered to facilitate the resonant tunneling along the horizontal direction. (c) Resonant tunneling (between the vibrational ground level of the left site and the vibrational excited level of the right site) can occur for the following two cases: (i) each of the left and right sites has exactly one particle, and the two particles are in the singlet state (see the left panel), (ii) the left site has one particle and the right site has zero particles (see the highlighted upper right panel). All other configurations are off-resonant, with negligible tunneling. After time $2\pi ∕t$, a geometric phase $\pi$ from the resonant tunneling is obtained for $S_{2,3}\otimes T_{1^{\prime },4^{\prime }}$ (the highlighted upper right panel), while only a trivial geometric phase (0 or $2\pi$) is obtained for the other three cases ($S_{2,3}\otimes S_{1^{\prime },4^{\prime }}$, $T_{2,3}\otimes S_{1^{\prime },4^{\prime }}$, and $T_{2,3}\otimes T_{1^{\prime },4^{\prime }}$).

Figure 7

(Color online) (a) Density plot of the lattice potential [see Eq. (15)], which generates an array of $2\times 2$ plaquettes in the $x\text{−}y$ plane with periodicity of $2\pi ∕k$. Using parameters $V^l=60E_R$, $V^s=9E_r$, and $V^{\prime }=10E_r$, with $E_R$ and $E_r$ photon recoil energy of the long and short lattices, respectively, one can achieve a parameter regime with $d∕J\approx 0.2$. (b) Energy levels of a single plaquette described by the Hamiltonian given in Eq. (14). In the plot we assume fermionic atoms, i.e., $J,d>0$ (fermions). For bosons $J,d<0$ the order of the energy levels is the reverse, i.e., the $S=2$ is the lowest energy state.

Figure 8

(Color online) (a) Parameters of the effective Hamiltonian for the general case $d\ne J$. At the point $d\approx 0.62J$, the Ising term in the effective Hamiltonian vanishes, ${\lambda }_z=\frac{1}{8}$ (see text). (b) The controlled-phase gate fidelity $F$ on a superplaquette as a function of $d∕J$ for different rations of $J^{\prime }∕J$. For same $d∕J$, the smaller $J^{\prime }∕J$ the higher the fidelity. There are four critical points at which the fidelity drops considerably; these are $d∕J\approx 0,0.5,0.62,1$. At $d∕J\approx 0$ and 0.5, one of the singlet states becomes degenerate with one $S=1$ state and consequently the effective Hamiltonian breaks down. At the $d∕J\approx 0.62$ the Ising term vanishes and at $d=J$ the rotating wave approximation used in the simplification of the effective Hamiltonian becomes invalid. The shadow regions are the ones where the achievable fidelity is higher than 0.98.

Figure 9

(Color online) For the $d<J$, the cluster state generation has to be applied in two steps (a) and (b).

Figure 10

(Color online) Examples of controlled-phase gate with the perturbative approach. (a) The values of $d∕J$ that satisfy the conditions stated in Eqs. (26). The shadow regions highlight the regime where the phase gate fidelity can be larger than 0.98 for $J^{\prime }∕J<0.1$. (b) The controlled-phase gate fidelity $F$ as a function of $J^{\prime }∕J$ can be calculated by exact numerical diagonalization. The red dashed (or blue solid) curve is for $(n,m)=(1,1)$ [or (3, 4)], which is also indicated by the red (blue) square in panel (a).

Figure 11

(Color online) (a) Smooth pulses with $L=20$ yielding to infidelities smaller than ${10}^{-7}$. (b) Infidelity $ϵ=1-F$ vs the deviation $\delta J∕J$ for the smooth pulses. The infidelity is constant at ${10}^{-7}$ for $\delta J∕J<{10}^{-4}$ and increases quadratically with $\delta J∕J$ for $\delta J∕J>{10}^{-3}$. (c) Quadratic dependence of the fidelity with $\delta J∕J$.

Figure 12

(Color online) The biased potential for the sites $L$ and $R$. The vibrational ground levels are $\mid L,a⟩$ and $\mid R,a⟩$, with energy difference △. The vibrational excited level for the right site is $\mid R,b⟩$ with excitation energy $\omega$.