Toward scalable information processing with ultracold polar molecules in an electric field: A numerical investigation

We numerically investigate the possibilities of driving quantum algorithms with laser pulses in a register of ultracold NaCs polar molecules in a static electric field. We focus on the possibilities of performing scalable logical operations by considering circuits that involve intermolecular gates (implemented on adjacent interacting molecules) to enable the transfer of information from one molecule to another during conditional laser-driven population inversions. We study the implementation of an arithmetic operation (the addition of 0 or 1 on a binary digit and a carry in) which requires population inversions only and the Deutsch-Jozsa algorithm which requires a control of the phases. Under typical experimental conditions, our simulations show that high-fidelity logical operations involving several qubits can be performed in a time scale of a few hundreds of microseconds, opening promising perspectives for the manipulation of a large number of qubits in these systems.

Figure 1

Top panel: Schematic of a polar-molecule-based quantum register [32]. Bottom panel: Permanent dipole moments ${\mu }_i$, $i=1$ and 2, of the molecules. Their orientations are characterized by the angles ${\theta }_i$ and ${\varphi }_i$. The electric field $E$ is parallel to the $Z$ axis connecting the center of mass of the molecules. In the simulation, the wave length of the lattice laser is $\lambda =600$ nm, so the intermolecular distance is 300 nm, the dipole moment is 4.6 D, and the dipole-dipole interaction is about a few tens of kHz.

Figure 2

Diagram of the lowest eigenvalues of the total time-independent Hamiltonian for three different cases: (a) without any electric field, (b) with electric fields $E^1>E^2$ without dipole-dipole interaction, and (c) same as (b) plus the dipole-dipole interaction. The eigenvectors $|{\mathrm{Ñ}}_1,{\mathrm{Ñ}}_2\rangle$ correlate adiabatically to the vector $|N_1,N_2\rangle$ with $v_1=v_2=0$. Only $m_N=0$ are considered. The $m_N$ and $v_i$ labels are omitted. Transitions in the first or second molecule are shown by solid blue or dotted red arrows, respectively. The dashed red arrow indicates the transition for the realization of a cnot operation. The subscripts labeling the frequencies indicate the rotational state of the neighboring molecule.

Figure 3

Evolution of the populations during the cnot gate driven by a $\pi$ pulse. When the control qubit is 0 (initial states $|00\rangle$ or $|01\rangle$), the final state of the qubits remains unchanged when the pulse is applied, although the population may vary during the process (top left and right panels, respectively). On the contrary, the second qubit is flipped when the initial state is $|10\rangle$ or $|11\rangle$ leading to final states $|11\rangle$ or $|10\rangle$(bottom left and right panels, respectively).

Figure 4

Weights $\left|p_k\right|{}^2$ of the product basis states $|N_1,N_2\rangle$ for the four lowest eigenvectors of the time-independent Hamiltonian showing the state mixing due to the static Stark electric field and the dipole-dipole coupling in the manifold $v=0$. The state mixture is similar in other $v$. Eigenstates are labeled after the main component of the product basis. Qubits $|10\rangle$ and $|11\rangle$ encoded in eigenstates $|{\mathrm{Ñ}}_1=1$, ${\mathrm{Ñ}}_2=0\rangle$ and $|{\mathrm{Ñ}}_1=1$, ${\mathrm{Ñ}}_2=1\rangle$, respectively, have a non-negligeable admixture of states with $N_i=3$ allowing electric dipole transitions to these states.

Figure 5

Logical circuits for the 0 adder (left panel) and 1 adder (right panel).

Figure 6

Encoding of the three qubits for the 0 adder in the states of the coupled basis set $|Q_1{Q}_2{Q}_3\rangle \leftrightarrow |{\mathrm{ṽ}}_1,{\mathrm{Ñ}}_1,{\mathrm{ṽ}}_2,{\mathrm{Ñ}}_2\rangle$. The active transitions for the toffoli and cnot gates (the latter is called cnot1 in the text) are shown by solid arrows and the corresponding unwanted transitions are shown by dashed arrows. The zero of energy is chosen at the ground state without Stark field and dipole-dipole coupling.

Figure 7

Population evolution for each computational basis state showing the realization of the gates of the 0 adder. Upper panel: toffoli gate. Lower panel: cnot1 gate. For both panels, simulations were performed considering an initial population of 1 in one single pure computational basis state at a time. For a better reading of the picture, populations have been weighted by different arbitrary factors. The picture does not present the evolution of the population for an arbitrary superposition of computational basis states. Note that the time scale is different from one panel to the other.

Figure 8

Population evolution of each computational basis state during the supplementary gates of the 1 adder. Upper panel: cnot2 gate. Lower panel: not gate. For both panels, simulations were performed considering an initial population of 1 in one single pure computational basis state at a time. For a better reading of the picture, populations have been weighted by different arbitrary factors. The picture does not present the evolution of the population for an arbitrary superposition of computational basis states. Note that the time scale is different from one panel to the other.

Figure 9

Population evolution of the computational basis state corresponding to the logical state $|110\rangle$ ($b_i=1$ and $c_i=1$ during the 1-add gate (toffoli-cnot1-cnot2-not gates) to give the final logical state $|111\rangle$ corresponding to $s_i=1$ and $c_{i+1}=1$.

Figure 10

Logical circuit for the Deutsch-Jozsa algorithm.

Figure 11

Evolution of the populations during the gates involving Hadamard gates in the Deutsch-Jozsa algorithm. Top panel: not-hadhad step (not gate on $|y\rangle$ and the two had gates on $|x\rangle$ and $|y\rangle$. Bottom panel: had gate on $|x\rangle$. The initial state is a superposed state with equal weights 0.25 on the computational basis set states $|\tilde{0}\tilde{0}\rangle$ (thick solid black curve), $|\tilde{0}\tilde{1}\rangle$ (thick solid green curve), $|\tilde{1}\tilde{0}\rangle$ (thin solid black curve), $|\tilde{1}\tilde{1}\rangle$ (thin solid green curve).

Figure 12

Amplitude of the optimal field for the not-hadhad gate (top left panel) and for the had gate (top right panel). Bottom panel: The corresponding Fourier transforms.

Figure 13

Evolution of the populations during the phase cnot gate starting from a superposition of states $|\tilde{0}\tilde{0}\rangle$ (red dashed curve), $|\tilde{0}\tilde{1}\rangle$ (solid red curve), $|\tilde{1}\tilde{0}\rangle$ (black dashed curve), and $|\tilde{1}\tilde{1}\rangle$ (black solid curve). Top panel: Gate driven by the $\pi$ pulse calculated in Fig. 3. The pulse optimized for a pure state does not lead to a high-fidelity operation when applied to a superposition of states. Bottom panel: High-fidelity gate optimized by optimal control. We checked that high-fidelity gates were obtained when the pulses were applied to other initial superpositions.

Figure 14

Amplitude of the optimal field for the cnot gate (top panel) and its Fourier transform (bottom panel).